2113:
2074:
3723:
2086:
2173:. At the same time George Paget Thomson and Alexander Reid at the University of Aberdeen were independently firing electrons at thin celluloid foils and later metal films, observing rings which can be similarly interpreted. (Alexander Reid, who was Thomson's graduate student, performed the first experiments but he died soon after in a motorcycle accident and is rarely mentioned.) Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be exhibited only by waves. Therefore, the presence of any
1293:
3384:
6767:. Electron optical systems use stabilized high voltage to give a narrow energy spread in combination with collimating (parallelizing) lenses and pointed filament sources to achieve good coherence. Because light at all frequencies travels the same velocity, longitudinal and temporal coherence are linked; in matter waves these are independent. For example, for atoms, velocity (energy) selection controls longitudinal coherence and pulsing or chopping controls temporal coherence.
12179:
3718:{\displaystyle {\begin{aligned}\mathbf {v} _{\mathrm {g} }&={\frac {\partial \omega }{\partial \mathbf {k} }}={\frac {\partial (E/\hbar )}{\partial (\mathbf {p} /\hbar )}}={\frac {\partial E}{\partial \mathbf {p} }}={\frac {\partial }{\partial \mathbf {p} }}\left({\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}\right)\\&={\frac {\mathbf {p} c^{2}}{\sqrt {p^{2}c^{2}+m_{0}^{2}c^{4}}}}\\&={\frac {\mathbf {p} c^{2}}{E}}.\end{aligned}}}
5122:
6102:
4889:
6342:
4618:
3921:
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1331:
When I conceived the first basic ideas of wave mechanics in 1923â1924, I was guided by the aim to perform a real physical synthesis, valid for all particles, of the coexistence of the wave and of the corpuscular aspects that
Einstein had introduced for photons in his theory of light quanta in
6500:
interacts weakly with air molecules. By contrast, strongly interacting particles like slow electrons and molecules require vacuum: the matter wave properties rapidly fade when they are exposed to even low pressures of gas. With special apparatus, high velocity electrons can be used to study
4457:
5960:
4884:
2368:
In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity. Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e., to certain
4468:
5682:
3800:
7456:
Matter-wave interfererometers generate nanostructures on molecular beams that can be read with nanometer accuracy and therefore be used for highly sensitive force measurements, from which one can deduce a plethora or properties of individualized complex molecules.
5875:
5723:
in the front, and the energy has been written more generally as a function of the wave vector. The various terms given before still apply, although the energy is no longer always proportional to the wave vector squared. A common approach is to define an
5117:{\displaystyle {\begin{aligned}&\lambda =\,\,{\frac {h}{\gamma m_{0}v}}\,=\,{\frac {h}{m_{0}v}}\,\,\,{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&f={\frac {\gamma \,m_{0}c^{2}}{h}}={\frac {m_{0}c^{2}}{h{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}},\end{aligned}}}
3316:
2842:
3049:
1607:
5546:
6773:
Optical manipulation of matter plays a critical role in matter wave optics: "Light waves can act as refractive, reflective, and absorptive structures for matter waves, just as glass interacts with light waves." Laser light momentum transfer can
2662:
2941:
6652:
11205:
Gerlich, Stefan; Fein, Yaakov Y.; Shayeghi, Armin; Köhler, Valentin; Mayor, Marcel; Arndt, Markus (2021), Friedrich, Bretislav; Schmidt-Böcking, Horst (eds.), "Otto Stern's Legacy in
Quantum Optics: Matter Waves and Deflectometry",
4363:
5331:
2479:
1217:
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5451:
3795:
126:
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for which the wavefunctions are plane waves. There are significant numbers of other matter waves, which can be broadly split into three classes: single-particle matter waves, collective matter waves and standing waves.
1504:
2019:
5392:
6176:
The wavelength is still described as the inverse of the modulus of the wavevector, although measurement is more complex. There are many cases where this approach is used to describe single-particle matter waves:
5572:
3372:
3136:
4057:
3999:
1688:
6855:
6422:
4894:
4775:
4171:
4114:
8241:
Darwin, Charles Galton. "Free motion in the wave mechanics." Proceedings of the Royal
Society of London. Series A, Containing Papers of a Mathematical and Physical Character 117.776 (1927): 258â293.
6432:. This can, and arguably should be, extended to many other cases. For instance, in early work de Broglie used the concept that an electron matter wave must be continuous in a ring to connect to the
6097:{\displaystyle \mathbf {j} (\mathbf {r} )={\frac {\hbar }{2mi}}\left(\psi ^{*}(\mathbf {r} )\mathbf {\nabla } \psi (\mathbf {r} )-\psi (\mathbf {r} )\mathbf {\nabla } \psi ^{*}(\mathbf {r} )\right)}
5766:
4736:
3805:
3389:
2790:
2667:
9381:
Eibenberger, Sandra; Gerlich, Stefan; Arndt, Markus; Mayor, Marcel; TĂŒxen, Jens (14 August 2013). "Matterâwave interference of particles selected from a molecular library with masses exceeding
3212:
2785:
2292:
in 1930, when a Na beam was diffracted off a surface of NaCl. The short de
Broglie wavelength of atoms prevented progress for many years until two technological breakthroughs revived interest:
2950:
10217:. Translated by Shearer, J. F.; Deans, Winifred Margaret (Third (augmented) edition, New York 1982 ed.). Providence, Rhode Island: AMS Chelsea Publishing, American Mathematical Society.
5262:
1525:
10842:
Grisenti, R. E.; W. Schöllkopf; J. P. Toennies; J. R. Manson; T. A. Savas; Henry I. Smith (2000). "He-atom diffraction from nanostructure transmission gratings: The role of imperfections".
7340:
explains how electrons escape from metals in an electrostatic field at energies less than classical predictions allow: the matter wave penetrates of the work function barrier in the metal.
6743:
6174:
5721:
4613:{\displaystyle \mathbf {v} _{\mathrm {p} }={\frac {E}{\mathbf {p} }}={\frac {mc^{2}}{m\mathbf {v} }}={\frac {\gamma m_{0}c^{2}}{\gamma m_{0}\mathbf {v} }}={\frac {c^{2}}{\mathbf {v} }}.}
6519:
while matter wave velocity varies strongly with frequency. The relationship between frequency (proportional to energy) and wavenumber or velocity (proportional to momentum) is called a
1846:
1088:
proposed that the thermal energy of oscillating atoms is divided into discrete portions, or quanta. Extending Planck's investigation in several ways, including its connection with the
1754:
1043:
10877:
Chapman, Michael S.; Christopher R. Ekstrom; Troy D. Hammond; Richard A. Rubenstein; Jörg
Schmiedmayer; Stefan Wehinger; David E. Pritchard (1995). "Optics and interferometry with Na
4351:
4306:
3916:{\displaystyle {\begin{aligned}\mathbf {v} _{\mathrm {g} }&={\frac {\mathbf {p} c^{2}}{E}}\\&={\frac {c^{2}}{\mathbf {v} _{\mathrm {p} }}}\\&=\mathbf {v} ,\end{aligned}}}
2881:
6555:
2120:
pattern building up electron by electron. Each white dot represents a single electron hitting a detector; with a statistically large number of electrons interference fringes appear.
2105:
used electron biprisms (essentially a wire placed in an electron microscope) and measured single electrons building up the diffraction pattern. Recently, a close copy of the famous
4261:
4216:
4678:
2098:
The de
Broglie hypothesis and the existence of matter waves has been confirmed for other elementary particles, neutral atoms and even molecules have been shown to be wave-like.
9254:
Pierre Cladé; Changhyun Ryu; Anand
Ramanathan; Kristian Helmerson; William D. Phillips (2008). "Observation of a 2D Bose Gas: From thermal to quasi-condensate to superfluid".
5160:
2638:
7695:
Hughes, A. Ll. "XXXIII. The photo-electric effect of some compounds." The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science 24.141 (1912): 380â390.
5460:
2112:
9544:
Hornberger, Klaus; Stefan
Uttenthaler; Björn Brezger; Lucia HackermĂŒller; Markus Arndt; Anton Zeilinger (2003). "Observation of Collisional Decoherence in Interferometry".
3177:
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5271:
2412:
1157:
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5184:
2039:
1131:
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5880:
2041:
is the uncertainty in the initial position. This position uncertainty creates uncertainty in velocity (the extra second term in the square root) consistent with
7304:
patterns emerge when energetic electrons reflect or penetrate ordered solids; analysis of the patterns leads to models of the atomic arrangement in the solids.
5208:
3200:
2557:
1927:
1907:
1887:
1426:
6479:. The approximation works well as long as the electric fields change more slowly than the de Broglie wavelength. Macroscopic apparatus fulfill this condition;
8987:
1932:
10982:
4642:
can either be interpreted as the speed of the particle or the group velocity of the corresponding matter waveâthe two are the same. Since the particle speed
9605:
HackermĂŒller, Lucia; Klaus
Hornberger; Björn Brezger; Anton Zeilinger; Markus Arndt (2004). "Decoherence of matter waves by thermal emission of radiation".
5569:
The more general description of matter waves corresponding to a single particle type (e.g. a single electron or neutron only) would have a form similar to
3321:
3085:
2765:{\displaystyle {\begin{aligned}&\lambda ={\frac {2\pi }{|\mathbf {k} |}}={\frac {h}{p}}\\&f={\frac {\omega }{2\pi }}={\frac {E}{h}}\end{aligned}}}
2307:
allowed cooling of neutral atoms down to nanokelvin temperatures. At these temperatures, the de Broglie wavelengths come into the micrometre range. Using
4004:
11423:
8497:
Merli, P. G., G. F. Missiroli, and G. Pozzi. "On the statistical aspect of electron interference phenomena." American Journal of Physics 44 (1976): 306
6353:(A), and matter waves (BâF). In (BâF), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of the
3960:
1626:
11085:
Fein, Yaakov Y.; Geyer, Philipp; Zwick, Patrick; KiaĆka, Filip; Pedalino, Sebastian; Mayor, Marcel; Gerlich, Stefan; Arndt, Markus (December 2019).
1398:
is to be measured, of course, in the rest frame of the energy packet. This hypothesis is the basis of our theory." (This frequency is also known as
6376:
5401:
3730:
56:
11976:
10661:
1849:
Position space probability density of an initially Gaussian state moving in one dimension at minimally uncertain, constant momentum in free space
6213:, where the component of the wavevector in one direction is complex. These are common when matter waves are being reflected, particularly for
4452:{\displaystyle \mathbf {v} _{\mathrm {p} }={\frac {\omega }{\mathbf {k} }}={\frac {E/\hbar }{\mathbf {p} /\hbar }}={\frac {E}{\mathbf {p} }}.}
11624:
5342:
392:
8507:
Tonomura, A.; Endo, J.; Matsuda, T.; Kawasaki, T.; Ezawa, H. (1989). "Demonstration of single-electron buildup of an interference pattern".
2177:
effects by matter demonstrated the wave-like nature of matter. The matter wave interpretation was placed onto a solid foundation in 1928 by
7330:
provides information about the chemistry and electronic structure of materials. Beams of electrons also lead to characteristic X-rays in
4691:
2244:, especially for biological materials. Neutrons were discovered in the early 1930s, and their diffraction was observed in 1936. In 1944,
893:
1767:
made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark,
6888:
to reach lower energies; shorter wavelengths make diffraction effects more difficult to discern. Therefore, many applications focus on
4879:{\displaystyle {\begin{aligned}E&=mc^{2}=\gamma m_{0}c^{2}\\\mathbf {p} &=m\mathbf {v} =\gamma m_{0}\mathbf {v} \end{aligned}}}
12147:
7970:
5231:
30:
This article is about wave-like phenomena exhibited by particles of matter. For elastic waves propagating through material media, see
7316:
6803:
1865:
in several idealized scenarios. For an unbound electron in free space he worked out the propagation of the wave, assuming an initial
11368:"Scientific Papers Presented to Max Born on his retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh"
11473:
1818:
of a particle is not part of the non-relativistic Schrödinger equation. The Schrödinger equation describes the time evolution of a
144:
6796:
The following subsections provide links to pages describing applications of matter waves as probes of materials or of fundamental
6142:
4119:
4062:
12159:
6456:
1776:
1410:
600:
2300:
allowing atoms to be slowed, increasing their de Broglie wavelength. The double-slit experiment on atoms was performed in 1991.
2240:) matches interatomic spacing and neutrons scatter strongly from hydrogen atoms. Consequently, neutron matter waves are used in
11843:
11416:
6509:. Neutrons, an important exception, interact primarily by collisions with nuclei, and thus travel several hundred feet in air.
5268:-independent. Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by:
5214:
in vacuum. This shows that as the velocity of a particle approaches zero (rest) the de Broglie wavelength approaches infinity.
373:
5677:{\displaystyle \psi (\mathbf {r} )=u(\mathbf {r} ,\mathbf {k} )\exp(i\mathbf {k} \cdot \mathbf {r} -iE(\mathbf {k} )t/\hbar )}
1723:
1072:
At the end of the 19th century, light was thought to consist of waves of electromagnetic fields which propagated according to
1012:
12213:
11777:
11305:
11233:
10467:
10436:
10222:
10197:
10172:
10108:
10083:
9975:
9926:
9901:
8805:
8577:
8169:
8137:
8054:
8029:
7385:
provides way to obtain structure of disordered systems that is sensitivity to light elements, isotopes and magnetic moments.
556:
17:
11700:
11316:
8662:
6269:
5725:
3139:
1779:), encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system â the trajectories of
479:
11007:
Shayeghi, A.; Rieser, P.; Richter, G.; Sezer, U.; Rodewald, J. H.; Geyer, P.; Martinez, T. J.; Arndt, M. (19 March 2020).
7446:
4746:
phase velocity does not violate special relativity, similar to the case above for non-isotropic media. See the article on
6655:
6520:
3206:
2944:
1314:
is spread out like a waveform; there is no definite position of the particle. As the amplitude increases above zero the
6468:
2079:
Original electron diffraction camera made and used by Nobel laureate G P Thomson and his student Alexander Reid in 1925
6486:
Beyond the equations of motion, other aspects of matter wave optics differ from the corresponding light optics cases.
11446:
11409:
11287:
9950:
8769:
7576:
7327:
139:
6800:. In most cases these involve some method of producing travelling matter waves which initially have the simple form
6746:
1866:
11556:
9002:
7435:
7312:
6502:
6433:
4748:
2161:
target. The diffracted electron intensity was measured, and was determined to have a similar angular dependence to
1318:
decreases, so the amplitude diminishes again, and vice versa. The result is an alternating amplitude: a wave. Top:
228:
10925:
6695:
5687:
12114:
11823:
11818:
11541:
6480:
886:
335:
315:
183:
6252:
where interactions with other electrons in the solid have been included. An electron quasiparticle has the same
12126:
11798:
7382:
7331:
7320:
6959:
6785:
5870:{\displaystyle {m_{ij}^{*}}^{-1}={\frac {1}{\hbar ^{2}}}{\frac {\partial ^{2}E}{\partial k_{i}\partial k_{j}}}}
2315:
atoms was explicitly measured and found to be consistent with the temperature measured by a different method.
2147:
2130:
2062:
1349:
also have wave-like properties. His thesis started from the hypothesis, "that to each portion of energy with a
575:
305:
10340:
5877:
so that in the simple case where all directions are the same the form is similar to that of a free wave above.
2405:
than solid objects. The simplest approach is to focus on the description in terms of plane matter waves for a
2327:
that otherwise might be supposed too large to undergo quantum mechanical effects. In 1999, a research team in
12104:
11881:
11803:
11586:
11456:
10876:
7504:
7343:
6784:
While single-particle free-space optical and matter wave equations are identical, multiparticle systems like
6272:
can differ substantially from that of a normal electron. Its electric field is also modified, as a result of
6214:
4311:
4266:
2185:, showing how this could explain the experimental results. His approach is similar to what is used in modern
1107:
1077:
1047:
Wave-like behavior of matter has been experimentally demonstrated, first for electrons in 1927 and for other
615:
353:
253:
11264:
9483:
Juffmann, Thomas; et al. (25 March 2012). "Real-time single-molecule imaging of quantum interference".
7799:
1810:. Frequencies of solutions of the non-relativistic Schrödinger equation differ from de Broglie waves by the
11838:
11772:
11767:
11738:
11451:
6365:
The third class are matter waves which have a wavevector, a wavelength and vary with time, but have a zero
6232:
Other classes of matter waves involve more than one particle, so are called collective waves and are often
3378:
3311:{\displaystyle \omega (\mathbf {k} )={\sqrt {k^{2}c^{2}+\left({\frac {m_{0}c^{2}}{\hbar }}\right)^{2}}}\,.}
2859:
2837:{\displaystyle {\begin{aligned}&\mathbf {p} =\hbar \mathbf {k} \\&E=\hbar \omega ,\\\end{aligned}}}
2390:
2073:
1835:
1509:
1101:
551:
546:
517:
368:
149:
10924:
Brezger, B.; HackermĂŒller, L.; Uttenthaler, S.; Petschinka, J.; Arndt, M.; Zeilinger, A. (February 2002).
7914:
4221:
4176:
3044:{\displaystyle \omega (\mathbf {k} )\approx {\frac {m_{0}c^{2}}{\hbar }}+{\frac {\hbar k^{2}}{2m_{0}}}\,.}
11906:
11813:
9020:
8998:
8191:
5265:
4645:
2091:
Example original electron diffraction photograph from the laboratory of G. P. Thomson, recorded 1925â1927
585:
330:
320:
9970:. International series in pure and applied physics (3. ed., 24. print ed.). New York: McGraw-Hill.
6763:
approach. As with light, transverse coherence (across the direction of propagation) can be increased by
1602:{\displaystyle v_{\text{g}}\equiv {\frac {\partial \omega }{\partial k}}={\frac {d\nu }{d(1/\lambda )}}}
12182:
11944:
11752:
11723:
8569:
8129:
7509:
7419:
6865:
5541:{\displaystyle \mathbf {U} =\gamma (c,{\mathbf {u} })=\gamma (c,v_{\mathrm {g} }{\hat {\mathbf {u} }})}
2276:
in relation to waveâparticle duality. The double-slit experiment was performed using neutrons in 1988.
1345:, in his 1924 PhD thesis, proposed that just as light has both wave-like and particle-like properties,
879:
531:
502:
5127:
2605:
11966:
11833:
11757:
11718:
11675:
11649:
11606:
11499:
7533:
7466:
7450:
6797:
2346:. More recent experiments prove the quantum nature of molecules made of 810 atoms and with a mass of
2102:
1788:
914:
536:
497:
450:
425:
348:
208:
11208:
Molecular Beams in Physics and Chemistry: From Otto Stern's Pioneering Exploits to Present-Day Feats
12033:
12013:
12003:
11993:
11949:
11524:
11394:
10423:, Graduate Texts in Mathematics, vol. 267, New York, NY: Springer New York, pp. 419â440,
9062:
9061:
Zeilinger, Anton; GĂ€hler, Roland; Shull, C. G.; Treimer, Wolfgang; Mampe, Walter (1 October 1988).
8597:
8160:
Philosophy of Quantum Mechanics: The interpretations of quantum mechanics in historical perspective
7486:
6506:
6273:
3144:
620:
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2506:
2484:
1136:
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11654:
11269:
10283:
9458:
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2226:
2117:
1282:
789:
507:
415:
12142:
11690:
7496:
6526:
6460:
5731:
4357:
phase velocity therefore does not violate special relativity, as it does not carry information.
4116:
as reported by R.W. Ditchburn in 1948 and J. L. Synge in 1952. Electromagnetic waves also obey
2936:{\displaystyle \mathbf {v_{g}} ={\frac {\partial \omega (\mathbf {k} )}{\partial \mathbf {k} }}}
2182:
1862:
1795:
465:
363:
129:
12053:
11828:
11808:
11733:
11601:
10803:
Moskowitz, Philip E.; Gould, Phillip L.; Atlas, Susan R.; Pritchard, David E. (1 August 1983).
9844:
9722:
8722:
6881:
6647:{\displaystyle \omega (k)\approx {\frac {m_{0}c^{2}}{\hbar }}+{\frac {\hbar k^{2}}{2m_{0}}}\,.}
6227:
6132:
2382:
2269:
2200:
demonstrated the particle nature of light, these experiments showed the wave nature of matter.
2166:
2106:
1799:
1772:
1073:
794:
512:
340:
310:
273:
11644:
7619:
420:
12078:
11591:
11571:
10073:
8888:"The early development of neutron diffraction: science in the wings of the Manhattan Project"
8886:
Mason, T. E.; Gawne, T. J.; Nagler, S. E.; Nestor, M. B.; Carpenter, J. M. (1 January 2013).
7388:
6107:
5169:
2253:
2024:
1854:
1784:
263:
248:
9667:
6755:
The visibility of diffraction features using an optical theory approach depends on the beam
1771:
decided to find a proper three-dimensional wave equation for the electron. He was guided by
1269:
as is done in the rest of this article. Einstein's postulate was verified experimentally by
12109:
12038:
11983:
11713:
11536:
11494:
11341:
11211:
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11030:
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10767:
10712:
10673:
10626:
10579:
10506:
10381:
10305:
10254:
10137:
10019:
9856:
9791:
9734:
9679:
9624:
9563:
9543:
9502:
9408:
9339:
9273:
9253:
9212:
9163:
9113:
9074:
9035:
8958:
8734:
8619:
8564:
8516:
8456:
8379:
8327:
8267:
8200:
8087:
7998:
7982:
7943:
7889:
7844:
7761:
7724:
7654:
7308:
7301:
7157:
6745:) and a quadratic part due to kinetic energy. The quadratic term causes rapid spreading of
6307:
6204:
6193:
3054:
2562:
2197:
2186:
2134:
2058:
2046:
1089:
1081:
580:
492:
218:
175:
11141:
6459:
to develop his wave mechanics for subatomic particles Consequently, wave solutions to the
6452:
6289:
of an electron in a state; it is most commonly used in the context of empty states in the
1768:
824:
8:
12094:
12063:
12008:
11988:
11896:
11853:
11708:
11634:
11561:
11551:
11463:
10557:
10416:
10103:. The history of modern physics (2nd ed.). Los Angeles (Calif.): Thomas publishers.
9319:
7399:
7368:
7353:
6756:
6350:
6261:
6237:
6189:
5955:
5396:
2265:
2209:
1048:
679:
487:
405:
233:
213:
165:
11345:
11215:
11167:
11102:
11034:
10954:
10894:
10855:
10820:
10771:
10716:
10677:
10630:
10583:
10510:
10385:
10309:
10258:
10141:
10023:
9860:
9795:
9738:
9683:
9628:
9604:
9567:
9506:
9412:
9343:
9277:
9216:
9167:
9117:
9078:
9039:
8962:
8738:
8631:
8623:
8520:
8460:
8383:
8331:
8271:
8204:
8091:
7986:
7947:
7893:
7848:
7765:
7728:
7658:
7428:
uses matter wave behavior to explain grazing angle atomic reflection, the basis of some
3945:
is the velocity of the center of mass of the particle, identical to the group velocity.
2085:
12154:
12023:
11921:
11629:
11576:
11468:
11331:
11187:
11153:
11122:
11059:
11020:
11008:
10974:
10940:
10736:
10614:
10569:
10538:
10496:
10321:
10295:
10126:"Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations"
10051:
9820:
9781:
9769:
9648:
9614:
9587:
9553:
9526:
9492:
9440:
9398:
9363:
9297:
9263:
9129:
8920:
8863:
8830:
8829:
Blakeley, Matthew P; Langan, Paul; Niimura, Nobuo; Podjarny, Alberto (1 October 2008).
8690:
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8609:
8480:
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8410:
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4462:
3185:
2542:
2285:
1912:
1892:
1872:
1811:
1414:
1399:
400:
325:
258:
170:
10841:
10804:
10370:"Description of States in Quantum Mechanics by Density Matrix and Operator Techniques"
12164:
12073:
12043:
11971:
11934:
11929:
11911:
11876:
11866:
11581:
11546:
11529:
11432:
11301:
11295:
11283:
11229:
11179:
11126:
11114:
11064:
11046:
10966:
10906:
10740:
10728:
10642:
10595:
10530:
10463:
10432:
10397:
10325:
10267:
10242:
10218:
10215:
Collected papers on wave mechanics: together with his Four lectures on wave mechanics
10193:
10168:
10104:
10079:
10043:
10035:
9971:
9946:
9922:
9897:
9891:
9872:
9825:
9807:
9750:
9703:
9695:
9640:
9579:
9518:
9432:
9424:
9355:
9289:
9236:
9228:
9181:
9176:
9151:
9133:
8925:
8907:
8868:
8850:
8811:
8801:
8775:
8765:
8694:
8682:
8635:
8573:
8532:
8472:
8415:
8397:
8345:
8285:
8228:
8216:
8165:
8133:
8103:
8050:
8025:
7781:
7678:
7670:
7623:
7608:
7572:
7491:
7406:, measure the difference in phase between atomic matter waves along different paths.
7372:
7347:
7337:
6441:
6425:
6358:
6136:
4743:
4354:
2536:
2261:
2193:
1803:
1692:
then integrating, de Broglie arrived as his formula for the relationship between the
910:
834:
809:
749:
744:
644:
610:
590:
188:
47:
11191:
11086:
10978:
10923:
10542:
10369:
9591:
9301:
8484:
2252:, recognized the potential for applying thermal neutrons from the newly operational
11891:
11886:
11743:
11639:
11382:
11349:
11219:
11175:
11171:
11106:
11054:
11038:
10958:
10898:
10859:
10824:
10788:
10783:
10775:
10720:
10681:
10634:
10587:
10522:
10514:
10424:
10389:
10313:
10262:
10145:
10055:
10027:
9864:
9815:
9799:
9742:
9687:
9652:
9632:
9571:
9530:
9510:
9444:
9416:
9367:
9347:
9285:
9281:
9220:
9171:
9121:
9082:
9043:
8966:
8915:
8899:
8858:
8842:
8742:
8674:
8627:
8524:
8464:
8405:
8387:
8335:
8297:
8275:
8208:
8186:
8095:
8075:
7990:
7951:
7897:
7866:
7852:
7828:
7769:
7732:
7662:
7449:
revealed by interference of matter waves from large molecules probes the limits of
6905:
6692:) in two parts: a constant part due to the de Broglie frequency of the rest mass (
6370:
6346:
6210:
6200:
5326:{\displaystyle \mathbf {K} =\left({\frac {\omega _{0}}{c^{2}}}\right)\mathbf {U} ,}
2293:
2245:
2233:
as they scatter from light atoms. The resulting de Broglie wavelength (around
2218:
2140:
1830:
of the wavefunction as a charge density. This approach was, however, unsuccessful.
1342:
1305:
1292:
1274:
943:
937:
839:
829:
819:
719:
699:
684:
654:
522:
410:
10962:
10755:
10615:"Nonequivalence between Stationary Matter Wave Optics and Stationary Light Optics"
10317:
9691:
9575:
8663:"Electron diffraction chez Thomson: early responses to quantum physics in Britain"
8647:
6493:
2474:{\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} -i\omega t},}
2311:
of atoms and a Ramsey interferometry technique, the de Broglie wavelength of cold
1096:
proposed in 1905 that light is also propagated and absorbed in quanta, now called
12208:
12121:
12048:
12028:
11998:
11961:
11956:
11861:
11685:
11224:
10428:
10125:
9323:
7715:
7591:
6901:
6472:
6253:
2779:
2583:
2257:
2241:
1827:
1711:
1278:
1260:
1212:{\displaystyle \left|\mathbf {p} \right|=p={\frac {E}{c}}={\frac {h}{\lambda }},}
1093:
1000:
864:
734:
714:
460:
300:
31:
11357:
10902:
10638:
9224:
9047:
4765:
1620:
1304:
amplitude is blue, imaginary part is green. The probability (shown as the color
12203:
12099:
12068:
12058:
11680:
11670:
11504:
11353:
11042:
10863:
10828:
10518:
9868:
9770:"Tomography of ultrarelativistic nuclei with polarized photon-gluon collisions"
9746:
9086:
8500:
7666:
7615:
7403:
6897:
6892:
structures, in parallel with applications of electromagnetic waves, especially
6861:
6760:
6516:
6464:
6437:
6366:
6286:
6257:
6185:
5211:
5187:
3954:
3180:
2875:
2871:
2858:
is the reduced Planck constant. The second equation is also referred to as the
2402:
2308:
2249:
1858:
1823:
1815:
1519:
1350:
1301:
1277:
and by A. L. Hughes in 1912 then more carefully including a measurement of the
799:
759:
739:
709:
689:
639:
605:
455:
445:
238:
11315:
Cronin, Alexander D.; Schmiedmayer, Jörg; Pritchard, David E. (28 July 2009).
11110:
10659:
10483:
Cronin, Alexander D.; Schmiedmayer, Jörg; Pritchard, David E. (28 July 2009).
10007:
8903:
8846:
8678:
7391:
is a neutron diffraction technique for measuring the structure of thin films.
6884:
is 154 pm.) Reaching longer wavelengths requires special techniques like
6341:
4886:
allows the equations for de Broglie wavelength and frequency to be written as
12197:
12018:
11871:
11762:
11596:
11566:
11519:
11118:
11050:
10732:
10646:
10534:
10401:
10393:
10039:
9876:
9811:
9754:
9699:
9428:
9232:
9185:
8949:
8911:
8854:
8815:
8779:
8746:
8686:
8639:
8596:
Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (13 March 2013).
8536:
8476:
8401:
8349:
8289:
8220:
8107:
7674:
7603:
7429:
7413:
7268:
7238:
7206:
7195:
7168:
7126:
7091:
7065:
7033:
7005:
6970:
6938:
6885:
6877:
6775:
6523:. Light waves in a vacuum have linear dispersion relation between frequency:
6497:
6476:
6336:
6294:
6281:
6233:
5557:
5455:
5446:{\displaystyle \mathbf {K} =\left({\frac {\omega }{c}},{\mathbf {k} }\right)}
5337:
3790:{\displaystyle \mathbf {v} _{\mathrm {p} }=E/\mathbf {p} =c^{2}/\mathbf {v} }
2656:
2406:
2353:
2324:
2304:
2297:
2144:
1512:
has proven more useful.) De Broglie identified the velocity of the particle,
1270:
936:
The concept that matter behaves like a wave was proposed by French physicist
859:
854:
784:
754:
724:
595:
541:
268:
243:
121:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
9200:
8434:
8099:
7653:(907). American Association for the Advancement of Science (AAAS): 783â784.
5947:{\displaystyle E(\mathbf {k} )={\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m^{*}}}}
2943:
The relationship between the angular frequency and wavevector is called the
2559:
is time. (Here the physics definition for the wave vector is used, which is
1619:). By applying the differentials to the energy equation and identifying the
11901:
11514:
11509:
11183:
11068:
10970:
10910:
10779:
10599:
10591:
10047:
9829:
9803:
9707:
9644:
9583:
9522:
9436:
9359:
9293:
9240:
8929:
8872:
8419:
8340:
8315:
7682:
7481:
7276:
7214:
7176:
7138:
7099:
7041:
6981:
6945:
6354:
6290:
2335:. The researchers calculated a de Broglie wavelength of the most probable C
1819:
1080:). In 1900, this division was questioned when, investigating the theory of
849:
844:
779:
764:
729:
223:
10149:
9514:
8392:
7934:
Espinosa, J.M. (1982). "Physical properties of de Broglie's phase waves".
7737:
7706:
1499:{\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}=h\nu }
11939:
11390:
11255:(Researches on the quantum theory), Thesis (Paris), 1924; L. de Broglie,
10945:
9619:
9558:
9459:"2000 atoms in two places at once: A new record in quantum superposition"
9201:"Young's double-slit experiment with atoms: A simple atom interferometer"
7529:
7409:
6323:
5223:
3374:
This relativistic form relates to the phase velocity as discussed below.
2370:
2174:
2162:
1764:
1323:
1226:
930:
814:
769:
704:
659:
10660:
Tonomura, Akira; Endo, J.; Matsuda, T.; Kawasaki, T.; Ezawa, H. (1989).
10526:
10031:
9636:
8591:
8589:
2014:{\displaystyle x_{0}+vt\pm {\sqrt {\sigma ^{2}+(ht/2\pi \sigma m)^{2}}}}
10724:
10556:
Akbari, Kamran; Di Giulio, Valerio; GarcĂa De Abajo, F. Javier (2022).
9420:
9125:
9101:
8212:
8153:
7773:
7501:
Theoretical and experimental justification for the Schrödinger equation
7471:
7357:
7334:
which can produce information about chemical content at the nanoscale.
7284:
7248:
6908:, and internal structure, presenting new challenges and opportunities.
6764:
6326:
is two electrons bound together so they behave as a single matter wave.
6181:
5387:{\displaystyle \mathbf {P} =\left({\frac {E}{c}},{\mathbf {p} }\right)}
3078:
is the rest mass. Applying the derivative gives the (non-relativistic)
2591:
2587:
2526:
2289:
2178:
2042:
1780:
1693:
1406:
1319:
1244:
1085:
981:
804:
774:
694:
669:
664:
649:
9145:
9143:
8971:
8944:
8468:
7901:
2057:
In 1927, matter waves were first experimentally confirmed to occur in
1845:
11793:
11489:
10700:
10662:"Demonstration of single-electron buildup of an interference pattern"
10484:
9317:
8831:"Neutron crystallography: opportunities, challenges, and limitations"
8586:
8559:
8280:
8255:
7915:
MacKinnon, E. (1976). De Broglie's thesis: a critical retrospective,
7857:
7832:
7222:
7184:
7146:
7107:
7075:
7049:
7015:
6989:
6953:
6873:
5228:
Using four-vectors, the de Broglie relations form a single equation:
3367:{\displaystyle \mathbf {v_{g}} ={\frac {\mathbf {k} c^{2}}{\omega }}}
2641:
2386:
2342:
2332:
2236:
2101:
The first electron wave interference patterns directly demonstrating
1839:
1240:
295:
11401:
10685:
8528:
7994:
7955:
7752:
de Broglie, Louis (1970). "The reinterpretation of wave mechanics".
7642:
3131:{\displaystyle \mathbf {v_{g}} ={\frac {\hbar \mathbf {k} }{m_{0}}}}
2870:
In the de Broglie hypothesis, the velocity of a particle equals the
11025:
10574:
10300:
9786:
9140:
8555:
6889:
6249:
6129:
5163:
2600:
2151:
1834:
proposed that the modulus squared of the wavefunction is instead a
1831:
1807:
1701:
1346:
1056:
991:
926:
674:
11367:
11336:
11158:
10501:
9497:
9403:
9351:
9268:
8614:
8451:
7634:
4052:{\displaystyle \mathbf {v_{p}} ={\frac {c^{2}}{\mathbf {v_{g}} }}}
1508:(Modern physics no longer uses this form of the total energy; the
1076:, while matter was thought to consist of localized particles (see
9893:
Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles
7026:
6447:
6315:
6302:
6265:
2273:
2214:
2155:
2109:
using electrons through physical apertures gave the movie shown.
9845:"New type of double-slit interference experiment at Fermi scale"
9723:"New type of double-slit interference experiment at Fermi scale"
8892:
Acta Crystallographica Section A: Foundations of Crystallography
7627:
7422:
uses He atom waves to image solid structures non-destructively.
7307:
They are used for imaging from the micron to atomic scale using
6373:. The simplest of these, similar to the notation above would be
3994:{\displaystyle \mathbf {v_{p}} ={\frac {\omega }{\mathbf {k} }}}
2323:
Recent experiments confirm the relations for molecules and even
2248:, with a background in X-ray scattering from his PhD work under
1683:{\displaystyle p={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}
1611:(The modern definition of group velocity uses angular frequency
9668:"Nondestructive Mass Selection of Small van der Waals Clusters"
8515:(2). American Association of Physics Teachers (AAPT): 117â120.
8187:"Schrödinger's original struggles with a complex wave function"
6893:
6869:
6850:{\displaystyle \exp(i\mathbf {k} \cdot \mathbf {r} -i\omega t)}
2328:
2312:
2158:
1236:
1097:
918:
10555:
8076:"An Undulatory Theory of the Mechanics of Atoms and Molecules"
961:
11210:, Cham: Springer International Publishing, pp. 547â573,
9021:"Observation of Gravitationally Induced Quantum Interference"
8828:
6417:{\displaystyle \cos(\mathbf {k} \cdot \mathbf {r} -\omega t)}
2170:
1315:
10802:
9666:
Schöllkopf, Wieland; Toennies, J. Peter (25 November 1994).
9380:
6496:
occur in air under many environmental conditions. Obviously
6436:
in the early approaches to quantum mechanics. In that sense
4166:{\displaystyle \mathbf {v_{p}} \cdot \mathbf {v_{g}} =c^{2}}
4109:{\displaystyle \mathbf {v_{p}} \cdot \mathbf {v_{g}} =c^{2}}
11314:
11142:"Macroscopicity of Mechanical Quantum Superposition States"
10482:
10005:
9995:. Commission of the European Communities. pp. 397â552.
9060:
8837:. Carbohydrates and glycoconjugates / Biophysical methods.
8506:
1826:
to each point in space. Schrödinger tried to interpret the
1052:
949:
922:
11006:
10805:"Diffraction of an Atomic Beam by Standing-Wave Radiation"
8019:
6880:
range from 62 to 520 pm, and the typical length of a
11204:
10753:
8435:"Testing the limits of quantum mechanical superpositions"
7360:, probes the electric and magnetic fields in thin films.
6125:
1361:
one may associate a periodic phenomenon of the frequency
11140:
Nimmrichter, Stefan; Hornberger, Klaus (18 April 2013).
7809:(English translation by A.F. Kracklauer, 2004. ed.)
6306:
is a quasiparticle where an electron interacts with the
2284:
Interference of atom matter waves was first observed by
917:. At all scales where measurements have been practical,
10192:(5th ed.). Upper Saddle River, NJ: Prentice Hall.
9018:
8943:
Snell, A. H.; Wilkinson, M. K.; Koehler, W. C. (1984).
7971:"De Broglie's relativistic phase waves and wave groups"
6490:
Sensitivity of matter waves to environmental condition.
5954:
In general the group velocity would be replaced by the
4764:, one for the relativistic mass energy and one for the
4353:, but only the group velocity carries information. The
2874:
of the matter wave. In isotropic media or a vacuum the
1758:
1263:. In the modern convention, frequency is symbolized by
9019:
Colella, R.; Overhauser, A. W.; Werner, S. A. (1975).
8885:
6345:
Some trajectories of a particle in a box according to
6319:
is an electron and hole pair which are bound together.
11139:
10008:"Production and application of electron vortex beams"
8942:
7880:
Medicus, H.A. (1974). "Fifty years of matter waves".
6806:
6698:
6664:
6658:
says the frequency in vacuum varies with wavenumber (
6558:
6529:
6379:
6145:
6110:
5963:
5883:
5769:
5734:
5690:
5575:
5463:
5404:
5345:
5274:
5234:
5196:
5172:
5130:
4892:
4773:
4694:
4684:), the phase velocity of matter waves always exceeds
4680:
for any particle that has nonzero mass (according to
4648:
4626:
4471:
4366:
4314:
4269:
4224:
4179:
4122:
4065:
4007:
3963:
3929:
3803:
3733:
3387:
3324:
3215:
3188:
3147:
3088:
3057:
2953:
2884:
2788:
2665:
2608:
2565:
2545:
2509:
2487:
2415:
2027:
1935:
1915:
1895:
1875:
1726:
1700:, associated with an electron and the modulus of its
1629:
1528:
1429:
1160:
1139:
1110:
1015:
59:
11317:"Optics and interferometry with atoms and molecules"
11084:
10756:"The Diffraction of Neutrons by Crystalline Powders"
10613:
Brukner, Äaslav; Zeilinger, Anton (6 October 1997).
10485:"Optics and interferometry with atoms and molecules"
10006:
Verbeeck, J.; Tian, H.; Schattschneider, P. (2010).
8595:
3138:
For comparison, the group velocity of light, with a
2061:
and Alexander Reid's diffraction experiment and the
958:
955:
946:
8796:Peng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011).
7707:"A Direct Photoelectric Determination of Planck's "
7641:Richardson, O. W.; Compton, Karl T. (17 May 1912).
7569:
A History of the Theories of Aether and Electricity
4731:{\displaystyle |\mathbf {v} _{\mathrm {p} }|>c,}
2357:. As of 2019, this has been pushed to molecules of
2331:demonstrated diffraction for molecules as large as
2049:The wave packet spreads out as show in the figure.
952:
11087:"Quantum superposition of molecules beyond 25 kDa"
11009:"Matter-wave interference of a native polypeptide"
9767:
9665:
8723:"Theorie der Beugung von Elektronen an Kristallen"
8157:
7607:
6849:
6778:and alter the internal excitation state of atoms.
6737:
6684:
6646:
6544:
6515:Light waves of all frequencies travel at the same
6416:
6168:
6116:
6096:
5946:
5869:
5755:
5715:
5676:
5540:
5445:
5386:
5325:
5256:
5202:
5178:
5154:
5116:
4878:
4730:
4672:
4634:
4612:
4451:
4345:
4300:
4255:
4210:
4165:
4108:
4051:
3993:
3937:
3915:
3789:
3717:
3366:
3310:
3194:
3171:
3130:
3070:
3043:
2935:
2836:
2764:
2632:
2574:
2551:
2517:
2495:
2473:
2033:
2013:
1921:
1901:
1881:
1748:
1682:
1601:
1498:
1211:
1147:
1125:
1037:
120:
10071:
9896:(2nd ed.). New York: John Wiley & Sons.
9063:"Single- and double-slit diffraction of neutrons"
8795:
8316:"Diffraction of Electrons by a Crystal of Nickel"
7640:
7350:to image the top atomic layer of solid surfaces.
6872:over 9 orders but the wavelengths are all in the
6135:. The momentum would then be described using the
1100:. These quanta would have an energy given by the
969:) in 1924, and so matter waves are also known as
12195:
10926:"MatterâWave Interferometer for Large Molecules"
10612:
10240:
9482:
9313:
9311:
9149:
8936:
8368:"Reflection of Electrons by a Crystal of Nickel"
7438:measurements rely on Rb atom wave interference.
6759:, which at the quantum level is equivalent to a
6297:. A hole has the opposite charge of an electron.
6285:is a quasiparticle which can be thought of as a
6264:) electron and, like a normal electron, it is a
5257:{\displaystyle \mathbf {P} =\hbar \mathbf {K} ,}
2192:This was a pivotal result in the development of
1409:equivalent to a moving body, de Broglie set the
11278:Tipler, Paul A. and Ralph A. Llewellyn (2003).
10101:The conceptual development of quantum mechanics
8798:High energy electron diffraction and microscopy
8433:Arndt, Markus; Hornberger, Klaus (April 2014).
8432:
8372:Proceedings of the National Academy of Sciences
8069:
8067:
6896:. Unlike light, matter wave particles may have
6791:
6552:. For matter waves the relation is non-linear:
5564:
10698:
10348:National Institute of Standards and Technology
10241:Adams, C.S; Sigel, M; Mlynek, J (1 May 1994).
10078:. Cambridge University Press. pp. 65â69.
9889:
9326:(14 October 1999). "Waveâparticle duality of C
9099:
8711:, Cambridge University Press 2004, pp. 156â157
8667:The British Journal for the History of Science
8550:
8548:
8546:
8365:
8313:
6448:Matter waves vs. electromagnetic waves (light)
5684:where now there is an additional spatial term
1763:Following up on de Broglie's ideas, physicist
11417:
10462:. London: Taylor & Francis. p. 117.
9476:
9308:
8709:Nobel Laureates and Twentieth-Century Physics
8598:"Controlled double-slit electron diffraction"
7562:
7560:
7558:
7556:
7554:
7356:, the electron matter wave analog of optical
6463:share many properties with results of light
5556:The preceding sections refer specifically to
4742:when the particle speed is relativistic. The
4001:Using the relativistic group velocity above:
3727:But (see below), since the phase velocity is
1775:'s analogy between mechanics and optics (see
887:
10457:
10417:"Systems and Subsystems, Multiple Particles"
10281:
10275:
9883:
9198:
8256:"Diffraction of Cathode Rays by a Thin Film"
8119:
8117:
8064:
7968:
7326:The measurements of the energy they lose in
6738:{\displaystyle \hbar \omega _{0}=m_{0}c^{2}}
6169:{\displaystyle \mathbf {p} =-i\hbar \nabla }
5716:{\displaystyle u(\mathbf {r} ,\mathbf {k} )}
2052:
115:
89:
11080:
11078:
10699:Estermann, I.; Stern, O. (1 January 1930).
10282:Schlosshauer, Maximilian (1 October 2019).
10212:
8543:
8253:
8073:
8044:
7571:. Vol. 2. Courier Dover Publications.
1308:) of finding the particle at a given point
933:just like a beam of light or a water wave.
11424:
11410:
11300:(5th ed.). Boston: Houghton Mifflin.
11282:. 4th ed. New York; W. H. Freeman and Co.
10067:
10065:
8249:
8247:
8146:
7827:
7751:
7594:(1917). Zur Quantentheorie der Strahlung,
7551:
7412:mimic many light optic devices, including
6221:
3205:As an alternative, using the relativistic
894:
880:
27:Quantum mechanical waves describing matter
11335:
11223:
11157:
11058:
11024:
10944:
10787:
10766:(8). American Physical Society: 830â841.
10573:
10500:
10299:
10266:
10162:
10075:Atomic and Electronic Structure of Solids
9990:
9961:
9959:
9819:
9785:
9618:
9557:
9496:
9402:
9267:
9175:
9104:(1930). "Beugung von Molekularstrahlen".
8988:"Early Development of Neutron Scattering"
8970:
8919:
8862:
8613:
8450:
8409:
8391:
8339:
8279:
8114:
8061:See the introduction to first 1926 paper.
7908:
7856:
7793:
7791:
7736:
7566:
7534:"Max Planck: the reluctant revolutionary"
6860:As shown in the table below, matter wave
6640:
6492:Many examples of electromagnetic (light)
5012:
4960:
4959:
4958:
4935:
4931:
4905:
4904:
4465:relations for energy and momentum yields
3304:
3037:
2947:. For the non-relativistic case this is:
2401:Waves have more complicated concepts for
2396:
1838:, a successful proposal now known as the
11075:
10458:Hawkes, Peter W.; Hawkes, P. W. (1972).
10236:
10234:
10123:
9993:Electron Microscopy in Materials Science
9150:Adams, C.S; Sigel, M; Mlynek, J (1994).
8753:
8554:
7933:
7704:
7602:
7567:Whittaker, Sir Edmund (1 January 1989).
6340:
6330:
2111:
1844:
1749:{\displaystyle \lambda ={\frac {h}{p}}.}
1291:
1287:
1038:{\displaystyle \lambda ={\frac {h}{p}}.}
11293:
11265:English translation by A.F. Kracklauer.
10460:Electron optics and electron microscopy
10167:. Englewood Cliffs, NJ: Prentice Hall.
10062:
8791:
8789:
8759:
8660:
8366:Davisson, C. J.; Germer, L. H. (1928).
8244:
7879:
7873:
6876:range, comparable to atomic spacings. (
6857:, then using these to probe materials.
5551:
4346:{\displaystyle |\mathbf {v_{p}} |>c}
4301:{\displaystyle |\mathbf {v_{g}} |<c}
2782:. The equations can also be written as
2590:.) The de Broglie equations relate the
2409:, that is a wave function described by
2268:throughout the 1940s. In the 1970s, a
925:-like behavior. For example, a beam of
14:
12196:
10558:"Optical manipulation of matter waves"
10187:
10098:
9965:
9956:
9916:
9199:Carnal, O.; Mlynek, J. (27 May 1991).
8361:
8359:
8309:
8307:
8164:. Wiley-Interscience. pp. 24â25.
8152:
7797:
7788:
6236:. Many of these occur in solids â see
1814:since the energy corresponding to the
11431:
11405:
11363:on 19 July 2011 – via Atomwave.
10231:
9921:. New York: Oxford University Press.
9768:STAR Collaboration (6 January 2023).
8985:
8835:Current Opinion in Structural Biology
8720:
8714:
8184:
8123:
8020:McEvoy, J. P.; Zarate, Oscar (2004).
7528:
4755:
1796:wave equation that now bears his name
11253:Recherches sur la théorie des quanta
10754:Wollan, E. O.; Shull, C. G. (1948).
10414:
10367:
10338:
10072:Efthimios Kaxiras (9 January 2003).
8786:
8314:Davisson, C.; Germer, L. H. (1927).
6457:Hamilton's optico-mechanical analogy
6192:, and are also used to describe the
4256:{\displaystyle |\mathbf {v_{g}} |=c}
4211:{\displaystyle |\mathbf {v_{p}} |=c}
1909:of the packet traveling at velocity
1777:Hamilton's optico-mechanical analogy
1759:Schrödinger's (matter) wave equation
1300:in one dimension â real part of the
1078:history of wave and particle duality
909:are a central part of the theory of
10988:from the original on 13 August 2007
9391:Physical Chemistry Chemical Physics
9247:
8800:. Oxford: Oxford University Press.
8356:
8304:
6196:of high-energy electrons by solids.
4673:{\displaystyle |\mathbf {v} |<c}
3377:For non-isotropic media we use the
2296:allowing precise small devices and
1794:In 1926, Schrödinger published the
24:
11380:
11245:
9842:
9720:
9001:. pp. 145â154. Archived from
6184:, which form the basis of much of
6163:
6111:
6064:
6028:
5851:
5838:
5824:
5515:
4708:
4480:
4375:
3957:in isotropic media is defined as:
3883:
3816:
3742:
3522:
3518:
3502:
3494:
3463:
3441:
3424:
3416:
3400:
2922:
2903:
1553:
1545:
990:, associated with a particle with
426:Sum-over-histories (path integral)
112:
86:
42:Part of a series of articles about
25:
12225:
11374:
10421:Quantum Theory for Mathematicians
10165:Introduction to quantum mechanics
9890:Resnick, R.; Eisberg, R. (1985).
8995:Nobel Lectures, Physics 1991â1995
8254:Thomson, G. P.; Reid, A. (1927).
8024:. Totem Books. pp. 110â114.
7328:electron energy loss spectroscopy
6598:
6160:
5668:
5243:
4425:
4410:
3948:
3479:
3455:
3287:
2995:
2865:
2821:
2802:
63:
12178:
12177:
9945:, Taylor & Francis, London,
7394:
6912:Various matter wave wavelengths
6828:
6820:
6398:
6390:
6147:
6082:
6056:
6039:
6020:
5973:
5965:
5916:
5891:
5706:
5698:
5653:
5636:
5628:
5608:
5600:
5583:
5525:
5486:
5465:
5433:
5406:
5374:
5347:
5316:
5276:
5247:
5236:
5155:{\displaystyle v=|\mathbf {v} |}
5143:
4868:
4847:
4832:
4702:
4655:
4628:
4601:
4578:
4526:
4495:
4474:
4440:
4416:
4390:
4369:
4326:
4322:
4281:
4277:
4236:
4232:
4191:
4187:
4144:
4140:
4129:
4125:
4087:
4083:
4072:
4068:
4041:
4037:
4014:
4010:
3985:
3970:
3966:
3931:
3902:
3877:
3833:
3810:
3783:
3760:
3736:
3688:
3607:
3526:
3506:
3470:
3428:
3394:
3344:
3331:
3327:
3223:
3111:
3095:
3091:
2961:
2926:
2913:
2891:
2887:
2806:
2795:
2694:
2633:{\displaystyle |\mathbf {p} |=p}
2615:
2511:
2489:
2450:
2442:
2423:
2084:
2072:
1166:
1141:
942:
11272:The wave nature of the electron
11198:
11133:
11000:
10917:
10870:
10835:
10796:
10747:
10701:"Beugung von Molekularstrahlen"
10692:
10653:
10606:
10549:
10476:
10451:
10408:
10361:
10341:"Neutron Scattering â A Primer"
10332:
10206:
10181:
10156:
10117:
10092:
9999:
9984:
9935:
9910:
9843:Ma, Yu-Gang (30 January 2023).
9836:
9761:
9721:Ma, Yu-Gang (30 January 2023).
9714:
9659:
9598:
9537:
9451:
9374:
9192:
9093:
9054:
9012:
8979:
8879:
8822:
8701:
8654:
8491:
8426:
8235:
8178:
8038:
8013:
7962:
7927:
7259:functionalized oligoporphyrins
6656:matter wave dispersion relation
6481:slow electrons moving in solids
6469:Kirchhoff's diffraction formula
6428:, and other cases such as in a
5217:
2539:with units of inverse time and
1062:
12127:Relativistic quantum mechanics
11176:10.1103/PhysRevLett.110.160403
9943:Introducing Special Relativity
9849:Nuclear Science and Techniques
9727:Nuclear Science and Techniques
9322:; C. Keller; G. van der Zouw;
9286:10.1103/PhysRevLett.102.170401
7821:
7807:Foundation of Louis de Broglie
7745:
7698:
7689:
7585:
7522:
7441:
7383:Small-angle neutron scattering
7379:and sensitivity to magnetism.
7332:energy dispersive spectroscopy
7296:
6844:
6813:
6568:
6562:
6411:
6386:
6086:
6078:
6060:
6052:
6043:
6035:
6024:
6016:
5977:
5969:
5895:
5887:
5710:
5694:
5671:
5657:
5649:
5621:
5612:
5596:
5587:
5579:
5535:
5529:
5500:
5491:
5475:
5148:
5138:
4715:
4696:
4660:
4650:
4360:For non-isotropic media, then
4333:
4316:
4288:
4271:
4243:
4226:
4198:
4181:
3482:
3466:
3458:
3444:
3227:
3219:
3157:
3151:
2965:
2957:
2917:
2909:
2699:
2689:
2620:
2610:
2582:times the wave vector used in
2427:
2419:
2221:with kinetic energy of around
2000:
1973:
1798:â the matter wave analogue of
1783:become sharp tracks that obey
1593:
1579:
576:Relativistic quantum mechanics
108:
101:
82:
13:
1:
12105:Quantum statistical mechanics
11882:Quantum differential calculus
11804:Delayed-choice quantum eraser
11587:Symmetry in quantum mechanics
11290:. pp. 203â4, 222â3, 236.
10963:10.1103/PhysRevLett.88.100404
10318:10.1016/j.physrep.2019.10.001
9692:10.1126/science.266.5189.1345
9576:10.1103/PhysRevLett.90.160401
8632:10.1088/1367-2630/15/3/033018
8608:(3). IOP Publishing: 033018.
8126:Schrödinger: Life and Thought
8049:. Friedrich Vieweg und Sohn.
7969:Brown, H.R.; Martins (1984).
7515:
7505:Thermal de Broglie wavelength
7416:, atom focusing zone plates.
7363:
7344:Scanning tunneling microscope
6771:Optically shaped matter waves
6215:grazing-incidence diffraction
5728:which in general is a tensor
3172:{\displaystyle \omega (k)=ck}
2503:is a position in real space,
1869:. Darwin showed that at time
1067:
616:Quantum statistical mechanics
12214:Foundational quantum physics
11225:10.1007/978-3-030-63963-1_24
10429:10.1007/978-1-4614-7116-5_19
10268:10.1016/0370-1573(94)90066-3
10163:Griffiths, David J. (1995).
10124:Mulliken, Robert S. (1932).
9177:10.1016/0370-1573(94)90066-3
8185:Karam, Ricardo (June 2020).
7453:and quantum macroscopicity.
6792:Applications of matter waves
6747:wave packets of matter waves
6685:{\displaystyle k=1/\lambda }
5565:Single-particle matter waves
4635:{\displaystyle \mathbf {v} }
3938:{\displaystyle \mathbf {v} }
2529:in units of inverse meters,
2518:{\displaystyle \mathbf {k} }
2496:{\displaystyle \mathbf {r} }
2381:Matter wave was detected in
2318:
2124:
1822:, a function that assigns a
1802:â and used it to derive the
1148:{\displaystyle \mathbf {p} }
7:
11907:Quantum stochastic calculus
11897:Quantum measurement problem
11819:MachâZehnder interferometer
11294:Zumdahl, Steven S. (2005).
10903:10.1103/PhysRevLett.74.4783
10666:American Journal of Physics
10639:10.1103/PhysRevLett.79.2599
10339:Pynn, Roger (1 July 1990).
10213:Schrödinger, Erwin (2001).
9966:Schiff, Leonard I. (1987).
9225:10.1103/PhysRevLett.66.2689
9162:(3). Elsevier BV: 143â210.
9048:10.1103/PhysRevLett.34.1472
8999:World Scientific Publishing
8509:American Journal of Physics
8192:American Journal of Physics
7460:
6444:are electron matter waves.
6424:These occur as part of the
2272:demonstrated the action of
2203:
586:Quantum information science
10:
12230:
11354:10.1103/RevModPhys.81.1051
11043:10.1038/s41467-020-15280-2
10864:10.1103/PhysRevA.61.033608
10829:10.1103/PhysRevLett.51.370
10519:10.1103/RevModPhys.81.1051
9869:10.1007/s41365-023-01167-6
9747:10.1007/s41365-023-01167-6
9087:10.1103/RevModPhys.60.1067
8570:Cambridge University Press
8130:Cambridge University Press
8022:Introducing Quantum Theory
7798:de Broglie, Louis Victor.
7667:10.1126/science.35.907.783
7643:"The Photoelectric Effect"
7601:: 121â128. Translated in
7420:Scanning helium microscopy
7233:polypeptide, Gramicidin A
6960:DavissonâGermer experiment
6782:Multi-particle experiments
6545:{\displaystyle \omega =ck}
6334:
6225:
5756:{\displaystyle m_{ij}^{*}}
5221:
4263:. Since for matter waves,
3080:matter wave group velocity
2207:
2131:DavissonâGermer experiment
2128:
2063:DavissonâGermer experiment
1853:The following year, 1927,
29:
12173:
12135:
12087:
11967:Quantum complexity theory
11945:Quantum cellular automata
11920:
11852:
11786:
11699:
11663:
11650:Path integral formulation
11617:
11482:
11439:
11324:Reviews of Modern Physics
11111:10.1038/s41567-019-0663-9
10489:Reviews of Modern Physics
10374:Reviews of Modern Physics
9991:Metherell, A. J. (1972).
9941:Williams, W.S.C. (2002).
9067:Reviews of Modern Physics
8904:10.1107/S0108767312036021
8847:10.1016/j.sbi.2008.06.009
8679:10.1017/S0007087410000026
7800:"On the Theory of Quanta"
7596:Physicalische Zeitschrift
7377:scattering cross sections
6882:carbonâcarbon single bond
6434:BohrâSommerfeld condition
6357:. The states (B,C,D) are
6137:kinetic momentum operator
2878:of a wave is defined by:
2376:
2053:Experimental confirmation
1789:principle of least action
12034:Quantum machine learning
12014:Quantum key distribution
12004:Quantum image processing
11994:Quantum error correction
11844:Wheeler's delayed choice
11395:University of Nottingham
11370:, 1953 (Oliver and Boyd)
10394:10.1103/RevModPhys.29.74
8993:. In Ekspong, G. (ed.).
8760:John M., Cowley (1995).
8747:10.1002/andp.19283921704
8074:Schrödinger, E. (1926).
8045:Schrödinger, E. (1984).
6274:electric field screening
4760:Using two formulas from
2860:PlanckâEinstein relation
2391:Bose-Einstein condensate
2279:
1510:energyâmomentum relation
1253:the speed of light, and
1102:PlanckâEinstein relation
621:Quantum machine learning
374:Wheeler's delayed-choice
11950:Quantum finite automata
11274:Nobel Lecture, 12, 1929
11146:Physical Review Letters
10933:Physical Review Letters
10883:Physical Review Letters
10809:Physical Review Letters
10789:2027/mdp.39015086506584
10619:Physical Review Letters
10415:Hall, Brian C. (2013),
10188:Levine, Ira N. (2000).
9256:Physical Review Letters
9205:Physical Review Letters
9028:Physical Review Letters
8661:Navarro, Jaume (2010).
8100:10.1103/PhysRev.28.1049
7404:optical interferometers
6440:around atoms, and also
6222:Collective matter waves
6117:{\displaystyle \nabla }
5179:{\displaystyle \gamma }
3207:dispersion relationship
2945:dispersion relationship
2397:Traveling matter waves
2383:van der Waals molecules
2118:double slit diffraction
2034:{\displaystyle \sigma }
1417:for that body equal to
1370:, such that one finds:
1126:{\displaystyle E=h\nu }
331:LeggettâGarg inequality
12054:Quantum neural network
10780:10.1103/PhysRev.73.830
10705:Zeitschrift fĂŒr Physik
10592:10.1126/sciadv.abq2659
9804:10.1126/sciadv.abq3903
8602:New Journal of Physics
8341:10.1103/physrev.30.705
7754:Foundations of Physics
7610:The Old Quantum Theory
7510:De BroglieâBohm theory
7375:through the different
7319:, and for surfaces at
6851:
6739:
6686:
6654:This non-relativistic
6648:
6546:
6418:
6362:
6246:electron quasiparticle
6228:List of quasiparticles
6170:
6118:
6098:
5948:
5871:
5757:
5717:
5678:
5542:
5447:
5388:
5327:
5258:
5204:
5180:
5156:
5118:
4880:
4732:
4674:
4636:
4614:
4453:
4347:
4302:
4257:
4212:
4167:
4110:
4053:
3995:
3939:
3917:
3791:
3719:
3368:
3312:
3196:
3173:
3132:
3072:
3045:
2937:
2838:
2766:
2634:
2599:to the modulus of the
2576:
2553:
2519:
2497:
2475:
2270:neutron interferometer
2139:In 1927 at Bell Labs,
2121:
2107:double-slit experiment
2065:, both for electrons.
2035:
2015:
1923:
1903:
1883:
1863:Schrödinger's equation
1850:
1773:William Rowan Hamilton
1750:
1684:
1603:
1500:
1340:
1327:
1213:
1149:
1133:and a momentum vector
1127:
1039:
122:
12079:Quantum teleportation
11607:Waveâparticle duality
11013:Nature Communications
10284:"Quantum decoherence"
10150:10.1103/PhysRev.41.49
9917:Holden, Alan (1971).
9515:10.1038/nnano.2012.34
9485:Nature Nanotechnology
9318:Arndt, M.; O. Nairz;
8986:Shull, C. G. (1997).
8393:10.1073/pnas.14.4.317
8124:Moore, W. J. (1992).
7738:10.1103/PhysRev.7.355
7705:Millikan, R. (1916).
7467:Wave-particle duality
7451:waveâparticle duality
7447:Quantum superposition
7389:Neutron reflectometry
7000:He atom, H2 molecule
6852:
6788:experiments are not.
6776:cool matter particles
6740:
6687:
6649:
6547:
6419:
6344:
6331:Standing matter waves
6205:electron vortex beams
6171:
6119:
6099:
5949:
5872:
5758:
5718:
5679:
5543:
5448:
5389:
5328:
5259:
5205:
5181:
5157:
5119:
4881:
4766:relativistic momentum
4752:for further details.
4733:
4675:
4637:
4615:
4454:
4348:
4303:
4258:
4213:
4168:
4111:
4054:
3996:
3940:
3918:
3792:
3720:
3369:
3313:
3197:
3174:
3133:
3073:
3071:{\displaystyle m_{0}}
3046:
2938:
2839:
2767:
2635:
2577:
2575:{\displaystyle 2\pi }
2554:
2520:
2498:
2476:
2129:Further information:
2115:
2103:waveâparticle duality
2036:
2016:
1924:
1904:
1884:
1848:
1751:
1685:
1621:relativistic momentum
1604:
1501:
1329:
1295:
1288:De Broglie hypothesis
1214:
1150:
1128:
1040:
978:de Broglie wavelength
915:waveâparticle duality
316:Elitzur–Vaidman
306:Davisson–Germer
123:
18:De Broglie wavelength
12110:Quantum field theory
12039:Quantum metamaterial
11984:Quantum cryptography
11714:Consistent histories
10099:Jammer, Max (1989).
8945:"Ernest Omar Wollan"
8565:Principles of Optics
8132:. pp. 219â220.
7497:Schrödinger equation
7487:KapitsaâDirac effect
7400:Atom interferometers
7309:electron microscopes
7302:Electron diffraction
7021:Estermann and Stern
6804:
6696:
6662:
6556:
6527:
6461:Schrödinger equation
6377:
6361:, but (E,F) are not.
6240:. Examples include:
6143:
6108:
5961:
5881:
5767:
5732:
5688:
5573:
5552:General matter waves
5461:
5402:
5343:
5272:
5232:
5194:
5170:
5128:
4890:
4771:
4692:
4646:
4624:
4469:
4364:
4312:
4267:
4222:
4177:
4120:
4063:
4005:
3961:
3927:
3801:
3731:
3385:
3322:
3213:
3186:
3145:
3086:
3055:
2951:
2882:
2786:
2663:
2649:to the total energy
2606:
2563:
2543:
2507:
2485:
2413:
2254:X-10 nuclear reactor
2198:photoelectric effect
2187:electron diffraction
2183:Schrödinger equation
2163:diffraction patterns
2135:Electron diffraction
2059:George Paget Thomson
2047:uncertainty relation
2025:
1933:
1913:
1893:
1873:
1867:Gaussian wave packet
1724:
1627:
1526:
1427:
1158:
1137:
1108:
1090:photoelectric effect
1082:black-body radiation
1059:in the years since.
1049:elementary particles
1013:
581:Quantum field theory
493:Consistent histories
130:Schrödinger equation
57:
12095:Quantum fluctuation
12064:Quantum programming
12024:Quantum logic gates
12009:Quantum information
11989:Quantum electronics
11464:Classical mechanics
11346:2009RvMP...81.1051C
11297:Chemical Principles
11270:Broglie, Louis de,
11216:2021mbpc.book..547G
11168:2013PhRvL.110p0403N
11103:2019NatPh..15.1242F
11035:2020NatCo..11.1447S
10955:2002PhRvL..88j0404B
10895:1995PhRvL..74.4783C
10856:2000PhRvA..61c3608G
10821:1983PhRvL..51..370M
10772:1948PhRv...73..830W
10717:1930ZPhy...61...95E
10678:1989AmJPh..57..117T
10631:1997PhRvL..79.2599B
10584:2022SciA....8.2659A
10511:2009RvMP...81.1051C
10386:1957RvMP...29...74F
10310:2019PhR...831....1S
10259:1994PhR...240..143A
10142:1932PhRv...41...49M
10032:10.1038/nature09366
10024:2010Natur.467..301V
9861:2023NuScT..34...16M
9796:2023SciA....9.3903.
9739:2023NuScT..34...16M
9684:1994Sci...266.1345S
9678:(5189): 1345â1348.
9637:10.1038/nature02276
9629:2004Natur.427..711H
9568:2003PhRvL..90p0401H
9507:2012NatNa...7..297J
9413:2013PCCP...1514696E
9344:1999Natur.401..680A
9278:2009PhRvL.102q0401C
9217:1991PhRvL..66.2689C
9168:1994PhR...240..143A
9118:1930ZPhy...61...95E
9079:1988RvMP...60.1067Z
9040:1975PhRvL..34.1472C
8963:1984PhT....37k.120S
8762:Diffraction physics
8739:1928AnP...392...55B
8624:2013NJPh...15c3018B
8521:1989AmJPh..57..117T
8461:2014NatPh..10..271A
8384:1928PNAS...14..317D
8332:1927PhRv...30..705D
8272:1927Natur.119Q.890T
8205:2020AmJPh..88..433K
8092:1926PhRv...28.1049S
7987:1984AmJPh..52.1130B
7948:1982AmJPh..50..357E
7894:1974PhT....27b..38M
7849:1923Natur.112..540D
7766:1970FoPh....1....5D
7729:1916PhRv....7..355M
7659:1912Sci....35..783R
7532:(1 December 2000).
7436:Quantum decoherence
7369:Neutron diffraction
7354:Electron holography
6913:
6866:orders of magnitude
6521:dispersion relation
6351:classical mechanics
6262:elementary particle
6238:Ashcroft and Mermin
6190:Ashcroft and Mermin
5956:probability current
5789:
5752:
4749:Dispersion (optics)
3660:
3576:
2266:neutron diffraction
2210:Neutron diffraction
1889:later the position
1836:probability density
1800:Maxwell's equations
1787:, an analog of the
1237:Greek letter lambda
1074:Maxwell's equations
369:Stern–Gerlach
166:Classical mechanics
12148:in popular culture
11930:Quantum algorithms
11778:Von NeumannâWigner
11758:Objective collapse
11469:Old quantum theory
11383:"de Broglie Waves"
10725:10.1007/BF01340293
9421:10.1039/c3cp51500a
9126:10.1007/bf01340293
8727:Annalen der Physik
8721:Bethe, H. (1928).
8213:10.1119/10.0000852
7833:"Waves and quanta"
7774:10.1007/BF00708650
7477:Compton wavelength
7426:Quantum reflection
6911:
6847:
6798:quantum properties
6735:
6682:
6644:
6542:
6442:molecular orbitals
6414:
6363:
6359:energy eigenstates
6166:
6114:
6094:
5944:
5867:
5772:
5753:
5735:
5713:
5674:
5538:
5443:
5384:
5323:
5254:
5200:
5176:
5152:
5114:
5112:
4876:
4874:
4762:special relativity
4756:Special relativity
4728:
4682:special relativity
4670:
4632:
4610:
4449:
4343:
4308:, it follows that
4298:
4253:
4208:
4163:
4106:
4049:
3991:
3935:
3913:
3911:
3787:
3715:
3713:
3646:
3562:
3364:
3308:
3192:
3169:
3128:
3068:
3041:
2933:
2834:
2832:
2762:
2760:
2659:as written above:
2630:
2572:
2549:
2515:
2493:
2471:
2286:Immanuel Estermann
2122:
2031:
2011:
1919:
1899:
1879:
1851:
1785:Fermat's principle
1746:
1680:
1599:
1496:
1415:special relativity
1328:
1209:
1145:
1123:
1035:
557:Von NeumannâWigner
537:Objective-collapse
336:Mach–Zehnder
326:Leggett inequality
321:Franck–Hertz
171:Old quantum theory
118:
12191:
12190:
12165:Quantum mysticism
12143:Schrödinger's cat
12074:Quantum simulator
12044:Quantum metrology
11972:Quantum computing
11935:Quantum amplifier
11912:Quantum spacetime
11877:Quantum cosmology
11867:Quantum chemistry
11582:Scattering theory
11530:Zero-point energy
11525:Degenerate levels
11433:Quantum mechanics
11307:978-0-618-37206-5
11235:978-3-030-63963-1
11097:(12): 1242â1245.
10889:(24): 4783â4786.
10844:Physical Review A
10625:(14): 2599â2603.
10469:978-0-85066-056-2
10438:978-1-4614-7115-8
10368:Fano, U. (1957).
10224:978-0-8218-3524-1
10199:978-0-13-685512-5
10190:Quantum chemistry
10174:978-0-13-124405-4
10110:978-0-88318-617-6
10085:978-0-521-52339-4
10018:(7313): 301â304.
9977:978-0-07-085643-1
9968:Quantum mechanics
9928:978-0-19-501497-6
9919:Stationary states
9903:978-0-471-87373-0
9613:(6976): 711â714.
9397:(35): 14696â700.
9338:(6754): 680â682.
9211:(21): 2689â2692.
9034:(23): 1472â1474.
8972:10.1063/1.2915947
8807:978-0-19-960224-7
8579:978-0-521-64222-4
8469:10.1038/nphys2863
8171:978-0-471-43958-5
8139:978-0-521-43767-7
8056:978-3-7001-0573-2
8031:978-1-84046-577-8
7981:(12): 1130â1140.
7902:10.1063/1.3128444
7492:Matter wave clock
7373:x-ray diffraction
7348:quantum tunneling
7338:Quantum tunneling
7294:
7293:
7081:Moskowitz et al.
7055:Wollan and Shull
6638:
6601:
6467:. In particular,
6426:particle in a box
5999:
5942:
5865:
5818:
5532:
5426:
5367:
5309:
5203:{\displaystyle c}
5105:
5102:
5100:
5037:
4993:
4991:
4956:
4929:
4738:which approaches
4605:
4583:
4531:
4499:
4444:
4429:
4394:
4047:
3989:
3889:
3851:
3706:
3672:
3671:
3587:
3531:
3511:
3486:
3433:
3362:
3302:
3290:
3209:for matter waves
3195:{\displaystyle c}
3126:
3035:
2998:
2931:
2756:
2743:
2717:
2704:
2552:{\displaystyle t}
2537:angular frequency
2309:Bragg diffraction
2264:, they developed
2262:Clifford G. Shull
2194:quantum mechanics
2181:, who solved the
2009:
1922:{\displaystyle v}
1902:{\displaystyle x}
1882:{\displaystyle t}
1857:(grandson of the
1812:Compton frequency
1769:Erwin Schrödinger
1741:
1678:
1677:
1675:
1597:
1560:
1536:
1485:
1484:
1482:
1400:Compton frequency
1204:
1191:
1030:
911:quantum mechanics
904:
903:
611:Scattering theory
591:Quantum computing
364:Schrödinger's cat
296:Bell's inequality
104:
79:
48:Quantum mechanics
16:(Redirected from
12221:
12181:
12180:
11892:Quantum geometry
11887:Quantum dynamics
11744:Superdeterminism
11640:Matrix mechanics
11495:Braâket notation
11426:
11419:
11412:
11403:
11402:
11398:
11364:
11362:
11356:. Archived from
11339:
11330:(3): 1051â1129.
11321:
11311:
11239:
11238:
11227:
11202:
11196:
11195:
11161:
11137:
11131:
11130:
11082:
11073:
11072:
11062:
11028:
11004:
10998:
10997:
10995:
10993:
10987:
10948:
10946:quant-ph/0202158
10930:
10921:
10915:
10914:
10874:
10868:
10867:
10839:
10833:
10832:
10800:
10794:
10793:
10791:
10751:
10745:
10744:
10696:
10690:
10689:
10657:
10651:
10650:
10610:
10604:
10603:
10577:
10568:(42): eabq2659.
10562:Science Advances
10553:
10547:
10546:
10504:
10495:(3): 1051â1129.
10480:
10474:
10473:
10455:
10449:
10448:
10447:
10445:
10412:
10406:
10405:
10365:
10359:
10358:
10356:
10354:
10345:
10336:
10330:
10329:
10303:
10279:
10273:
10272:
10270:
10238:
10229:
10228:
10210:
10204:
10203:
10185:
10179:
10178:
10160:
10154:
10153:
10121:
10115:
10114:
10096:
10090:
10089:
10069:
10060:
10059:
10003:
9997:
9996:
9988:
9982:
9981:
9963:
9954:
9939:
9933:
9932:
9914:
9908:
9907:
9887:
9881:
9880:
9840:
9834:
9833:
9823:
9789:
9774:Science Advances
9765:
9759:
9758:
9718:
9712:
9711:
9663:
9657:
9656:
9622:
9620:quant-ph/0402146
9602:
9596:
9595:
9561:
9559:quant-ph/0303093
9541:
9535:
9534:
9500:
9480:
9474:
9473:
9471:
9469:
9455:
9449:
9448:
9406:
9388:
9386:
9378:
9372:
9371:
9315:
9306:
9305:
9271:
9251:
9245:
9244:
9196:
9190:
9189:
9179:
9147:
9138:
9137:
9097:
9091:
9090:
9073:(4): 1067â1073.
9058:
9052:
9051:
9025:
9016:
9010:
9009:
9007:
8992:
8983:
8977:
8976:
8974:
8940:
8934:
8933:
8923:
8883:
8877:
8876:
8866:
8826:
8820:
8819:
8793:
8784:
8783:
8757:
8751:
8750:
8718:
8712:
8705:
8699:
8698:
8658:
8652:
8651:
8617:
8593:
8584:
8583:
8552:
8541:
8540:
8504:
8498:
8495:
8489:
8488:
8454:
8430:
8424:
8423:
8413:
8395:
8363:
8354:
8353:
8343:
8311:
8302:
8301:
8283:
8281:10.1038/119890a0
8251:
8242:
8239:
8233:
8232:
8182:
8176:
8175:
8163:
8150:
8144:
8143:
8121:
8112:
8111:
8086:(6): 1049â1070.
8071:
8062:
8060:
8047:Collected papers
8042:
8036:
8035:
8017:
8011:
8010:
8008:
8006:
7997:. Archived from
7966:
7960:
7959:
7931:
7925:
7912:
7906:
7905:
7877:
7871:
7870:
7860:
7858:10.1038/112540a0
7825:
7819:
7818:
7816:
7814:
7804:
7795:
7786:
7785:
7749:
7743:
7742:
7740:
7702:
7696:
7693:
7687:
7686:
7638:
7632:
7631:
7613:
7589:
7583:
7582:
7564:
7549:
7548:
7546:
7544:
7526:
7287:
7279:
7271:
7266:
7254:Shayeghi et al.
7251:
7241:
7225:
7217:
7209:
7187:
7179:
7171:
7149:
7141:
7136:
7129:
7113:Grisenti et al.
7110:
7102:
7094:
7078:
7068:
7052:
7044:
7036:
7018:
7008:
6995:Tonomura et al.
6992:
6984:
6979:
6956:
6948:
6914:
6910:
6906:magnetic moments
6878:Atomic diameters
6856:
6854:
6853:
6848:
6831:
6823:
6744:
6742:
6741:
6736:
6734:
6733:
6724:
6723:
6711:
6710:
6691:
6689:
6688:
6683:
6678:
6653:
6651:
6650:
6645:
6639:
6637:
6636:
6635:
6622:
6621:
6620:
6607:
6602:
6597:
6596:
6595:
6586:
6585:
6575:
6551:
6549:
6548:
6543:
6423:
6421:
6420:
6415:
6401:
6393:
6371:probability flux
6310:of nearby atoms.
6211:Evanescent waves
6201:angular momentum
6188:as described in
6175:
6173:
6172:
6167:
6150:
6123:
6121:
6120:
6115:
6103:
6101:
6100:
6095:
6093:
6089:
6085:
6077:
6076:
6067:
6059:
6042:
6031:
6023:
6015:
6014:
6000:
5998:
5984:
5976:
5968:
5953:
5951:
5950:
5945:
5943:
5941:
5940:
5939:
5926:
5925:
5924:
5919:
5913:
5912:
5902:
5894:
5876:
5874:
5873:
5868:
5866:
5864:
5863:
5862:
5850:
5849:
5836:
5832:
5831:
5821:
5819:
5817:
5816:
5804:
5799:
5798:
5790:
5788:
5783:
5762:
5760:
5759:
5754:
5751:
5746:
5722:
5720:
5719:
5714:
5709:
5701:
5683:
5681:
5680:
5675:
5667:
5656:
5639:
5631:
5611:
5603:
5586:
5547:
5545:
5544:
5539:
5534:
5533:
5528:
5523:
5520:
5519:
5518:
5490:
5489:
5468:
5452:
5450:
5449:
5444:
5442:
5438:
5437:
5436:
5427:
5419:
5409:
5393:
5391:
5390:
5385:
5383:
5379:
5378:
5377:
5368:
5360:
5350:
5332:
5330:
5329:
5324:
5319:
5314:
5310:
5308:
5307:
5298:
5297:
5288:
5279:
5263:
5261:
5260:
5255:
5250:
5239:
5209:
5207:
5206:
5201:
5185:
5183:
5182:
5177:
5161:
5159:
5158:
5153:
5151:
5146:
5141:
5123:
5121:
5120:
5115:
5113:
5106:
5104:
5103:
5101:
5099:
5098:
5089:
5088:
5079:
5071:
5065:
5064:
5063:
5054:
5053:
5043:
5038:
5033:
5032:
5031:
5022:
5021:
5007:
4998:
4994:
4992:
4990:
4989:
4980:
4979:
4970:
4962:
4957:
4955:
4951:
4950:
4937:
4930:
4928:
4924:
4923:
4907:
4896:
4885:
4883:
4882:
4877:
4875:
4871:
4866:
4865:
4850:
4835:
4826:
4825:
4816:
4815:
4800:
4799:
4737:
4735:
4734:
4729:
4718:
4713:
4712:
4711:
4705:
4699:
4679:
4677:
4676:
4671:
4663:
4658:
4653:
4641:
4639:
4638:
4633:
4631:
4619:
4617:
4616:
4611:
4606:
4604:
4599:
4598:
4589:
4584:
4582:
4581:
4576:
4575:
4562:
4561:
4560:
4551:
4550:
4537:
4532:
4530:
4529:
4520:
4519:
4518:
4505:
4500:
4498:
4490:
4485:
4484:
4483:
4477:
4458:
4456:
4455:
4450:
4445:
4443:
4435:
4430:
4428:
4424:
4419:
4413:
4409:
4400:
4395:
4393:
4385:
4380:
4379:
4378:
4372:
4352:
4350:
4349:
4344:
4336:
4331:
4330:
4329:
4319:
4307:
4305:
4304:
4299:
4291:
4286:
4285:
4284:
4274:
4262:
4260:
4259:
4254:
4246:
4241:
4240:
4239:
4229:
4217:
4215:
4214:
4209:
4201:
4196:
4195:
4194:
4184:
4172:
4170:
4169:
4164:
4162:
4161:
4149:
4148:
4147:
4134:
4133:
4132:
4115:
4113:
4112:
4107:
4105:
4104:
4092:
4091:
4090:
4077:
4076:
4075:
4059:This shows that
4058:
4056:
4055:
4050:
4048:
4046:
4045:
4044:
4034:
4033:
4024:
4019:
4018:
4017:
4000:
3998:
3997:
3992:
3990:
3988:
3980:
3975:
3974:
3973:
3944:
3942:
3941:
3936:
3934:
3922:
3920:
3919:
3914:
3912:
3905:
3894:
3890:
3888:
3887:
3886:
3880:
3874:
3873:
3864:
3856:
3852:
3847:
3846:
3845:
3836:
3830:
3821:
3820:
3819:
3813:
3796:
3794:
3793:
3788:
3786:
3781:
3776:
3775:
3763:
3758:
3747:
3746:
3745:
3739:
3724:
3722:
3721:
3716:
3714:
3707:
3702:
3701:
3700:
3691:
3685:
3677:
3673:
3670:
3669:
3659:
3654:
3642:
3641:
3632:
3631:
3622:
3621:
3620:
3619:
3610:
3604:
3596:
3592:
3588:
3586:
3585:
3575:
3570:
3558:
3557:
3548:
3547:
3538:
3532:
3530:
3529:
3517:
3512:
3510:
3509:
3500:
3492:
3487:
3485:
3478:
3473:
3461:
3454:
3439:
3434:
3432:
3431:
3422:
3414:
3405:
3404:
3403:
3397:
3373:
3371:
3370:
3365:
3363:
3358:
3357:
3356:
3347:
3341:
3336:
3335:
3334:
3317:
3315:
3314:
3309:
3303:
3301:
3300:
3295:
3291:
3286:
3285:
3284:
3275:
3274:
3264:
3254:
3253:
3244:
3243:
3234:
3226:
3201:
3199:
3198:
3193:
3178:
3176:
3175:
3170:
3137:
3135:
3134:
3129:
3127:
3125:
3124:
3115:
3114:
3105:
3100:
3099:
3098:
3077:
3075:
3074:
3069:
3067:
3066:
3050:
3048:
3047:
3042:
3036:
3034:
3033:
3032:
3019:
3018:
3017:
3004:
2999:
2994:
2993:
2992:
2983:
2982:
2972:
2964:
2942:
2940:
2939:
2934:
2932:
2930:
2929:
2920:
2916:
2901:
2896:
2895:
2894:
2857:
2843:
2841:
2840:
2835:
2833:
2813:
2809:
2798:
2792:
2777:
2771:
2769:
2768:
2763:
2761:
2757:
2749:
2744:
2742:
2731:
2722:
2718:
2710:
2705:
2703:
2702:
2697:
2692:
2686:
2678:
2669:
2654:
2648:
2639:
2637:
2636:
2631:
2623:
2618:
2613:
2598:
2581:
2579:
2578:
2573:
2558:
2556:
2555:
2550:
2534:
2524:
2522:
2521:
2516:
2514:
2502:
2500:
2499:
2494:
2492:
2480:
2478:
2477:
2472:
2467:
2466:
2453:
2445:
2426:
2364:
2362:
2356:
2351:
2345:
2294:microlithography
2246:Ernest O. Wollan
2239:
2232:
2224:
2219:nuclear reactors
2141:Clinton Davisson
2088:
2076:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
2010:
2008:
2007:
1986:
1969:
1968:
1959:
1945:
1944:
1928:
1926:
1925:
1920:
1908:
1906:
1905:
1900:
1888:
1886:
1885:
1880:
1859:famous biologist
1755:
1753:
1752:
1747:
1742:
1734:
1719:
1709:
1699:
1689:
1687:
1686:
1681:
1679:
1676:
1674:
1673:
1664:
1663:
1654:
1646:
1645:
1637:
1618:
1615:and wave number
1614:
1608:
1606:
1605:
1600:
1598:
1596:
1589:
1574:
1566:
1561:
1559:
1551:
1543:
1538:
1537:
1534:
1518:, with the wave
1517:
1505:
1503:
1502:
1497:
1486:
1483:
1481:
1480:
1471:
1470:
1461:
1453:
1452:
1451:
1450:
1437:
1422:
1397:
1389:. The frequency
1388:
1369:
1360:
1338:
1313:
1298:de Broglie waves
1275:O. W. Richardson
1268:
1258:
1252:
1234:
1224:
1218:
1216:
1215:
1210:
1205:
1197:
1192:
1184:
1173:
1169:
1154:
1152:
1151:
1146:
1144:
1132:
1130:
1129:
1124:
1044:
1042:
1041:
1036:
1031:
1023:
1008:
998:
989:
971:de Broglie waves
968:
967:
964:
963:
960:
957:
954:
951:
948:
938:Louis de Broglie
913:, being half of
896:
889:
882:
523:Superdeterminism
176:Braâket notation
127:
125:
124:
119:
111:
106:
105:
97:
85:
80:
78:
67:
39:
38:
21:
12229:
12228:
12224:
12223:
12222:
12220:
12219:
12218:
12194:
12193:
12192:
12187:
12169:
12155:Wigner's friend
12131:
12122:Quantum gravity
12083:
12069:Quantum sensing
12049:Quantum network
12029:Quantum machine
11999:Quantum imaging
11962:Quantum circuit
11957:Quantum channel
11916:
11862:Quantum biology
11848:
11824:ElitzurâVaidman
11799:DavissonâGermer
11782:
11734:Hidden-variable
11724:de BroglieâBohm
11701:Interpretations
11695:
11659:
11613:
11500:Complementarity
11478:
11435:
11430:
11381:Bowley, Roger.
11377:
11360:
11319:
11308:
11251:L. de Broglie,
11248:
11246:Further reading
11243:
11242:
11236:
11203:
11199:
11138:
11134:
11083:
11076:
11005:
11001:
10991:
10989:
10985:
10928:
10922:
10918:
10880:
10875:
10871:
10840:
10836:
10801:
10797:
10760:Physical Review
10752:
10748:
10697:
10693:
10686:10.1119/1.16104
10658:
10654:
10611:
10607:
10554:
10550:
10481:
10477:
10470:
10456:
10452:
10443:
10441:
10439:
10413:
10409:
10366:
10362:
10352:
10350:
10343:
10337:
10333:
10288:Physics Reports
10280:
10276:
10247:Physics Reports
10239:
10232:
10225:
10211:
10207:
10200:
10186:
10182:
10175:
10161:
10157:
10130:Physical Review
10122:
10118:
10111:
10097:
10093:
10086:
10070:
10063:
10004:
10000:
9989:
9985:
9978:
9964:
9957:
9940:
9936:
9929:
9915:
9911:
9904:
9888:
9884:
9841:
9837:
9766:
9762:
9719:
9715:
9664:
9660:
9603:
9599:
9546:Phys. Rev. Lett
9542:
9538:
9481:
9477:
9467:
9465:
9457:
9456:
9452:
9384:
9382:
9379:
9375:
9329:
9320:J. Voss-Andreae
9316:
9309:
9252:
9248:
9197:
9193:
9156:Physics Reports
9148:
9141:
9100:Estermann, I.;
9098:
9094:
9059:
9055:
9023:
9017:
9013:
9008:on 19 May 2017.
9005:
8990:
8984:
8980:
8941:
8937:
8884:
8880:
8827:
8823:
8808:
8794:
8787:
8772:
8758:
8754:
8719:
8715:
8706:
8702:
8659:
8655:
8594:
8587:
8580:
8553:
8544:
8529:10.1119/1.16104
8505:
8501:
8496:
8492:
8431:
8427:
8364:
8357:
8320:Physical Review
8312:
8305:
8252:
8245:
8240:
8236:
8183:
8179:
8172:
8151:
8147:
8140:
8122:
8115:
8080:Physical Review
8072:
8065:
8057:
8043:
8039:
8032:
8018:
8014:
8004:
8002:
8001:on 29 July 2020
7995:10.1119/1.13743
7967:
7963:
7956:10.1119/1.12844
7932:
7928:
7913:
7909:
7878:
7874:
7826:
7822:
7812:
7810:
7802:
7796:
7789:
7750:
7746:
7716:Physical Review
7703:
7699:
7694:
7690:
7639:
7635:
7590:
7586:
7579:
7565:
7552:
7542:
7540:
7527:
7523:
7518:
7463:
7444:
7397:
7366:
7299:
7282:
7274:
7264:
7262:
7246:
7236:
7228:Brezger et al.
7220:
7212:
7204:
7199:
7182:
7174:
7166:
7161:
7152:Chapman et al.
7144:
7134:
7132:
7124:
7121:
7105:
7097:
7089:
7073:
7063:
7047:
7039:
7031:
7013:
7003:
6987:
6977:
6975:
6951:
6943:
6902:electric charge
6827:
6819:
6805:
6802:
6801:
6794:
6729:
6725:
6719:
6715:
6706:
6702:
6697:
6694:
6693:
6674:
6663:
6660:
6659:
6631:
6627:
6623:
6616:
6612:
6608:
6606:
6591:
6587:
6581:
6577:
6576:
6574:
6557:
6554:
6553:
6528:
6525:
6524:
6473:electron optics
6471:works well for
6450:
6438:atomic orbitals
6397:
6389:
6378:
6375:
6374:
6339:
6333:
6268:. However, its
6260:as a "normal" (
6230:
6224:
6146:
6144:
6141:
6140:
6109:
6106:
6105:
6081:
6072:
6068:
6063:
6055:
6038:
6027:
6019:
6010:
6006:
6005:
6001:
5988:
5983:
5972:
5964:
5962:
5959:
5958:
5935:
5931:
5927:
5920:
5915:
5914:
5908:
5904:
5903:
5901:
5890:
5882:
5879:
5878:
5858:
5854:
5845:
5841:
5837:
5827:
5823:
5822:
5820:
5812:
5808:
5803:
5791:
5784:
5776:
5771:
5770:
5768:
5765:
5764:
5747:
5739:
5733:
5730:
5729:
5705:
5697:
5689:
5686:
5685:
5663:
5652:
5635:
5627:
5607:
5599:
5582:
5574:
5571:
5570:
5567:
5554:
5524:
5522:
5521:
5514:
5513:
5509:
5485:
5484:
5464:
5462:
5459:
5458:
5432:
5431:
5418:
5417:
5413:
5405:
5403:
5400:
5399:
5397:Four-wavevector
5373:
5372:
5359:
5358:
5354:
5346:
5344:
5341:
5340:
5315:
5303:
5299:
5293:
5289:
5287:
5283:
5275:
5273:
5270:
5269:
5246:
5235:
5233:
5230:
5229:
5226:
5220:
5195:
5192:
5191:
5171:
5168:
5167:
5147:
5142:
5137:
5129:
5126:
5125:
5111:
5110:
5094:
5090:
5084:
5080:
5078:
5070:
5066:
5059:
5055:
5049:
5045:
5044:
5042:
5027:
5023:
5017:
5013:
5008:
5006:
4996:
4995:
4985:
4981:
4975:
4971:
4969:
4961:
4946:
4942:
4941:
4936:
4919:
4915:
4911:
4906:
4893:
4891:
4888:
4887:
4873:
4872:
4867:
4861:
4857:
4846:
4836:
4831:
4828:
4827:
4821:
4817:
4811:
4807:
4795:
4791:
4781:
4774:
4772:
4769:
4768:
4758:
4714:
4707:
4706:
4701:
4700:
4695:
4693:
4690:
4689:
4659:
4654:
4649:
4647:
4644:
4643:
4627:
4625:
4622:
4621:
4600:
4594:
4590:
4588:
4577:
4571:
4567:
4563:
4556:
4552:
4546:
4542:
4538:
4536:
4525:
4521:
4514:
4510:
4506:
4504:
4494:
4489:
4479:
4478:
4473:
4472:
4470:
4467:
4466:
4439:
4434:
4420:
4415:
4414:
4405:
4401:
4399:
4389:
4384:
4374:
4373:
4368:
4367:
4365:
4362:
4361:
4332:
4325:
4321:
4320:
4315:
4313:
4310:
4309:
4287:
4280:
4276:
4275:
4270:
4268:
4265:
4264:
4242:
4235:
4231:
4230:
4225:
4223:
4220:
4219:
4197:
4190:
4186:
4185:
4180:
4178:
4175:
4174:
4157:
4153:
4143:
4139:
4138:
4128:
4124:
4123:
4121:
4118:
4117:
4100:
4096:
4086:
4082:
4081:
4071:
4067:
4066:
4064:
4061:
4060:
4040:
4036:
4035:
4029:
4025:
4023:
4013:
4009:
4008:
4006:
4003:
4002:
3984:
3979:
3969:
3965:
3964:
3962:
3959:
3958:
3951:
3930:
3928:
3925:
3924:
3910:
3909:
3901:
3892:
3891:
3882:
3881:
3876:
3875:
3869:
3865:
3863:
3854:
3853:
3841:
3837:
3832:
3831:
3829:
3822:
3815:
3814:
3809:
3808:
3804:
3802:
3799:
3798:
3782:
3777:
3771:
3767:
3759:
3754:
3741:
3740:
3735:
3734:
3732:
3729:
3728:
3712:
3711:
3696:
3692:
3687:
3686:
3684:
3675:
3674:
3665:
3661:
3655:
3650:
3637:
3633:
3627:
3623:
3615:
3611:
3606:
3605:
3603:
3594:
3593:
3581:
3577:
3571:
3566:
3553:
3549:
3543:
3539:
3537:
3533:
3525:
3521:
3516:
3505:
3501:
3493:
3491:
3474:
3469:
3462:
3450:
3440:
3438:
3427:
3423:
3415:
3413:
3406:
3399:
3398:
3393:
3392:
3388:
3386:
3383:
3382:
3379:Energyâmomentum
3352:
3348:
3343:
3342:
3340:
3330:
3326:
3325:
3323:
3320:
3319:
3296:
3280:
3276:
3270:
3266:
3265:
3263:
3259:
3258:
3249:
3245:
3239:
3235:
3233:
3222:
3214:
3211:
3210:
3187:
3184:
3183:
3146:
3143:
3142:
3120:
3116:
3110:
3106:
3104:
3094:
3090:
3089:
3087:
3084:
3083:
3062:
3058:
3056:
3053:
3052:
3028:
3024:
3020:
3013:
3009:
3005:
3003:
2988:
2984:
2978:
2974:
2973:
2971:
2960:
2952:
2949:
2948:
2925:
2921:
2912:
2902:
2900:
2890:
2886:
2885:
2883:
2880:
2879:
2868:
2845:
2831:
2830:
2811:
2810:
2805:
2794:
2789:
2787:
2784:
2783:
2780:Planck constant
2773:
2759:
2758:
2748:
2735:
2730:
2720:
2719:
2709:
2698:
2693:
2688:
2687:
2679:
2677:
2666:
2664:
2661:
2660:
2650:
2644:
2619:
2614:
2609:
2607:
2604:
2603:
2594:
2584:crystallography
2564:
2561:
2560:
2544:
2541:
2540:
2530:
2510:
2508:
2505:
2504:
2488:
2486:
2483:
2482:
2449:
2441:
2437:
2433:
2422:
2414:
2411:
2410:
2399:
2379:
2360:
2358:
2349:
2347:
2340:
2338:
2321:
2282:
2258:crystallography
2242:crystallography
2234:
2230:
2222:
2212:
2206:
2137:
2127:
2096:
2095:
2094:
2093:
2092:
2089:
2081:
2080:
2077:
2055:
2026:
2023:
2022:
2003:
1999:
1982:
1964:
1960:
1958:
1940:
1936:
1934:
1931:
1930:
1914:
1911:
1910:
1894:
1891:
1890:
1874:
1871:
1870:
1828:modulus squared
1804:energy spectrum
1761:
1733:
1725:
1722:
1721:
1715:
1712:Planck constant
1705:
1697:
1669:
1665:
1659:
1655:
1653:
1638:
1636:
1628:
1625:
1624:
1616:
1612:
1585:
1575:
1567:
1565:
1552:
1544:
1542:
1533:
1529:
1527:
1524:
1523:
1522:in free space:
1513:
1476:
1472:
1466:
1462:
1460:
1446:
1442:
1438:
1436:
1428:
1425:
1424:
1418:
1396:
1390:
1384:
1377:
1371:
1368:
1362:
1359:
1353:
1339:
1336:
1309:
1296:Propagation of
1290:
1283:Robert Millikan
1279:Planck constant
1264:
1261:Planck constant
1254:
1248:
1230:
1227:Greek letter nu
1220:
1196:
1183:
1165:
1161:
1159:
1156:
1155:
1140:
1138:
1135:
1134:
1109:
1106:
1105:
1094:Albert Einstein
1070:
1065:
1022:
1014:
1011:
1010:
1004:
1001:Planck constant
994:
985:
945:
941:
900:
871:
870:
869:
634:
626:
625:
571:
570:Advanced topics
563:
562:
561:
513:Hidden-variable
503:de BroglieâBohm
482:
480:Interpretations
472:
471:
470:
440:
432:
431:
430:
388:
380:
379:
378:
345:
301:CHSH inequality
290:
282:
281:
280:
209:Complementarity
203:
195:
194:
193:
161:
132:
107:
96:
95:
81:
71:
66:
58:
55:
54:
35:
32:Mechanical wave
28:
23:
22:
15:
12:
11:
5:
12227:
12217:
12216:
12211:
12206:
12189:
12188:
12186:
12185:
12174:
12171:
12170:
12168:
12167:
12162:
12157:
12152:
12151:
12150:
12139:
12137:
12133:
12132:
12130:
12129:
12124:
12119:
12118:
12117:
12107:
12102:
12100:Casimir effect
12097:
12091:
12089:
12085:
12084:
12082:
12081:
12076:
12071:
12066:
12061:
12059:Quantum optics
12056:
12051:
12046:
12041:
12036:
12031:
12026:
12021:
12016:
12011:
12006:
12001:
11996:
11991:
11986:
11981:
11980:
11979:
11969:
11964:
11959:
11954:
11953:
11952:
11942:
11937:
11932:
11926:
11924:
11918:
11917:
11915:
11914:
11909:
11904:
11899:
11894:
11889:
11884:
11879:
11874:
11869:
11864:
11858:
11856:
11850:
11849:
11847:
11846:
11841:
11836:
11834:Quantum eraser
11831:
11826:
11821:
11816:
11811:
11806:
11801:
11796:
11790:
11788:
11784:
11783:
11781:
11780:
11775:
11770:
11765:
11760:
11755:
11750:
11749:
11748:
11747:
11746:
11731:
11726:
11721:
11716:
11711:
11705:
11703:
11697:
11696:
11694:
11693:
11688:
11683:
11678:
11673:
11667:
11665:
11661:
11660:
11658:
11657:
11652:
11647:
11642:
11637:
11632:
11627:
11621:
11619:
11615:
11614:
11612:
11611:
11610:
11609:
11604:
11594:
11589:
11584:
11579:
11574:
11569:
11564:
11559:
11554:
11549:
11544:
11539:
11534:
11533:
11532:
11527:
11522:
11517:
11507:
11505:Density matrix
11502:
11497:
11492:
11486:
11484:
11480:
11479:
11477:
11476:
11471:
11466:
11461:
11460:
11459:
11449:
11443:
11441:
11437:
11436:
11429:
11428:
11421:
11414:
11406:
11400:
11399:
11376:
11375:External links
11373:
11372:
11371:
11365:
11312:
11306:
11291:
11280:Modern Physics
11276:
11267:
11247:
11244:
11241:
11240:
11234:
11197:
11152:(16): 160403.
11132:
11091:Nature Physics
11074:
10999:
10939:(10): 100404.
10916:
10878:
10869:
10834:
10815:(5): 370â373.
10795:
10746:
10691:
10672:(2): 117â120.
10652:
10605:
10548:
10475:
10468:
10450:
10437:
10407:
10360:
10331:
10274:
10253:(3): 143â210.
10230:
10223:
10205:
10198:
10180:
10173:
10155:
10116:
10109:
10091:
10084:
10061:
9998:
9983:
9976:
9955:
9934:
9927:
9909:
9902:
9882:
9835:
9760:
9713:
9658:
9597:
9552:(16): 160401.
9536:
9491:(5): 297â300.
9475:
9450:
9373:
9327:
9307:
9262:(17): 170401.
9246:
9191:
9139:
9092:
9053:
9011:
8978:
8935:
8878:
8841:(5): 593â600.
8821:
8806:
8785:
8770:
8752:
8733:(17): 55â129.
8713:
8700:
8673:(2): 245â275.
8653:
8585:
8578:
8542:
8499:
8490:
8445:(4): 271â277.
8439:Nature Physics
8425:
8378:(4): 317â322.
8355:
8326:(6): 705â740.
8303:
8243:
8234:
8199:(6): 433â438.
8177:
8170:
8145:
8138:
8113:
8063:
8055:
8037:
8030:
8012:
7961:
7942:(4): 357â362.
7926:
7907:
7872:
7829:de Broglie, L.
7820:
7787:
7744:
7723:(3): 355â388.
7697:
7688:
7633:
7616:Pergamon Press
7584:
7577:
7550:
7520:
7519:
7517:
7514:
7513:
7512:
7507:
7502:
7499:
7494:
7489:
7484:
7479:
7474:
7469:
7462:
7459:
7443:
7440:
7430:atomic mirrors
7396:
7393:
7365:
7362:
7298:
7295:
7292:
7291:
7288:
7280:
7272:
7260:
7256:
7255:
7252:
7244:
7242:
7234:
7230:
7229:
7226:
7218:
7210:
7202:
7197:
7192:
7191:
7188:
7180:
7172:
7164:
7159:
7154:
7153:
7150:
7142:
7130:
7122:
7119:
7115:
7114:
7111:
7103:
7095:
7087:
7083:
7082:
7079:
7071:
7069:
7061:
7057:
7056:
7053:
7045:
7037:
7029:
7023:
7022:
7019:
7011:
7009:
7001:
6997:
6996:
6993:
6985:
6973:
6967:
6963:
6962:
6957:
6949:
6941:
6935:
6931:
6930:
6927:
6924:
6923:kinetic energy
6921:
6918:
6864:ranges over 6
6846:
6843:
6840:
6837:
6834:
6830:
6826:
6822:
6818:
6815:
6812:
6809:
6793:
6790:
6761:density matrix
6732:
6728:
6722:
6718:
6714:
6709:
6705:
6701:
6681:
6677:
6673:
6670:
6667:
6643:
6634:
6630:
6626:
6619:
6615:
6611:
6605:
6600:
6594:
6590:
6584:
6580:
6573:
6570:
6567:
6564:
6561:
6541:
6538:
6535:
6532:
6517:speed of light
6449:
6446:
6413:
6410:
6407:
6404:
6400:
6396:
6392:
6388:
6385:
6382:
6367:group velocity
6332:
6329:
6328:
6327:
6320:
6311:
6298:
6277:
6270:effective mass
6244:In solids, an
6234:quasiparticles
6223:
6220:
6219:
6218:
6208:
6197:
6186:band structure
6165:
6162:
6159:
6156:
6153:
6149:
6113:
6092:
6088:
6084:
6080:
6075:
6071:
6066:
6062:
6058:
6054:
6051:
6048:
6045:
6041:
6037:
6034:
6030:
6026:
6022:
6018:
6013:
6009:
6004:
5997:
5994:
5991:
5987:
5982:
5979:
5975:
5971:
5967:
5938:
5934:
5930:
5923:
5918:
5911:
5907:
5900:
5897:
5893:
5889:
5886:
5861:
5857:
5853:
5848:
5844:
5840:
5835:
5830:
5826:
5815:
5811:
5807:
5802:
5797:
5794:
5787:
5782:
5779:
5775:
5750:
5745:
5742:
5738:
5726:effective mass
5712:
5708:
5704:
5700:
5696:
5693:
5673:
5670:
5666:
5662:
5659:
5655:
5651:
5648:
5645:
5642:
5638:
5634:
5630:
5626:
5623:
5620:
5617:
5614:
5610:
5606:
5602:
5598:
5595:
5592:
5589:
5585:
5581:
5578:
5566:
5563:
5558:free particles
5553:
5550:
5549:
5548:
5537:
5531:
5527:
5517:
5512:
5508:
5505:
5502:
5499:
5496:
5493:
5488:
5483:
5480:
5477:
5474:
5471:
5467:
5453:
5441:
5435:
5430:
5425:
5422:
5416:
5412:
5408:
5394:
5382:
5376:
5371:
5366:
5363:
5357:
5353:
5349:
5322:
5318:
5313:
5306:
5302:
5296:
5292:
5286:
5282:
5278:
5253:
5249:
5245:
5242:
5238:
5222:Main article:
5219:
5216:
5212:speed of light
5199:
5188:Lorentz factor
5175:
5150:
5145:
5140:
5136:
5133:
5109:
5097:
5093:
5087:
5083:
5077:
5074:
5069:
5062:
5058:
5052:
5048:
5041:
5036:
5030:
5026:
5020:
5016:
5011:
5005:
5002:
4999:
4997:
4988:
4984:
4978:
4974:
4968:
4965:
4954:
4949:
4945:
4940:
4934:
4927:
4922:
4918:
4914:
4910:
4903:
4900:
4897:
4895:
4870:
4864:
4860:
4856:
4853:
4849:
4845:
4842:
4839:
4837:
4834:
4830:
4829:
4824:
4820:
4814:
4810:
4806:
4803:
4798:
4794:
4790:
4787:
4784:
4782:
4780:
4777:
4776:
4757:
4754:
4727:
4724:
4721:
4717:
4710:
4704:
4698:
4669:
4666:
4662:
4657:
4652:
4630:
4609:
4603:
4597:
4593:
4587:
4580:
4574:
4570:
4566:
4559:
4555:
4549:
4545:
4541:
4535:
4528:
4524:
4517:
4513:
4509:
4503:
4497:
4493:
4488:
4482:
4476:
4448:
4442:
4438:
4433:
4427:
4423:
4418:
4412:
4408:
4404:
4398:
4392:
4388:
4383:
4377:
4371:
4342:
4339:
4335:
4328:
4324:
4318:
4297:
4294:
4290:
4283:
4279:
4273:
4252:
4249:
4245:
4238:
4234:
4228:
4207:
4204:
4200:
4193:
4189:
4183:
4160:
4156:
4152:
4146:
4142:
4137:
4131:
4127:
4103:
4099:
4095:
4089:
4085:
4080:
4074:
4070:
4043:
4039:
4032:
4028:
4022:
4016:
4012:
3987:
3983:
3978:
3972:
3968:
3955:phase velocity
3950:
3949:Phase velocity
3947:
3933:
3908:
3904:
3900:
3897:
3895:
3893:
3885:
3879:
3872:
3868:
3862:
3859:
3857:
3855:
3850:
3844:
3840:
3835:
3828:
3825:
3823:
3818:
3812:
3807:
3806:
3785:
3780:
3774:
3770:
3766:
3762:
3757:
3753:
3750:
3744:
3738:
3710:
3705:
3699:
3695:
3690:
3683:
3680:
3678:
3676:
3668:
3664:
3658:
3653:
3649:
3645:
3640:
3636:
3630:
3626:
3618:
3614:
3609:
3602:
3599:
3597:
3595:
3591:
3584:
3580:
3574:
3569:
3565:
3561:
3556:
3552:
3546:
3542:
3536:
3528:
3524:
3520:
3515:
3508:
3504:
3499:
3496:
3490:
3484:
3481:
3477:
3472:
3468:
3465:
3460:
3457:
3453:
3449:
3446:
3443:
3437:
3430:
3426:
3421:
3418:
3412:
3409:
3407:
3402:
3396:
3391:
3390:
3381:form instead:
3361:
3355:
3351:
3346:
3339:
3333:
3329:
3307:
3299:
3294:
3289:
3283:
3279:
3273:
3269:
3262:
3257:
3252:
3248:
3242:
3238:
3232:
3229:
3225:
3221:
3218:
3191:
3181:speed of light
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3123:
3119:
3113:
3109:
3103:
3097:
3093:
3065:
3061:
3040:
3031:
3027:
3023:
3016:
3012:
3008:
3002:
2997:
2991:
2987:
2981:
2977:
2970:
2967:
2963:
2959:
2956:
2928:
2924:
2919:
2915:
2911:
2908:
2905:
2899:
2893:
2889:
2876:group velocity
2872:group velocity
2867:
2866:Group velocity
2864:
2829:
2826:
2823:
2820:
2817:
2814:
2812:
2808:
2804:
2801:
2797:
2793:
2791:
2755:
2752:
2747:
2741:
2738:
2734:
2729:
2726:
2723:
2721:
2716:
2713:
2708:
2701:
2696:
2691:
2685:
2682:
2676:
2673:
2670:
2668:
2629:
2626:
2622:
2617:
2612:
2571:
2568:
2548:
2513:
2491:
2470:
2465:
2462:
2459:
2456:
2452:
2448:
2444:
2440:
2436:
2432:
2429:
2425:
2421:
2418:
2398:
2395:
2378:
2375:
2336:
2325:macromolecules
2320:
2317:
2281:
2278:
2250:Arthur Compton
2217:, produced in
2205:
2202:
2196:. Just as the
2126:
2123:
2090:
2083:
2082:
2078:
2071:
2070:
2069:
2068:
2067:
2054:
2051:
2030:
2006:
2002:
1998:
1995:
1992:
1989:
1985:
1981:
1978:
1975:
1972:
1967:
1963:
1957:
1954:
1951:
1948:
1943:
1939:
1918:
1898:
1878:
1824:complex number
1760:
1757:
1745:
1740:
1737:
1732:
1729:
1710:, through the
1672:
1668:
1662:
1658:
1652:
1649:
1644:
1641:
1635:
1632:
1595:
1592:
1588:
1584:
1581:
1578:
1573:
1570:
1564:
1558:
1555:
1550:
1547:
1541:
1532:
1520:group velocity
1495:
1492:
1489:
1479:
1475:
1469:
1465:
1459:
1456:
1449:
1445:
1441:
1435:
1432:
1394:
1382:
1375:
1366:
1357:
1334:
1289:
1286:
1247:of the light,
1208:
1203:
1200:
1195:
1190:
1187:
1182:
1179:
1176:
1172:
1168:
1164:
1143:
1122:
1119:
1116:
1113:
1069:
1066:
1064:
1061:
1034:
1029:
1026:
1021:
1018:
902:
901:
899:
898:
891:
884:
876:
873:
872:
868:
867:
862:
857:
852:
847:
842:
837:
832:
827:
822:
817:
812:
807:
802:
797:
792:
787:
782:
777:
772:
767:
762:
757:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
636:
635:
632:
631:
628:
627:
624:
623:
618:
613:
608:
606:Density matrix
603:
598:
593:
588:
583:
578:
572:
569:
568:
565:
564:
560:
559:
554:
549:
544:
539:
534:
529:
528:
527:
526:
525:
510:
505:
500:
495:
490:
484:
483:
478:
477:
474:
473:
469:
468:
463:
458:
453:
448:
442:
441:
438:
437:
434:
433:
429:
428:
423:
418:
413:
408:
403:
397:
396:
395:
389:
386:
385:
382:
381:
377:
376:
371:
366:
360:
359:
358:
357:
356:
354:Delayed-choice
349:Quantum eraser
344:
343:
338:
333:
328:
323:
318:
313:
308:
303:
298:
292:
291:
288:
287:
284:
283:
279:
278:
277:
276:
266:
261:
256:
251:
246:
241:
239:Quantum number
236:
231:
226:
221:
216:
211:
205:
204:
201:
200:
197:
196:
192:
191:
186:
180:
179:
178:
173:
168:
162:
159:
158:
155:
154:
153:
152:
147:
142:
134:
133:
128:
117:
114:
110:
103:
100:
94:
91:
88:
84:
77:
74:
70:
65:
62:
51:
50:
44:
43:
26:
9:
6:
4:
3:
2:
12226:
12215:
12212:
12210:
12207:
12205:
12202:
12201:
12199:
12184:
12176:
12175:
12172:
12166:
12163:
12161:
12158:
12156:
12153:
12149:
12146:
12145:
12144:
12141:
12140:
12138:
12134:
12128:
12125:
12123:
12120:
12116:
12113:
12112:
12111:
12108:
12106:
12103:
12101:
12098:
12096:
12093:
12092:
12090:
12086:
12080:
12077:
12075:
12072:
12070:
12067:
12065:
12062:
12060:
12057:
12055:
12052:
12050:
12047:
12045:
12042:
12040:
12037:
12035:
12032:
12030:
12027:
12025:
12022:
12020:
12019:Quantum logic
12017:
12015:
12012:
12010:
12007:
12005:
12002:
12000:
11997:
11995:
11992:
11990:
11987:
11985:
11982:
11978:
11975:
11974:
11973:
11970:
11968:
11965:
11963:
11960:
11958:
11955:
11951:
11948:
11947:
11946:
11943:
11941:
11938:
11936:
11933:
11931:
11928:
11927:
11925:
11923:
11919:
11913:
11910:
11908:
11905:
11903:
11900:
11898:
11895:
11893:
11890:
11888:
11885:
11883:
11880:
11878:
11875:
11873:
11872:Quantum chaos
11870:
11868:
11865:
11863:
11860:
11859:
11857:
11855:
11851:
11845:
11842:
11840:
11839:SternâGerlach
11837:
11835:
11832:
11830:
11827:
11825:
11822:
11820:
11817:
11815:
11812:
11810:
11807:
11805:
11802:
11800:
11797:
11795:
11792:
11791:
11789:
11785:
11779:
11776:
11774:
11773:Transactional
11771:
11769:
11766:
11764:
11763:Quantum logic
11761:
11759:
11756:
11754:
11751:
11745:
11742:
11741:
11740:
11737:
11736:
11735:
11732:
11730:
11727:
11725:
11722:
11720:
11717:
11715:
11712:
11710:
11707:
11706:
11704:
11702:
11698:
11692:
11689:
11687:
11684:
11682:
11679:
11677:
11674:
11672:
11669:
11668:
11666:
11662:
11656:
11653:
11651:
11648:
11646:
11643:
11641:
11638:
11636:
11633:
11631:
11628:
11626:
11623:
11622:
11620:
11616:
11608:
11605:
11603:
11600:
11599:
11598:
11597:Wave function
11595:
11593:
11590:
11588:
11585:
11583:
11580:
11578:
11575:
11573:
11572:Superposition
11570:
11568:
11567:Quantum state
11565:
11563:
11560:
11558:
11555:
11553:
11550:
11548:
11545:
11543:
11540:
11538:
11535:
11531:
11528:
11526:
11523:
11521:
11520:Excited state
11518:
11516:
11513:
11512:
11511:
11508:
11506:
11503:
11501:
11498:
11496:
11493:
11491:
11488:
11487:
11485:
11481:
11475:
11472:
11470:
11467:
11465:
11462:
11458:
11455:
11454:
11453:
11450:
11448:
11445:
11444:
11442:
11438:
11434:
11427:
11422:
11420:
11415:
11413:
11408:
11407:
11404:
11396:
11392:
11388:
11387:Sixty Symbols
11384:
11379:
11378:
11369:
11366:
11359:
11355:
11351:
11347:
11343:
11338:
11333:
11329:
11325:
11318:
11313:
11309:
11303:
11299:
11298:
11292:
11289:
11288:0-7167-4345-0
11285:
11281:
11277:
11275:
11273:
11268:
11266:
11263:, 22 (1925).
11262:
11258:
11254:
11250:
11249:
11237:
11231:
11226:
11221:
11217:
11213:
11209:
11201:
11193:
11189:
11185:
11181:
11177:
11173:
11169:
11165:
11160:
11155:
11151:
11147:
11143:
11136:
11128:
11124:
11120:
11116:
11112:
11108:
11104:
11100:
11096:
11092:
11088:
11081:
11079:
11070:
11066:
11061:
11056:
11052:
11048:
11044:
11040:
11036:
11032:
11027:
11022:
11018:
11014:
11010:
11003:
10984:
10980:
10976:
10972:
10968:
10964:
10960:
10956:
10952:
10947:
10942:
10938:
10934:
10927:
10920:
10912:
10908:
10904:
10900:
10896:
10892:
10888:
10884:
10873:
10865:
10861:
10857:
10853:
10850:(3): 033608.
10849:
10845:
10838:
10830:
10826:
10822:
10818:
10814:
10810:
10806:
10799:
10790:
10785:
10781:
10777:
10773:
10769:
10765:
10761:
10757:
10750:
10742:
10738:
10734:
10730:
10726:
10722:
10718:
10714:
10711:(1): 95â125.
10710:
10707:(in German).
10706:
10702:
10695:
10687:
10683:
10679:
10675:
10671:
10667:
10663:
10656:
10648:
10644:
10640:
10636:
10632:
10628:
10624:
10620:
10616:
10609:
10601:
10597:
10593:
10589:
10585:
10581:
10576:
10571:
10567:
10563:
10559:
10552:
10544:
10540:
10536:
10532:
10528:
10524:
10520:
10516:
10512:
10508:
10503:
10498:
10494:
10490:
10486:
10479:
10471:
10465:
10461:
10454:
10440:
10434:
10430:
10426:
10422:
10418:
10411:
10403:
10399:
10395:
10391:
10387:
10383:
10379:
10375:
10371:
10364:
10349:
10342:
10335:
10327:
10323:
10319:
10315:
10311:
10307:
10302:
10297:
10293:
10289:
10285:
10278:
10269:
10264:
10260:
10256:
10252:
10248:
10244:
10243:"Atom optics"
10237:
10235:
10226:
10220:
10216:
10209:
10201:
10195:
10191:
10184:
10176:
10170:
10166:
10159:
10151:
10147:
10143:
10139:
10135:
10131:
10127:
10120:
10112:
10106:
10102:
10095:
10087:
10081:
10077:
10076:
10068:
10066:
10057:
10053:
10049:
10045:
10041:
10037:
10033:
10029:
10025:
10021:
10017:
10013:
10009:
10002:
9994:
9987:
9979:
9973:
9969:
9962:
9960:
9952:
9951:0-415-27761-2
9948:
9944:
9938:
9930:
9924:
9920:
9913:
9905:
9899:
9895:
9894:
9886:
9878:
9874:
9870:
9866:
9862:
9858:
9854:
9850:
9846:
9839:
9831:
9827:
9822:
9817:
9813:
9809:
9805:
9801:
9797:
9793:
9788:
9783:
9779:
9775:
9771:
9764:
9756:
9752:
9748:
9744:
9740:
9736:
9732:
9728:
9724:
9717:
9709:
9705:
9701:
9697:
9693:
9689:
9685:
9681:
9677:
9673:
9669:
9662:
9654:
9650:
9646:
9642:
9638:
9634:
9630:
9626:
9621:
9616:
9612:
9608:
9601:
9593:
9589:
9585:
9581:
9577:
9573:
9569:
9565:
9560:
9555:
9551:
9547:
9540:
9532:
9528:
9524:
9520:
9516:
9512:
9508:
9504:
9499:
9494:
9490:
9486:
9479:
9464:
9460:
9454:
9446:
9442:
9438:
9434:
9430:
9426:
9422:
9418:
9414:
9410:
9405:
9400:
9396:
9392:
9377:
9369:
9365:
9361:
9357:
9353:
9352:10.1038/44348
9349:
9345:
9341:
9337:
9333:
9325:
9321:
9314:
9312:
9303:
9299:
9295:
9291:
9287:
9283:
9279:
9275:
9270:
9265:
9261:
9257:
9250:
9242:
9238:
9234:
9230:
9226:
9222:
9218:
9214:
9210:
9206:
9202:
9195:
9187:
9183:
9178:
9173:
9169:
9165:
9161:
9157:
9153:
9152:"Atom optics"
9146:
9144:
9135:
9131:
9127:
9123:
9119:
9115:
9111:
9107:
9103:
9096:
9088:
9084:
9080:
9076:
9072:
9068:
9064:
9057:
9049:
9045:
9041:
9037:
9033:
9029:
9022:
9015:
9004:
9000:
8996:
8989:
8982:
8973:
8968:
8964:
8960:
8956:
8952:
8951:
8950:Physics Today
8946:
8939:
8931:
8927:
8922:
8917:
8913:
8909:
8905:
8901:
8897:
8893:
8889:
8882:
8874:
8870:
8865:
8860:
8856:
8852:
8848:
8844:
8840:
8836:
8832:
8825:
8817:
8813:
8809:
8803:
8799:
8792:
8790:
8781:
8777:
8773:
8771:0-444-82218-6
8767:
8763:
8756:
8748:
8744:
8740:
8736:
8732:
8729:(in German).
8728:
8724:
8717:
8710:
8707:Mauro Dardo,
8704:
8696:
8692:
8688:
8684:
8680:
8676:
8672:
8668:
8664:
8657:
8649:
8645:
8641:
8637:
8633:
8629:
8625:
8621:
8616:
8611:
8607:
8603:
8599:
8592:
8590:
8581:
8575:
8571:
8567:
8566:
8561:
8557:
8551:
8549:
8547:
8538:
8534:
8530:
8526:
8522:
8518:
8514:
8510:
8503:
8494:
8486:
8482:
8478:
8474:
8470:
8466:
8462:
8458:
8453:
8448:
8444:
8440:
8436:
8429:
8421:
8417:
8412:
8407:
8403:
8399:
8394:
8389:
8385:
8381:
8377:
8373:
8369:
8362:
8360:
8351:
8347:
8342:
8337:
8333:
8329:
8325:
8321:
8317:
8310:
8308:
8299:
8295:
8291:
8287:
8282:
8277:
8273:
8269:
8266:(3007): 890.
8265:
8261:
8257:
8250:
8248:
8238:
8230:
8226:
8222:
8218:
8214:
8210:
8206:
8202:
8198:
8194:
8193:
8188:
8181:
8173:
8167:
8162:
8161:
8155:
8149:
8141:
8135:
8131:
8127:
8120:
8118:
8109:
8105:
8101:
8097:
8093:
8089:
8085:
8081:
8077:
8070:
8068:
8058:
8052:
8048:
8041:
8033:
8027:
8023:
8016:
8000:
7996:
7992:
7988:
7984:
7980:
7976:
7972:
7965:
7957:
7953:
7949:
7945:
7941:
7937:
7930:
7923:
7921:
7918:
7911:
7903:
7899:
7895:
7891:
7887:
7883:
7882:Physics Today
7876:
7868:
7864:
7859:
7854:
7850:
7846:
7843:(2815): 540.
7842:
7838:
7834:
7830:
7824:
7808:
7801:
7794:
7792:
7783:
7779:
7775:
7771:
7767:
7763:
7759:
7755:
7748:
7739:
7734:
7730:
7726:
7722:
7718:
7717:
7712:
7710:
7701:
7692:
7684:
7680:
7676:
7672:
7668:
7664:
7660:
7656:
7652:
7648:
7644:
7637:
7629:
7625:
7621:
7617:
7612:
7611:
7605:
7600:
7597:
7593:
7588:
7580:
7578:0-486-26126-3
7574:
7570:
7563:
7561:
7559:
7557:
7555:
7539:
7538:Physics World
7535:
7531:
7525:
7521:
7511:
7508:
7506:
7503:
7500:
7498:
7495:
7493:
7490:
7488:
7485:
7483:
7480:
7478:
7475:
7473:
7470:
7468:
7465:
7464:
7458:
7454:
7452:
7448:
7439:
7437:
7433:
7431:
7427:
7423:
7421:
7417:
7415:
7411:
7407:
7405:
7402:, similar to
7401:
7395:Neutral atoms
7392:
7390:
7386:
7384:
7380:
7378:
7374:
7370:
7361:
7359:
7355:
7351:
7349:
7345:
7341:
7339:
7335:
7333:
7329:
7324:
7322:
7318:
7314:
7310:
7305:
7303:
7289:
7286:
7281:
7278:
7273:
7270:
7261:
7258:
7257:
7253:
7250:
7245:
7243:
7240:
7235:
7232:
7231:
7227:
7224:
7219:
7216:
7211:
7208:
7203:
7201:
7194:
7193:
7190:Arndt et al.
7189:
7186:
7181:
7178:
7173:
7170:
7165:
7163:
7156:
7155:
7151:
7148:
7143:
7140:
7131:
7128:
7123:
7117:
7116:
7112:
7109:
7104:
7101:
7096:
7093:
7088:
7085:
7084:
7080:
7077:
7072:
7070:
7067:
7062:
7059:
7058:
7054:
7051:
7046:
7043:
7038:
7035:
7030:
7028:
7025:
7024:
7020:
7017:
7012:
7010:
7007:
7002:
6999:
6998:
6994:
6991:
6986:
6983:
6974:
6972:
6968:
6965:
6964:
6961:
6958:
6955:
6950:
6947:
6942:
6940:
6936:
6933:
6932:
6928:
6925:
6922:
6919:
6916:
6915:
6909:
6907:
6903:
6899:
6895:
6891:
6887:
6886:laser cooling
6883:
6879:
6875:
6871:
6867:
6863:
6858:
6841:
6838:
6835:
6832:
6824:
6816:
6810:
6807:
6799:
6789:
6787:
6783:
6779:
6777:
6772:
6768:
6766:
6762:
6758:
6754:
6750:
6748:
6730:
6726:
6720:
6716:
6712:
6707:
6703:
6699:
6679:
6675:
6671:
6668:
6665:
6657:
6641:
6632:
6628:
6624:
6617:
6613:
6609:
6603:
6592:
6588:
6582:
6578:
6571:
6565:
6559:
6539:
6536:
6533:
6530:
6522:
6518:
6514:
6510:
6508:
6504:
6499:
6498:visible light
6495:
6491:
6487:
6484:
6482:
6478:
6477:atomic optics
6474:
6470:
6466:
6462:
6458:
6454:
6445:
6443:
6439:
6435:
6431:
6427:
6408:
6405:
6402:
6394:
6383:
6380:
6372:
6368:
6360:
6356:
6352:
6348:
6347:Newton's laws
6343:
6338:
6337:Standing wave
6325:
6321:
6318:
6317:
6312:
6309:
6305:
6304:
6299:
6296:
6295:semiconductor
6292:
6288:
6284:
6283:
6278:
6275:
6271:
6267:
6263:
6259:
6255:
6251:
6247:
6243:
6242:
6241:
6239:
6235:
6229:
6216:
6212:
6209:
6206:
6202:
6198:
6195:
6191:
6187:
6183:
6180:
6179:
6178:
6157:
6154:
6151:
6138:
6134:
6131:
6127:
6090:
6073:
6069:
6049:
6046:
6032:
6011:
6007:
6002:
5995:
5992:
5989:
5985:
5980:
5957:
5936:
5932:
5928:
5921:
5909:
5905:
5898:
5884:
5859:
5855:
5846:
5842:
5833:
5828:
5813:
5809:
5805:
5800:
5795:
5792:
5785:
5780:
5777:
5773:
5748:
5743:
5740:
5736:
5727:
5702:
5691:
5664:
5660:
5646:
5643:
5640:
5632:
5624:
5618:
5615:
5604:
5593:
5590:
5576:
5562:
5559:
5510:
5506:
5503:
5497:
5494:
5481:
5478:
5472:
5469:
5457:
5456:Four-velocity
5454:
5439:
5428:
5423:
5420:
5414:
5410:
5398:
5395:
5380:
5369:
5364:
5361:
5355:
5351:
5339:
5338:Four-momentum
5336:
5335:
5334:
5320:
5311:
5304:
5300:
5294:
5290:
5284:
5280:
5267:
5251:
5240:
5225:
5215:
5213:
5197:
5189:
5173:
5165:
5134:
5131:
5107:
5095:
5091:
5085:
5081:
5075:
5072:
5067:
5060:
5056:
5050:
5046:
5039:
5034:
5028:
5024:
5018:
5014:
5009:
5003:
5000:
4986:
4982:
4976:
4972:
4966:
4963:
4952:
4947:
4943:
4938:
4932:
4925:
4920:
4916:
4912:
4908:
4901:
4898:
4862:
4858:
4854:
4851:
4843:
4840:
4838:
4822:
4818:
4812:
4808:
4804:
4801:
4796:
4792:
4788:
4785:
4783:
4778:
4767:
4763:
4753:
4751:
4750:
4745:
4741:
4725:
4722:
4719:
4687:
4683:
4667:
4664:
4620:The variable
4607:
4595:
4591:
4585:
4572:
4568:
4564:
4557:
4553:
4547:
4543:
4539:
4533:
4522:
4515:
4511:
4507:
4501:
4491:
4486:
4464:
4459:
4446:
4436:
4431:
4421:
4406:
4402:
4396:
4386:
4381:
4358:
4356:
4340:
4337:
4295:
4292:
4250:
4247:
4205:
4202:
4158:
4154:
4150:
4135:
4101:
4097:
4093:
4078:
4030:
4026:
4020:
3981:
3976:
3956:
3946:
3906:
3898:
3896:
3870:
3866:
3860:
3858:
3848:
3842:
3838:
3826:
3824:
3778:
3772:
3768:
3764:
3755:
3751:
3748:
3725:
3708:
3703:
3697:
3693:
3681:
3679:
3666:
3662:
3656:
3651:
3647:
3643:
3638:
3634:
3628:
3624:
3616:
3612:
3600:
3598:
3589:
3582:
3578:
3572:
3567:
3563:
3559:
3554:
3550:
3544:
3540:
3534:
3513:
3497:
3488:
3475:
3451:
3447:
3435:
3419:
3410:
3408:
3380:
3375:
3359:
3353:
3349:
3337:
3305:
3297:
3292:
3281:
3277:
3271:
3267:
3260:
3255:
3250:
3246:
3240:
3236:
3230:
3216:
3208:
3203:
3189:
3182:
3166:
3163:
3160:
3154:
3148:
3141:
3121:
3117:
3107:
3101:
3081:
3063:
3059:
3038:
3029:
3025:
3021:
3014:
3010:
3006:
3000:
2989:
2985:
2979:
2975:
2968:
2954:
2946:
2906:
2897:
2877:
2873:
2863:
2861:
2856:
2852:
2848:
2827:
2824:
2818:
2815:
2799:
2781:
2776:
2753:
2750:
2745:
2739:
2736:
2732:
2727:
2724:
2714:
2711:
2706:
2683:
2680:
2674:
2671:
2658:
2657:free particle
2653:
2647:
2643:
2627:
2624:
2602:
2597:
2593:
2589:
2585:
2569:
2566:
2546:
2538:
2533:
2528:
2468:
2463:
2460:
2457:
2454:
2446:
2438:
2434:
2430:
2416:
2408:
2407:free particle
2404:
2394:
2392:
2388:
2384:
2374:
2372:
2366:
2355:
2344:
2334:
2330:
2326:
2316:
2314:
2310:
2306:
2305:laser cooling
2301:
2299:
2298:laser cooling
2295:
2291:
2287:
2277:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2238:
2231:0.025 eV
2228:
2220:
2216:
2211:
2201:
2199:
2195:
2190:
2188:
2184:
2180:
2176:
2172:
2168:
2165:predicted by
2164:
2160:
2157:
2153:
2149:
2146:
2145:Lester Germer
2142:
2136:
2132:
2119:
2114:
2110:
2108:
2104:
2099:
2087:
2075:
2066:
2064:
2060:
2050:
2048:
2044:
2028:
2004:
1996:
1993:
1990:
1987:
1983:
1979:
1976:
1970:
1965:
1961:
1955:
1952:
1949:
1946:
1941:
1937:
1916:
1896:
1876:
1868:
1864:
1860:
1856:
1847:
1843:
1841:
1837:
1833:
1829:
1825:
1821:
1817:
1813:
1809:
1805:
1801:
1797:
1792:
1790:
1786:
1782:
1778:
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1770:
1766:
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1713:
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1703:
1695:
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1609:
1590:
1586:
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1576:
1571:
1568:
1562:
1556:
1548:
1539:
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1511:
1506:
1493:
1490:
1487:
1477:
1473:
1467:
1463:
1457:
1454:
1447:
1443:
1439:
1433:
1430:
1421:
1416:
1412:
1408:
1403:
1401:
1393:
1387:
1381:
1374:
1365:
1356:
1352:
1348:
1344:
1333:
1325:
1321:
1317:
1312:
1307:
1303:
1299:
1294:
1285:
1284:
1280:
1276:
1272:
1271:K. T. Compton
1267:
1262:
1257:
1251:
1246:
1242:
1239:) denote the
1238:
1233:
1228:
1223:
1206:
1201:
1198:
1193:
1188:
1185:
1180:
1177:
1174:
1170:
1162:
1120:
1117:
1114:
1111:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1060:
1058:
1054:
1050:
1045:
1032:
1027:
1024:
1019:
1016:
1007:
1002:
997:
993:
988:
983:
979:
974:
972:
966:
939:
934:
932:
928:
924:
920:
916:
912:
908:
897:
892:
890:
885:
883:
878:
877:
875:
874:
866:
863:
861:
858:
856:
853:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
826:
823:
821:
818:
816:
813:
811:
808:
806:
803:
801:
798:
796:
793:
791:
788:
786:
783:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
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673:
671:
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663:
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658:
656:
653:
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646:
643:
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638:
637:
630:
629:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
597:
596:Quantum chaos
594:
592:
589:
587:
584:
582:
579:
577:
574:
573:
567:
566:
558:
555:
553:
552:Transactional
550:
548:
545:
543:
542:Quantum logic
540:
538:
535:
533:
530:
524:
521:
520:
519:
516:
515:
514:
511:
509:
506:
504:
501:
499:
496:
494:
491:
489:
486:
485:
481:
476:
475:
467:
464:
462:
459:
457:
454:
452:
449:
447:
444:
443:
436:
435:
427:
424:
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419:
417:
414:
412:
409:
407:
404:
402:
399:
398:
394:
391:
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384:
383:
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372:
370:
367:
365:
362:
361:
355:
352:
351:
350:
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346:
342:
339:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
293:
286:
285:
275:
272:
271:
270:
269:Wave function
267:
265:
262:
260:
257:
255:
252:
250:
249:Superposition
247:
245:
242:
240:
237:
235:
232:
230:
227:
225:
222:
220:
217:
215:
212:
210:
207:
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199:
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187:
185:
182:
181:
177:
174:
172:
169:
167:
164:
163:
157:
156:
151:
148:
146:
143:
141:
138:
137:
136:
135:
131:
98:
92:
75:
72:
68:
60:
53:
52:
49:
46:
45:
41:
40:
37:
33:
19:
11902:Quantum mind
11814:FranckâHertz
11676:KleinâGordon
11625:Formulations
11618:Formulations
11547:Interference
11537:Entanglement
11515:Ground state
11510:Energy level
11483:Fundamentals
11447:Introduction
11386:
11358:the original
11327:
11323:
11296:
11279:
11271:
11260:
11256:
11252:
11207:
11200:
11149:
11145:
11135:
11094:
11090:
11016:
11012:
11002:
10990:. Retrieved
10936:
10932:
10919:
10886:
10882:
10881:molecules".
10872:
10847:
10843:
10837:
10812:
10808:
10798:
10763:
10759:
10749:
10708:
10704:
10694:
10669:
10665:
10655:
10622:
10618:
10608:
10565:
10561:
10551:
10527:1721.1/52372
10492:
10488:
10478:
10459:
10453:
10442:, retrieved
10420:
10410:
10380:(1): 74â93.
10377:
10373:
10363:
10351:. Retrieved
10347:
10334:
10291:
10287:
10277:
10250:
10246:
10214:
10208:
10189:
10183:
10164:
10158:
10136:(1): 49â71.
10133:
10129:
10119:
10100:
10094:
10074:
10015:
10011:
10001:
9992:
9986:
9967:
9942:
9937:
9918:
9912:
9892:
9885:
9852:
9848:
9838:
9777:
9773:
9763:
9730:
9726:
9716:
9675:
9671:
9661:
9610:
9606:
9600:
9549:
9545:
9539:
9488:
9484:
9478:
9468:25 September
9466:. Retrieved
9462:
9453:
9394:
9390:
9376:
9335:
9331:
9324:A. Zeilinger
9259:
9255:
9249:
9208:
9204:
9194:
9159:
9155:
9109:
9105:
9095:
9070:
9066:
9056:
9031:
9027:
9014:
9003:the original
8994:
8981:
8954:
8948:
8938:
8898:(1): 37â44.
8895:
8891:
8881:
8838:
8834:
8824:
8797:
8764:. Elsevier.
8761:
8755:
8730:
8726:
8716:
8708:
8703:
8670:
8666:
8656:
8605:
8601:
8563:
8512:
8508:
8502:
8493:
8442:
8438:
8428:
8375:
8371:
8323:
8319:
8263:
8259:
8237:
8196:
8190:
8180:
8159:
8148:
8125:
8083:
8079:
8046:
8040:
8021:
8015:
8003:. Retrieved
7999:the original
7978:
7974:
7964:
7939:
7935:
7929:
7919:
7917:Am. J. Phys.
7916:
7910:
7888:(2): 38â45.
7885:
7881:
7875:
7840:
7836:
7823:
7811:. Retrieved
7806:
7757:
7753:
7747:
7720:
7714:
7708:
7700:
7691:
7650:
7646:
7636:
7609:
7604:ter Haar, D.
7598:
7595:
7592:Einstein, A.
7587:
7568:
7541:. Retrieved
7537:
7530:Kragh, Helge
7524:
7482:Faraday wave
7455:
7445:
7434:
7424:
7418:
7408:
7398:
7387:
7381:
7371:complements
7367:
7352:
7342:
7336:
7325:
7321:low energies
7313:transmission
7306:
7300:
7290:Fein et al.
7060:Sodium atom
6859:
6795:
6781:
6780:
6770:
6769:
6752:
6751:
6512:
6511:
6489:
6488:
6485:
6451:
6364:
6355:wavefunction
6314:
6308:polarization
6301:
6291:valence band
6280:
6245:
6231:
5568:
5555:
5227:
5218:Four-vectors
4759:
4747:
4744:superluminal
4739:
4685:
4463:relativistic
4460:
4359:
4355:superluminal
3952:
3726:
3376:
3204:
3079:
2869:
2854:
2850:
2846:
2774:
2651:
2645:
2595:
2531:
2400:
2380:
2373:mechanisms.
2367:
2339:velocity as
2322:
2303:Advances in
2302:
2283:
2260:. Joined by
2213:
2191:
2189:approaches.
2150:slow-moving
2138:
2116:Matter wave
2100:
2097:
2056:
1855:C. G. Darwin
1852:
1820:wavefunction
1793:
1762:
1716:
1706:
1691:
1610:
1514:
1507:
1419:
1411:total energy
1405:To find the
1404:
1391:
1385:
1379:
1372:
1363:
1354:
1341:
1330:
1310:
1297:
1265:
1255:
1249:
1231:
1221:
1071:
1063:Introduction
1046:
1005:
999:through the
995:
986:
977:
975:
970:
935:
907:Matter waves
906:
905:
451:KleinâGordon
387:Formulations
224:Energy level
219:Entanglement
202:Fundamentals
189:Interference
140:Introduction
36:
12160:EPR paradox
11940:Quantum bus
11809:Double-slit
11787:Experiments
11753:Many-worlds
11691:Schrödinger
11655:Phase space
11645:Schrödinger
11635:Interaction
11592:Uncertainty
11562:Nonlocality
11557:Measurement
11552:Decoherence
11542:Hamiltonian
11391:Brady Haran
11019:(1): 1447.
9112:(1â2): 95.
9102:Stern, Otto
8957:(11): 120.
8154:Jammer, Max
8005:16 December
7975:Am. J. Phys
7936:Am. J. Phys
7922:: 1047â1055
7813:25 February
7760:(1): 5â15.
7618:. pp.
7410:Atom optics
7098:0.065
7040:0.025
6786:coincidence
6765:collimation
6513:Dispersion.
6494:diffraction
6465:wave optics
6453:Schrödinger
6324:Cooper pair
6199:Waves with
6194:diffraction
5224:Four-vector
2527:wave vector
2371:decoherence
2175:diffraction
2156:crystalline
1861:) explored
1765:Peter Debye
1351:proper mass
1324:wave packet
1281:in 1916 by
1235:(lowercase
1225:(lowercase
840:von Neumann
825:Schrödinger
601:EPR paradox
532:Many-worlds
466:Schrödinger
421:Schrödinger
416:Phase-space
406:Interaction
311:Double-slit
289:Experiments
264:Uncertainty
234:Nonlocality
229:Measurement
214:Decoherence
184:Hamiltonian
12198:Categories
12088:Extensions
11922:Technology
11768:Relational
11719:Copenhagen
11630:Heisenberg
11577:Tunnelling
11440:Background
11257:Ann. Phys.
11026:1910.14538
10575:2203.07257
10301:1911.06282
9787:2204.01625
7516:References
7472:Bohr model
7358:holography
7346:leverages
7237:1860
6929:reference
6926:wavelength
6335:See also:
6226:See also:
6182:Bloch wave
4461:Using the
4173:, as both
3140:dispersion
2592:wavelength
2588:wavevector
2387:rho mesons
2333:fullerenes
2290:Otto Stern
2229:to around
2227:thermalize
2223:1 MeV
2208:See also:
2179:Hans Bethe
2043:Heisenberg
1781:light rays
1694:wavelength
1407:wavelength
1343:De Broglie
1337:de Broglie
1322:. Bottom:
1320:plane wave
1245:wavelength
1086:Max Planck
1068:Background
1051:, neutral
982:wavelength
931:diffracted
835:Sommerfeld
750:Heisenberg
745:Gutzwiller
685:de Broglie
633:Scientists
547:Relational
498:Copenhagen
401:Heisenberg
259:Tunnelling
160:Background
11794:Bell test
11664:Equations
11490:Born rule
11337:0712.3703
11159:1205.3447
11127:256703296
11119:1745-2481
11051:2041-1723
10929:(reprint)
10741:121757478
10733:0044-3328
10647:0031-9007
10535:0034-6861
10502:0712.3703
10444:13 August
10402:0034-6861
10326:208006050
10040:1476-4687
9953:, p. 192.
9877:2210-3147
9855:(1): 16.
9812:2375-2548
9755:2210-3147
9733:(1): 16.
9700:0036-8075
9498:1402.1867
9429:1463-9084
9404:1310.8343
9387: amu
9269:0805.3519
9233:1079-7114
9186:0370-1573
9134:121757478
8912:0108-7673
8855:0959-440X
8816:656767858
8780:247191522
8695:171025814
8687:0007-0874
8640:1367-2630
8615:1210.6243
8537:0002-9505
8477:1745-2473
8452:1410.0270
8402:0027-8424
8350:0031-899X
8290:0028-0836
8229:219513834
8221:0002-9505
8108:0031-899X
7782:122931010
7675:0036-8075
7442:Molecules
7297:Electrons
7247:360
7213:0.2
7205:841
7200:fullerene
7175:0.2
7167:720
7162:fullerene
7145:459
7048:181
6952:167
6874:picometre
6839:ω
6833:−
6825:⋅
6811:
6757:coherence
6753:Coherence
6704:ω
6700:ℏ
6680:λ
6610:ℏ
6599:ℏ
6572:≈
6560:ω
6531:ω
6406:ω
6403:−
6395:⋅
6384:
6164:∇
6161:ℏ
6155:−
6112:∇
6074:∗
6070:ψ
6065:∇
6050:ψ
6047:−
6033:ψ
6029:∇
6012:∗
6008:ψ
5986:ℏ
5937:∗
5906:ℏ
5852:∂
5839:∂
5825:∂
5810:ℏ
5793:−
5786:∗
5763:given by
5749:∗
5669:ℏ
5641:−
5633:⋅
5619:
5577:ψ
5530:^
5498:γ
5473:γ
5421:ω
5291:ω
5264:which is
5244:ℏ
5174:γ
5076:−
5010:γ
4967:−
4913:γ
4899:λ
4855:γ
4805:γ
4565:γ
4540:γ
4426:ℏ
4411:ℏ
4387:ω
4136:⋅
4079:⋅
3982:ω
3523:∂
3519:∂
3503:∂
3495:∂
3480:ℏ
3464:∂
3456:ℏ
3442:∂
3425:∂
3420:ω
3417:∂
3360:ω
3288:ℏ
3217:ω
3179:, is the
3149:ω
3108:ℏ
3007:ℏ
2996:ℏ
2969:≈
2955:ω
2923:∂
2907:ω
2904:∂
2825:ω
2822:ℏ
2803:ℏ
2740:π
2733:ω
2684:π
2672:λ
2642:frequency
2570:π
2461:ω
2455:−
2447:⋅
2417:ψ
2341:2.5
2319:Molecules
2235:180
2152:electrons
2125:Electrons
2029:σ
1994:σ
1991:π
1962:σ
1956:±
1929:would be
1840:Born rule
1816:rest mass
1728:λ
1651:−
1591:λ
1572:ν
1554:∂
1549:ω
1546:∂
1540:≡
1494:ν
1458:−
1347:electrons
1241:frequency
1202:λ
1121:ν
1057:molecules
1017:λ
927:electrons
921:exhibits
865:Zeilinger
710:Ehrenfest
439:Equations
116:⟩
113:Ψ
102:^
90:⟩
87:Ψ
64:ℏ
12183:Category
11977:Timeline
11729:Ensemble
11709:Bayesian
11602:Collapse
11474:Glossary
11457:Timeline
11393:for the
11259:(Paris)
11192:12088376
11184:23679586
11069:32193414
10992:30 April
10983:Archived
10979:19793304
10971:11909334
10911:10058598
10600:36260664
10543:28009912
10294:: 1â57.
10048:20844532
9830:36598973
9708:17772840
9645:14973478
9592:31057272
9584:12731960
9523:22447163
9463:phys.org
9437:23900710
9360:18494170
9302:19465661
9294:19518764
9241:10043591
8930:23250059
8873:18656544
8562:(1999).
8560:Wolf, E.
8556:Born, M.
8485:56438353
8420:16587341
8156:(1974).
7831:(1923).
7683:17792421
7628:66029628
7606:(1967).
7461:See also
7364:Neutrons
7317:scanning
7315:, using
7283:53
7275:17
7125:23
7106:56
7074:20
7064:23
7014:50
6980:10
6966:Electron
6944:54
6934:Electron
6890:material
6483:do not.
6475:and for
6455:applied
6250:electron
6203:such as
6133:operator
6130:gradient
5164:velocity
4688:, i.e.,
2601:momentum
2403:velocity
2363: Da
2215:Neutrons
2204:Neutrons
1832:Max Born
1808:hydrogen
1702:momentum
1335:â
992:momentum
790:Millikan
715:Einstein
700:Davisson
655:Blackett
640:Aharonov
508:Ensemble
488:Bayesian
393:Overview
274:Collapse
254:Symmetry
145:Glossary
12136:Related
12115:History
11854:Science
11686:Rydberg
11452:History
11342:Bibcode
11212:Bibcode
11164:Bibcode
11099:Bibcode
11060:7081299
11031:Bibcode
10951:Bibcode
10891:Bibcode
10852:Bibcode
10817:Bibcode
10768:Bibcode
10713:Bibcode
10674:Bibcode
10627:Bibcode
10580:Bibcode
10507:Bibcode
10382:Bibcode
10353:24 June
10306:Bibcode
10255:Bibcode
10138:Bibcode
10056:2970408
10020:Bibcode
9857:Bibcode
9821:9812379
9792:Bibcode
9735:Bibcode
9680:Bibcode
9672:Science
9653:3482856
9625:Bibcode
9564:Bibcode
9531:5918772
9503:Bibcode
9445:3944699
9409:Bibcode
9368:4424892
9340:Bibcode
9274:Bibcode
9213:Bibcode
9164:Bibcode
9114:Bibcode
9106:Z. Phys
9075:Bibcode
9036:Bibcode
8959:Bibcode
8921:3526866
8864:2586829
8735:Bibcode
8620:Bibcode
8517:Bibcode
8457:Bibcode
8411:1085484
8380:Bibcode
8328:Bibcode
8298:4122313
8268:Bibcode
8201:Bibcode
8088:Bibcode
7983:Bibcode
7944:Bibcode
7890:Bibcode
7867:4082518
7845:Bibcode
7762:Bibcode
7725:Bibcode
7655:Bibcode
7647:Science
7620:167â183
7414:mirrors
7221:2
7183:5
7090:4
7032:1
7027:Neutron
7004:4
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3541:p
3535:(
3527:p
3514:=
3507:p
3498:E
3489:=
3483:)
3476:/
3471:p
3467:(
3459:)
3452:/
3448:E
3445:(
3436:=
3429:k
3411:=
3401:g
3395:v
3354:2
3350:c
3345:k
3338:=
3332:g
3328:v
3306:.
3298:2
3293:)
3282:2
3278:c
3272:0
3268:m
3261:(
3256:+
3251:2
3247:c
3241:2
3237:k
3231:=
3228:)
3224:k
3220:(
3190:c
3167:k
3164:c
3161:=
3158:)
3155:k
3152:(
3122:0
3118:m
3112:k
3102:=
3096:g
3092:v
3064:0
3060:m
3039:.
3030:0
3026:m
3022:2
3015:2
3011:k
3001:+
2990:2
2986:c
2980:0
2976:m
2966:)
2962:k
2958:(
2927:k
2918:)
2914:k
2910:(
2898:=
2892:g
2888:v
2855:Ï
2851:h
2847:ħ
2828:,
2819:=
2816:E
2807:k
2800:=
2796:p
2775:h
2754:h
2751:E
2746:=
2737:2
2728:=
2725:f
2715:p
2712:h
2707:=
2700:|
2695:k
2690:|
2681:2
2675:=
2652:E
2646:f
2628:p
2625:=
2621:|
2616:p
2611:|
2596:λ
2567:2
2547:t
2532:Ï
2512:k
2490:r
2469:,
2464:t
2458:i
2451:r
2443:k
2439:i
2435:e
2431:=
2428:)
2424:r
2420:(
2005:2
2001:)
1997:m
1988:2
1984:/
1980:t
1977:h
1974:(
1971:+
1966:2
1953:t
1950:v
1947:+
1942:0
1938:x
1917:v
1897:x
1877:t
1744:.
1739:p
1736:h
1731:=
1717:h
1707:p
1698:λ
1671:2
1667:c
1661:2
1657:v
1648:1
1643:v
1640:m
1634:=
1631:p
1617:k
1613:Ï
1594:)
1587:/
1583:1
1580:(
1577:d
1569:d
1563:=
1557:k
1535:g
1531:v
1515:v
1491:h
1488:=
1478:2
1474:c
1468:2
1464:v
1455:1
1448:2
1444:c
1440:m
1434:=
1431:E
1395:0
1392:Μ
1386:c
1383:0
1380:m
1376:0
1367:0
1364:Μ
1358:0
1355:m
1326:.
1311:x
1266:f
1256:h
1250:c
1232:λ
1222:Μ
1207:,
1199:h
1194:=
1189:c
1186:E
1181:=
1178:p
1175:=
1171:|
1167:p
1163:|
1142:p
1118:h
1115:=
1112:E
1033:.
1028:p
1025:h
1020:=
1006:h
996:p
987:λ
965:/
959:r
956:b
953:Ë
950:É
947:d
944:/
940:(
895:e
888:t
881:v
109:|
99:H
93:=
83:|
76:t
73:d
69:d
61:i
34:.
20:)
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