2441:), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian. To solve for the transition amplitude, one needs to average over (integrate over) all the vacuum modes, since one must consider the probabilities that the emitted photon occupies various parts of phase space equally. The "spontaneously" emitted photon has infinite different modes to propagate into, thus the probability of the atom re-absorbing the photon and returning to the original state is negligible, making the atomic decay practically irreversible. Such irreversible time evolution of the atom-vacuum system is responsible for the apparent spontaneous decay of an excited atom. If one were to keep track of all the vacuum modes, the combined atom-vacuum system would undergo unitary time evolution, making the decay process reversible.
1281:
1877:, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a
2470:. The rate of emission depends on two factors: an 'atomic part', which describes the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as
2678:
2395:, which are the wavenumber and polarization of the emitted photon, respectively. As mentioned above, the emitted photon has a chance to be emitted with different wavenumbers and polarizations, and the resulting wavefunction is a superposition of these possibilities. To calculate the probability of the atom at the ground state (
1908:
Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the different
3284:
only occurs under emission of light. In this case one speaks of full radiative decay and this means that the quantum efficiency is 100%. Besides radiative decay, which occurs under the emission of light, there is a second decay mechanism; nonradiative decay. To determine the total decay rate
1900:
of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it.
919:. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms (or molecules) are excited by some means other than heating, the spontaneous emission is called
990:
interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave the nonintuitive prediction that the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field. These experiments gave rise to
2457:
in 1930 in a landmark paper. The
Weisskopf-Wigner calculation remains the standard approach to spontaneous radiation emission in atomic and molecular physics. Dirac had also developed the same calculation a couple of years prior to the paper by Wigner and Weisskopf.
2108:
2480:
2805:
3537:), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition. This final transition is the transition over the
3520:
1917:
In the presence of electromagnetic vacuum modes, the combined atom-vacuum system is explained by the superposition of the wavefunctions of the excited state atom with no photon and the ground state atom with a single emitted photon:
3194:
stands for position operator. (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to
4178:
A. F. van Driel, G. Allan, C. Delerue, P. Lodahl, W. L. Vos and D. Vanmaekelbergh, Frequency-dependent spontaneous emission rate from CdSe and CdTe nanocrystals: Influence of dark states, Physical Review
Letters, 95, 236804
1913:
offered by the electromagnetic field is infinitely larger than that offered by the atom. This infinite degree of freedom for the emission of the photon results in the apparent irreversible decay, i.e., spontaneous emission.
3366:
108:
1610:
3533:. Nonradiative relaxation occurs when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance,
1433:
3117:
1863:
3632:
2673:{\displaystyle \Gamma _{\text{rad}}(\omega )={\frac {\omega ^{3}n|\mu _{12}|^{2}}{3\pi \varepsilon _{0}\hbar c^{3}}}={\frac {4\alpha \omega ^{3}n|\langle 1|\mathbf {r} |2\rangle |^{2}}{3c^{2}}}}
1924:
3040:
2686:
3150:
1159:
1692:
3450:
3423:
3396:
3310:
1770:
1218:
2218:
2910:
2353:
3458:
1743:
1104:
3192:
2145:
3254:
3220:
2307:
2439:
1797:
2879:
2257:
970:
by several decades. Later, after the formal discovery of quantum mechanics in 1926, the rate of spontaneous emission was accurately described from first principles by
2982:
2828:
1078:
2934:
1490:
1463:
1182:
954:
and is fundamentally a quantum process. According to the
American Physical Society, the first person to correctly predict the phenomenon of spontaneous emission was
1055:
1028:
2174:
1642:
1333:
1262:
2280:
3282:
3170:
2958:
2848:
2393:
2373:
1712:
1662:
1353:
1304:
1238:
4041:
Sharafiev, Aleksei; Juan, Mathieu L.; Gargiulo, Oscar; Zanner, Maximilian; Wögerer, Stephanie; GarcĂa-Ripoll, Juan JosĂ©; Kirchmair, Gerhard (2021-06-10).
4201:
1896:, which can mix with the excited stationary states of the atom. As a result of this interaction, the "stationary state" of the atom is no longer a true
1881:, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as
4181:
Phys. Rev. Lett. 95, 236804 (2005) - Frequency-Dependent
Spontaneous Emission Rate from CdSe and CdTe Nanocrystals: Influence of Dark States (aps.org)
3318:
38:
3452:
the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved:
1504:
374:
875:
3887:
Jaynes, E. T.; Cummings, F. W. (1963). "Comparison of quantum and semiclassical radiation theories with application to the beam maser".
3553:. Specifically, since electrons decay slowly from them, they can be deliberately piled up in this state without too much loss and then
935:. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called
4122:
1361:
923:. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced (
3045:
126:
1805:
1492:
is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the
582:
355:
3852:
Welton, Theodore Allen (1948). "Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field".
538:
3951:
2103:{\displaystyle |\psi (t)\rangle =a(t)e^{-i\omega _{0}t}|e;0\rangle +\sum _{k,s}b_{ks}(t)e^{-i\omega _{k}t}|g;1_{ks}\rangle }
461:
1272:
of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for
2991:
4235:
3694:
3961:
3934:
3684:
2800:{\displaystyle {\frac {|\mu _{12}|^{2}}{\pi \varepsilon _{0}\hbar c}}=4\alpha |\langle 1|\mathbf {r} |2\rangle |^{2}}
121:
3545:
generally cannot support large vibrations without destroying bonds (which generally doesn't happen for relaxation).
978:. Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the
210:
3924:
868:
317:
297:
165:
3122:
1112:
2442:
992:
557:
287:
1667:
4198:
3768:
597:
335:
235:
3428:
2445:
is one such system where the vacuum modes are modified resulting in the reversible decay process, see also
533:
528:
499:
350:
131:
4169:
B. Henderson and G. Imbusch, Optical
Spectroscopy of Inorganic Solids (Clarendon Press, Oxford, UK, 1989).
3546:
3401:
3374:
3288:
1748:
4225:
1191:
567:
312:
302:
3515:{\displaystyle {\text{QE}}={\frac {\Gamma _{\text{rad}}}{\Gamma _{\text{nrad}}+\Gamma _{\text{rad}}}}.}
2179:
951:
931:
etc.). If the excitation is effected by the absorption of radiation the spontaneous emission is called
861:
513:
484:
4180:
4017:
2888:
2312:
3611:
983:
518:
479:
432:
407:
330:
190:
4024:, Advances in Atomic, Molecular, and Optical Physics, vol. 59, Academic Press, pp. 175–221
1721:
4240:
2882:
1086:
602:
3175:
2116:
3606:
3232:
3198:
2985:
2937:
2285:
1882:
1185:
975:
963:
771:
489:
397:
2398:
2176:
are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude,
1874:
1775:
447:
345:
111:
4230:
4016:
Berman, Paul R.; Ford, George W. (2010-01-01), Arimondo, E.; Berman, P. R.; Lin, C. C. (eds.),
3913:
R. Loudon, The
Quantum Theory of Light, 3rd ed. (Oxford University Press Inc., New York, 2001).
2857:
2467:
2223:
776:
494:
322:
292:
255:
1057:, releasing the difference in energy between the two states as a photon. The photon will have
402:
2967:
2813:
1494:
1063:
959:
245:
230:
3662:
Straumann, Norbert (23 Mar 2017). "Einstein in 1916: "On the
Quantum Theory of Radiation"".
3256:-frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.
2919:
1468:
1441:
1167:
4043:"Visualizing the emission of a single photon with frequency and time resolved spectroscopy"
3861:
3818:
3783:
3733:
3586:
3229:
can be tuned continuously by changing their size. This property has been used to check the
1878:
1276:. An energy level diagram illustrating the process of spontaneous emission is shown below:
1033:
1006:
562:
474:
200:
157:
4210:
2150:
1718:. After one lifetime, the number of excited states decays to 36.8% of its original value (
1618:
1309:
1247:
806:
8:
3571:
3566:
3554:
2913:
2851:
2449:. The theory of the spontaneous emission under the QED framework was first calculated by
1902:
1273:
1030:, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy
944:
924:
661:
469:
387:
215:
195:
147:
4083:
3865:
3822:
3787:
3737:
2262:
4134:
4054:
4042:
3834:
3663:
3541:
in semiconductors. Large nonradiative transitions do not occur frequently because the
3267:
3155:
2943:
2833:
2378:
2358:
1697:
1647:
1338:
1289:
1223:
904:
382:
307:
240:
152:
974:
in his quantum theory of radiation, the precursor to the theory which he later called
4152:
4103:
3998:
3957:
3930:
3690:
3576:
3542:
3264:
In the rate-equation above, it is assumed that decay of the number of excited states
1909:
directions in which the photon can be emitted. Equivalently, one might say that the
1715:
1058:
979:
928:
892:
816:
791:
731:
726:
626:
592:
572:
170:
29:
4144:
4095:
4064:
3990:
3896:
3869:
3838:
3826:
3791:
3764:
3741:
3601:
2450:
987:
821:
811:
801:
701:
681:
666:
636:
504:
392:
4018:"Chapter 5 - Spontaneous Decay, Unitarity, and the Weisskopf–Wigner Approximation"
4205:
2446:
1241:
955:
936:
846:
716:
696:
442:
282:
2466:
The rate of spontaneous emission (i.e., the radiative rate) can be described by
3979:"Berechnung der natĂĽrlichen Linienbreite auf Grund der Diracschen Lichttheorie"
3809:
Weisskopf, Viktor (1935). "Probleme der neueren
Quantentheorie des Elektrons".
3596:
3361:{\displaystyle \Gamma _{\text{tot}}=\Gamma _{\text{rad}}+\Gamma _{\text{nrad}}}
2961:
995:(CQED), the study of effects of mirrors and cavities on radiative corrections.
967:
781:
741:
721:
691:
671:
621:
587:
437:
427:
220:
4068:
962:. Einstein's quantum theory of radiation anticipated ideas later expressed in
958:
in a series of papers starting in 1916, culminating in what is now called the
4219:
4156:
4107:
4002:
3591:
3581:
3534:
2454:
1269:
908:
841:
836:
766:
736:
706:
577:
523:
250:
225:
103:{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }
3873:
2474:, the rate of spontaneous emission in the dipole approximation is given by:
1694:
is the radiative decay rate of the transition. The number of excited states
3900:
3746:
3721:
3259:
3226:
1889:
932:
920:
912:
831:
826:
761:
746:
711:
205:
1910:
1873:
Spontaneous transitions were not explainable within the framework of the
1605:{\displaystyle N(t)=N(0)e^{-A_{21}t}=N(0)e^{-\Gamma _{\!{\text{rad}}}t},}
943:
start via spontaneous emission, then during continuous operation work by
796:
751:
686:
641:
4099:
4199:
Detail calculation of the
Spontaneous Emission: Weisskopf-Wigner Theory
3994:
3830:
3549:
form a very important feature that is exploited in the construction of
2471:
1897:
1893:
971:
786:
756:
676:
651:
646:
631:
4148:
3956:. Cambridge UK: Cambridge University Press. p. §1.5.2 pp. 22–23.
3633:"This Month in Physics History: Einstein Predicts Stimulated Emission"
1280:
3978:
1888:
In quantum electrodynamics (or QED), the electromagnetic field has a
1265:
277:
3795:
4059:
3668:
896:
656:
4139:
1003:
If a light source ('the atom') is in an excited state with energy
3538:
3225:
In contrast with atoms, which have a discrete emission spectrum,
3949:
3722:"The Quantum Theory of the Emission and Absorption of Radiation"
1428:{\displaystyle {\frac {\partial N(t)}{\partial t}}=-A_{21}N(t),}
3526:
1081:
916:
3112:{\displaystyle |\mu _{12}|=|\langle 1|\mathbf {d} |2\rangle |}
1498:, and has units s. The above equation can be solved to give:
3550:
1644:
is the initial number of light sources in the excited state,
940:
4082:
Stenholm, Stig
Torsten; Suominen, Kalle-Antti (1998-04-27).
1858:{\displaystyle A_{21}=\Gamma _{21}={\frac {1}{\tau _{21}}}.}
1286:
If the number of light sources in the excited state at time
4040:
3926:
Spontaneous emission and laser oscillation in microcavities
3530:
900:
3922:
3686:
Einstein and the Quantum: The Quest of the Valiant Swabian
3042:
stands for the definition of the transition dipole moment
1465:
is the rate of spontaneous emission. In the rate-equation
915:) and emits a quantized amount of energy in the form of a
3260:
Radiative and nonradiative decay: the quantum efficiency
2259:
are the ground state atom with a single photon (of mode
4123:"P. A. M. Dirac and the discovery of quantum mechanics"
939:. Figurines that glow in the dark are phosphorescent.
3525:
In nonradiative relaxation, the energy is released as
4084:"Weisskopf-Wigner decay of excited oscillator states"
3461:
3431:
3404:
3377:
3321:
3312:, radiative and nonradiative rates should be summed:
3291:
3270:
3235:
3201:
3178:
3158:
3125:
3048:
2994:
2970:
2946:
2922:
2891:
2860:
2836:
2816:
2689:
2483:
2401:
2381:
2361:
2315:
2288:
2265:
2226:
2182:
2153:
2119:
1927:
1808:
1778:
1751:
1724:
1700:
1670:
1650:
1621:
1507:
1471:
1444:
1364:
1341:
1312:
1292:
1250:
1226:
1194:
1170:
1115:
1089:
1066:
1036:
1009:
41:
4022:Advances In Atomic, Molecular, and Optical Physics
3514:
3444:
3417:
3390:
3360:
3304:
3276:
3248:
3214:
3186:
3164:
3144:
3111:
3035:{\displaystyle |\langle 1|\mathbf {r} |2\rangle |}
3034:
2976:
2952:
2928:
2904:
2873:
2842:
2822:
2799:
2672:
2433:
2387:
2367:
2347:
2301:
2274:
2251:
2212:
2168:
2139:
2102:
1857:
1791:
1764:
1737:
1706:
1686:
1656:
1636:
1604:
1484:
1457:
1427:
1347:
1327:
1298:
1256:
1232:
1212:
1176:
1153:
1098:
1072:
1049:
1022:
102:
2355:is the frequency of the photon. The sum is over
1676:
1586:
4217:
4081:
3950:Marian O Scully & M. Suhail Zubairy (1997).
1901:Spontaneous emission in free space depends upon
1714:thus decays exponentially with time, similar to
3976:
2461:
3886:
3689:(First ed.). Princeton University Press.
2282:) wavefunction and its probability amplitude,
869:
3101:
3077:
3024:
3000:
2782:
2758:
2637:
2613:
2207:
2134:
2097:
2003:
1945:
950:Spontaneous emission cannot be explained by
97:
71:
982:of the electromagnetic field. In 1963, the
4015:
3923:Hiroyuki Yokoyama & Ujihara K (1995).
3145:{\displaystyle \mathbf {d} =q\mathbf {r} }
1772:is inversely proportional to the lifetime
1154:{\displaystyle E_{2}-E_{1}=\hbar \omega ,}
876:
862:
4138:
4120:
4058:
3916:
3808:
3759:
3757:
3745:
3667:
3661:
986:was developed describing the system of a
4211:Britney's Guide to Semiconductor Physics
3977:Weisskopf, V.; Wigner, E. (1930-01-01).
3655:
3630:
3557:can be used to boost an optical signal.
2309:is the atomic transition frequency, and
1687:{\displaystyle \Gamma _{\!{\text{rad}}}}
3943:
3763:
3715:
3713:
4218:
3851:
3754:
3719:
3682:
3445:{\displaystyle \Gamma _{\text{nrad}}}
3929:. Boca Raton: CRC Press. p. 6.
3710:
3683:Stone, A. Douglas (6 October 2013).
3676:
3418:{\displaystyle \Gamma _{\text{rad}}}
3391:{\displaystyle \Gamma _{\text{tot}}}
3305:{\displaystyle \Gamma _{\text{tot}}}
1765:{\displaystyle \Gamma _{\text{rad}}}
3720:Dirac, Paul Adrien Maurice (1927).
1213:{\displaystyle \hbar \omega =h\nu }
911:to a lower energy state (e.g., its
13:
3624:
3497:
3484:
3473:
3433:
3406:
3379:
3349:
3336:
3323:
3293:
2485:
1823:
1753:
1672:
1582:
1385:
1368:
408:Sum-over-histories (path integral)
94:
68:
24:Part of a series of articles about
14:
4252:
4192:
2735:
2567:
2213:{\displaystyle |g;1_{ks}\rangle }
1745:-time). The radiative decay rate
1142:
45:
3425:is the radiative decay rate and
3180:
3138:
3127:
3089:
3012:
2905:{\displaystyle \varepsilon _{0}}
2770:
2625:
2348:{\displaystyle \omega _{k}=c|k|}
1279:
952:classical electromagnetic theory
4172:
4163:
4114:
4075:
4034:
4009:
3970:
3631:Tretkoff, Ernie (August 2005).
998:
16:Quantum mechanical state change
4121:Gottfried, Kurt (2011-03-01).
3907:
3880:
3845:
3802:
3637:American Physical Society News
3105:
3094:
3084:
3073:
3065:
3050:
3028:
3017:
3007:
2996:
2787:
2775:
2765:
2754:
2710:
2694:
2642:
2630:
2620:
2609:
2539:
2523:
2500:
2494:
2443:Cavity quantum electrodynamics
2421:
2416:
2410:
2403:
2341:
2333:
2246:
2240:
2184:
2163:
2157:
2121:
2074:
2044:
2038:
1990:
1960:
1954:
1942:
1936:
1929:
1738:{\displaystyle {\frac {1}{e}}}
1631:
1625:
1570:
1564:
1532:
1526:
1517:
1511:
1419:
1413:
1380:
1374:
1322:
1316:
993:cavity quantum electrodynamics
558:Relativistic quantum mechanics
90:
83:
64:
1:
3617:
3172:is the elementary charge and
1099:{\displaystyle \hbar \omega }
598:Quantum statistical mechanics
3187:{\displaystyle \mathbf {r} }
2462:Rate of spontaneous emission
2140:{\displaystyle |e;0\rangle }
7:
4127:American Journal of Physics
3769:"Why spontaneous emission?"
3560:
3249:{\displaystyle \omega ^{3}}
3215:{\displaystyle \omega ^{3}}
3119:for dipole moment operator
2830:is the emission frequency,
2302:{\displaystyle \omega _{0}}
568:Quantum information science
10:
4257:
2434:{\displaystyle |b(t)|^{2}}
1792:{\displaystyle \tau _{21}}
891:is the process in which a
4236:Electromagnetic radiation
4069:10.22331/q-2021-06-10-474
3529:, more commonly known as
3398:is the total decay rate,
2874:{\displaystyle \mu _{12}}
2252:{\displaystyle b_{ks}(t)}
1868:
2883:transition dipole moment
603:Quantum machine learning
356:Wheeler's delayed-choice
3889:Proceedings of the IEEE
3874:10.1103/PhysRev.74.1157
3607:Vacuum Rabi oscillation
2986:fine-structure constant
2977:{\displaystyle \alpha }
2938:reduced Planck constant
2823:{\displaystyle \omega }
1883:quantum electrodynamics
1186:reduced Planck constant
1073:{\displaystyle \omega }
976:quantum electrodynamics
964:quantum electrodynamics
313:Leggett–Garg inequality
3983:Zeitschrift fĂĽr Physik
3901:10.1109/PROC.1963.1664
3747:10.1098/rspa.1927.0039
3516:
3446:
3419:
3392:
3362:
3306:
3278:
3250:
3216:
3188:
3166:
3146:
3113:
3036:
2978:
2954:
2930:
2929:{\displaystyle \hbar }
2906:
2875:
2844:
2824:
2801:
2674:
2435:
2389:
2369:
2349:
2303:
2276:
2253:
2214:
2170:
2141:
2104:
1859:
1793:
1766:
1739:
1708:
1688:
1658:
1638:
1606:
1495:Einstein A coefficient
1486:
1485:{\displaystyle A_{21}}
1459:
1458:{\displaystyle A_{21}}
1429:
1349:
1329:
1300:
1258:
1234:
1214:
1178:
1177:{\displaystyle \hbar }
1155:
1100:
1074:
1051:
1024:
960:Einstein A Coefficient
104:
3612:Jaynes–Cummings model
3517:
3447:
3420:
3393:
3363:
3307:
3279:
3251:
3217:
3189:
3167:
3147:
3114:
3037:
2979:
2955:
2931:
2907:
2876:
2845:
2825:
2802:
2675:
2436:
2390:
2370:
2350:
2304:
2277:
2254:
2215:
2171:
2142:
2105:
1860:
1794:
1767:
1740:
1709:
1689:
1659:
1639:
1607:
1487:
1460:
1430:
1350:
1330:
1301:
1259:
1235:
1215:
1179:
1156:
1101:
1075:
1052:
1050:{\displaystyle E_{1}}
1025:
1023:{\displaystyle E_{2}}
984:Jaynes–Cummings model
298:Elitzur–Vaidman
288:Davisson–Germer
105:
3587:Atomic spectral line
3459:
3429:
3402:
3375:
3319:
3289:
3268:
3233:
3199:
3176:
3156:
3123:
3046:
2992:
2968:
2944:
2920:
2889:
2858:
2834:
2814:
2687:
2481:
2399:
2379:
2359:
2313:
2286:
2263:
2224:
2180:
2169:{\displaystyle a(t)}
2151:
2117:
1925:
1879:quantum field theory
1875:Schrödinger equation
1806:
1776:
1749:
1722:
1698:
1668:
1648:
1637:{\displaystyle N(0)}
1619:
1505:
1469:
1442:
1362:
1339:
1335:, the rate at which
1328:{\displaystyle N(t)}
1310:
1290:
1257:{\displaystyle \nu }
1248:
1224:
1192:
1168:
1113:
1087:
1064:
1034:
1007:
909:excited energy state
889:Spontaneous emission
563:Quantum field theory
475:Consistent histories
112:Schrödinger equation
39:
4100:10.1364/OE.2.000378
3866:1948PhRv...74.1157W
3823:1935NW.....23..631W
3811:Naturwissenschaften
3788:1984AmJPh..52..340M
3738:1927RSPSA.114..243D
3572:Stimulated emission
3567:Absorption (optics)
3555:stimulated emission
2914:vacuum permittivity
2852:index of refraction
2468:Fermi's golden rule
1903:vacuum fluctuations
1274:stimulated emission
945:stimulated emission
925:electroluminescence
907:) transits from an
351:Stern–Gerlach
148:Classical mechanics
4226:Physical phenomena
4204:2016-03-04 at the
3995:10.1007/BF01336768
3831:10.1007/BF01492012
3547:Meta-stable states
3512:
3442:
3415:
3388:
3358:
3302:
3274:
3246:
3212:
3184:
3162:
3142:
3109:
3032:
2974:
2950:
2926:
2902:
2871:
2840:
2820:
2797:
2670:
2431:
2385:
2365:
2345:
2299:
2275:{\displaystyle ks}
2272:
2249:
2210:
2166:
2137:
2100:
2024:
1855:
1789:
1762:
1735:
1704:
1684:
1654:
1634:
1602:
1482:
1455:
1425:
1345:
1325:
1296:
1254:
1230:
1210:
1174:
1151:
1096:
1070:
1047:
1020:
905:subatomic particle
895:system (such as a
893:quantum mechanical
539:Von Neumann–Wigner
519:Objective-collapse
318:Mach–Zehnder
308:Leggett inequality
303:Franck–Hertz
153:Old quantum theory
100:
4149:10.1119/1.3536639
3765:Milonni, Peter W.
3577:Emission spectrum
3543:crystal structure
3507:
3503:
3490:
3479:
3465:
3439:
3412:
3385:
3355:
3342:
3329:
3299:
3277:{\displaystyle N}
3165:{\displaystyle q}
2988:. The expression
2953:{\displaystyle c}
2843:{\displaystyle n}
2742:
2668:
2581:
2491:
2388:{\displaystyle s}
2368:{\displaystyle k}
2009:
1850:
1759:
1733:
1716:radioactive decay
1707:{\displaystyle N}
1680:
1657:{\displaystyle t}
1590:
1392:
1348:{\displaystyle N}
1299:{\displaystyle t}
1233:{\displaystyle h}
1059:angular frequency
980:zero-point energy
929:chemiluminescence
886:
885:
593:Scattering theory
573:Quantum computing
346:Schrödinger's cat
278:Bell's inequality
86:
61:
30:Quantum mechanics
4248:
4183:
4176:
4170:
4167:
4161:
4160:
4142:
4118:
4112:
4111:
4079:
4073:
4072:
4062:
4038:
4032:
4031:
4030:
4029:
4013:
4007:
4006:
3974:
3968:
3967:
3947:
3941:
3940:
3920:
3914:
3911:
3905:
3904:
3884:
3878:
3877:
3849:
3843:
3842:
3806:
3800:
3799:
3773:
3761:
3752:
3751:
3749:
3732:(767): 243–265.
3717:
3708:
3707:
3705:
3703:
3680:
3674:
3673:
3671:
3659:
3653:
3652:
3650:
3648:
3628:
3602:Photonic crystal
3521:
3519:
3518:
3513:
3508:
3506:
3505:
3504:
3501:
3492:
3491:
3488:
3481:
3480:
3477:
3471:
3466:
3463:
3451:
3449:
3448:
3443:
3441:
3440:
3437:
3424:
3422:
3421:
3416:
3414:
3413:
3410:
3397:
3395:
3394:
3389:
3387:
3386:
3383:
3367:
3365:
3364:
3359:
3357:
3356:
3353:
3344:
3343:
3340:
3331:
3330:
3327:
3311:
3309:
3308:
3303:
3301:
3300:
3297:
3283:
3281:
3280:
3275:
3255:
3253:
3252:
3247:
3245:
3244:
3221:
3219:
3218:
3213:
3211:
3210:
3193:
3191:
3190:
3185:
3183:
3171:
3169:
3168:
3163:
3151:
3149:
3148:
3143:
3141:
3130:
3118:
3116:
3115:
3110:
3108:
3097:
3092:
3087:
3076:
3068:
3063:
3062:
3053:
3041:
3039:
3038:
3033:
3031:
3020:
3015:
3010:
2999:
2983:
2981:
2980:
2975:
2959:
2957:
2956:
2951:
2935:
2933:
2932:
2927:
2911:
2909:
2908:
2903:
2901:
2900:
2880:
2878:
2877:
2872:
2870:
2869:
2849:
2847:
2846:
2841:
2829:
2827:
2826:
2821:
2806:
2804:
2803:
2798:
2796:
2795:
2790:
2778:
2773:
2768:
2757:
2743:
2741:
2734:
2733:
2720:
2719:
2718:
2713:
2707:
2706:
2697:
2691:
2679:
2677:
2676:
2671:
2669:
2667:
2666:
2665:
2652:
2651:
2650:
2645:
2633:
2628:
2623:
2612:
2604:
2603:
2587:
2582:
2580:
2579:
2578:
2566:
2565:
2549:
2548:
2547:
2542:
2536:
2535:
2526:
2518:
2517:
2507:
2493:
2492:
2489:
2451:Victor Weisskopf
2440:
2438:
2437:
2432:
2430:
2429:
2424:
2406:
2394:
2392:
2391:
2386:
2374:
2372:
2371:
2366:
2354:
2352:
2351:
2346:
2344:
2336:
2325:
2324:
2308:
2306:
2305:
2300:
2298:
2297:
2281:
2279:
2278:
2273:
2258:
2256:
2255:
2250:
2239:
2238:
2219:
2217:
2216:
2211:
2206:
2205:
2187:
2175:
2173:
2172:
2167:
2146:
2144:
2143:
2138:
2124:
2109:
2107:
2106:
2101:
2096:
2095:
2077:
2072:
2071:
2067:
2066:
2037:
2036:
2023:
1993:
1988:
1987:
1983:
1982:
1932:
1905:to get started.
1864:
1862:
1861:
1856:
1851:
1849:
1848:
1836:
1831:
1830:
1818:
1817:
1798:
1796:
1795:
1790:
1788:
1787:
1771:
1769:
1768:
1763:
1761:
1760:
1757:
1744:
1742:
1741:
1736:
1734:
1726:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1683:
1682:
1681:
1678:
1664:is the time and
1663:
1661:
1660:
1655:
1643:
1641:
1640:
1635:
1611:
1609:
1608:
1603:
1598:
1597:
1593:
1592:
1591:
1588:
1557:
1556:
1552:
1551:
1491:
1489:
1488:
1483:
1481:
1480:
1464:
1462:
1461:
1456:
1454:
1453:
1434:
1432:
1431:
1426:
1409:
1408:
1393:
1391:
1383:
1366:
1354:
1352:
1351:
1346:
1334:
1332:
1331:
1326:
1305:
1303:
1302:
1297:
1283:
1263:
1261:
1260:
1255:
1239:
1237:
1236:
1231:
1219:
1217:
1216:
1211:
1183:
1181:
1180:
1175:
1160:
1158:
1157:
1152:
1138:
1137:
1125:
1124:
1105:
1103:
1102:
1097:
1079:
1077:
1076:
1071:
1056:
1054:
1053:
1048:
1046:
1045:
1029:
1027:
1026:
1021:
1019:
1018:
878:
871:
864:
505:Superdeterminism
158:Bra–ket notation
109:
107:
106:
101:
93:
88:
87:
79:
67:
62:
60:
49:
21:
20:
4256:
4255:
4251:
4250:
4249:
4247:
4246:
4245:
4241:Charge carriers
4216:
4215:
4206:Wayback Machine
4195:
4189:
4187:
4186:
4177:
4173:
4168:
4164:
4119:
4115:
4080:
4076:
4039:
4035:
4027:
4025:
4014:
4010:
3975:
3971:
3964:
3948:
3944:
3937:
3921:
3917:
3912:
3908:
3885:
3881:
3850:
3846:
3817:(37): 631–637.
3807:
3803:
3796:10.1119/1.13886
3771:
3762:
3755:
3718:
3711:
3701:
3699:
3697:
3681:
3677:
3660:
3656:
3646:
3644:
3629:
3625:
3620:
3563:
3500:
3496:
3487:
3483:
3482:
3476:
3472:
3470:
3462:
3460:
3457:
3456:
3436:
3432:
3430:
3427:
3426:
3409:
3405:
3403:
3400:
3399:
3382:
3378:
3376:
3373:
3372:
3352:
3348:
3339:
3335:
3326:
3322:
3320:
3317:
3316:
3296:
3292:
3290:
3287:
3286:
3269:
3266:
3265:
3262:
3240:
3236:
3234:
3231:
3230:
3206:
3202:
3200:
3197:
3196:
3179:
3177:
3174:
3173:
3157:
3154:
3153:
3137:
3126:
3124:
3121:
3120:
3104:
3093:
3088:
3083:
3072:
3064:
3058:
3054:
3049:
3047:
3044:
3043:
3027:
3016:
3011:
3006:
2995:
2993:
2990:
2989:
2969:
2966:
2965:
2945:
2942:
2941:
2921:
2918:
2917:
2896:
2892:
2890:
2887:
2886:
2865:
2861:
2859:
2856:
2855:
2835:
2832:
2831:
2815:
2812:
2811:
2791:
2786:
2785:
2774:
2769:
2764:
2753:
2729:
2725:
2721:
2714:
2709:
2708:
2702:
2698:
2693:
2692:
2690:
2688:
2685:
2684:
2661:
2657:
2653:
2646:
2641:
2640:
2629:
2624:
2619:
2608:
2599:
2595:
2588:
2586:
2574:
2570:
2561:
2557:
2550:
2543:
2538:
2537:
2531:
2527:
2522:
2513:
2509:
2508:
2506:
2488:
2484:
2482:
2479:
2478:
2464:
2447:Quantum revival
2425:
2420:
2419:
2402:
2400:
2397:
2396:
2380:
2377:
2376:
2360:
2357:
2356:
2340:
2332:
2320:
2316:
2314:
2311:
2310:
2293:
2289:
2287:
2284:
2283:
2264:
2261:
2260:
2231:
2227:
2225:
2222:
2221:
2198:
2194:
2183:
2181:
2178:
2177:
2152:
2149:
2148:
2120:
2118:
2115:
2114:
2088:
2084:
2073:
2062:
2058:
2051:
2047:
2029:
2025:
2013:
1989:
1978:
1974:
1967:
1963:
1928:
1926:
1923:
1922:
1871:
1844:
1840:
1835:
1826:
1822:
1813:
1809:
1807:
1804:
1803:
1783:
1779:
1777:
1774:
1773:
1756:
1752:
1750:
1747:
1746:
1725:
1723:
1720:
1719:
1699:
1696:
1695:
1677:
1675:
1671:
1669:
1666:
1665:
1649:
1646:
1645:
1620:
1617:
1616:
1587:
1585:
1581:
1577:
1573:
1547:
1543:
1539:
1535:
1506:
1503:
1502:
1476:
1472:
1470:
1467:
1466:
1449:
1445:
1443:
1440:
1439:
1404:
1400:
1384:
1367:
1365:
1363:
1360:
1359:
1340:
1337:
1336:
1311:
1308:
1307:
1291:
1288:
1287:
1249:
1246:
1245:
1242:Planck constant
1225:
1222:
1221:
1193:
1190:
1189:
1169:
1166:
1165:
1133:
1129:
1120:
1116:
1114:
1111:
1110:
1088:
1085:
1084:
1065:
1062:
1061:
1041:
1037:
1035:
1032:
1031:
1014:
1010:
1008:
1005:
1004:
1001:
956:Albert Einstein
937:phosphorescence
882:
853:
852:
851:
616:
608:
607:
553:
552:Advanced topics
545:
544:
543:
495:Hidden-variable
485:de Broglie–Bohm
464:
462:Interpretations
454:
453:
452:
422:
414:
413:
412:
370:
362:
361:
360:
327:
283:CHSH inequality
272:
264:
263:
262:
191:Complementarity
185:
177:
176:
175:
143:
114:
89:
78:
77:
63:
53:
48:
40:
37:
36:
17:
12:
11:
5:
4254:
4244:
4243:
4238:
4233:
4228:
4214:
4213:
4208:
4194:
4193:External links
4191:
4185:
4184:
4171:
4162:
4133:(3): 261–266.
4113:
4088:Optics Express
4074:
4033:
4008:
3969:
3962:
3953:Quantum optics
3942:
3935:
3915:
3906:
3879:
3844:
3801:
3753:
3709:
3696:978-0691139685
3695:
3675:
3654:
3622:
3621:
3619:
3616:
3615:
3614:
3609:
3604:
3599:
3597:Purcell effect
3594:
3589:
3584:
3579:
3574:
3569:
3562:
3559:
3535:semiconductors
3523:
3522:
3511:
3499:
3495:
3486:
3475:
3469:
3435:
3408:
3381:
3369:
3368:
3351:
3347:
3338:
3334:
3325:
3295:
3273:
3261:
3258:
3243:
3239:
3209:
3205:
3182:
3161:
3140:
3136:
3133:
3129:
3107:
3103:
3100:
3096:
3091:
3086:
3082:
3079:
3075:
3071:
3067:
3061:
3057:
3052:
3030:
3026:
3023:
3019:
3014:
3009:
3005:
3002:
2998:
2973:
2962:speed of light
2960:is the vacuum
2949:
2925:
2899:
2895:
2868:
2864:
2839:
2819:
2808:
2807:
2794:
2789:
2784:
2781:
2777:
2772:
2767:
2763:
2760:
2756:
2752:
2749:
2746:
2740:
2737:
2732:
2728:
2724:
2717:
2712:
2705:
2701:
2696:
2681:
2680:
2664:
2660:
2656:
2649:
2644:
2639:
2636:
2632:
2627:
2622:
2618:
2615:
2611:
2607:
2602:
2598:
2594:
2591:
2585:
2577:
2573:
2569:
2564:
2560:
2556:
2553:
2546:
2541:
2534:
2530:
2525:
2521:
2516:
2512:
2505:
2502:
2499:
2496:
2487:
2463:
2460:
2428:
2423:
2418:
2415:
2412:
2409:
2405:
2384:
2364:
2343:
2339:
2335:
2331:
2328:
2323:
2319:
2296:
2292:
2271:
2268:
2248:
2245:
2242:
2237:
2234:
2230:
2209:
2204:
2201:
2197:
2193:
2190:
2186:
2165:
2162:
2159:
2156:
2136:
2133:
2130:
2127:
2123:
2111:
2110:
2099:
2094:
2091:
2087:
2083:
2080:
2076:
2070:
2065:
2061:
2057:
2054:
2050:
2046:
2043:
2040:
2035:
2032:
2028:
2022:
2019:
2016:
2012:
2008:
2005:
2002:
1999:
1996:
1992:
1986:
1981:
1977:
1973:
1970:
1966:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1931:
1870:
1867:
1866:
1865:
1854:
1847:
1843:
1839:
1834:
1829:
1825:
1821:
1816:
1812:
1786:
1782:
1755:
1732:
1729:
1703:
1674:
1653:
1633:
1630:
1627:
1624:
1613:
1612:
1601:
1596:
1584:
1580:
1576:
1572:
1569:
1566:
1563:
1560:
1555:
1550:
1546:
1542:
1538:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1479:
1475:
1452:
1448:
1436:
1435:
1424:
1421:
1418:
1415:
1412:
1407:
1403:
1399:
1396:
1390:
1387:
1382:
1379:
1376:
1373:
1370:
1344:
1324:
1321:
1318:
1315:
1295:
1264:is the linear
1253:
1229:
1209:
1206:
1203:
1200:
1197:
1173:
1162:
1161:
1150:
1147:
1144:
1141:
1136:
1132:
1128:
1123:
1119:
1095:
1092:
1069:
1044:
1040:
1017:
1013:
1000:
997:
988:two-level atom
968:quantum optics
884:
883:
881:
880:
873:
866:
858:
855:
854:
850:
849:
844:
839:
834:
829:
824:
819:
814:
809:
804:
799:
794:
789:
784:
779:
774:
769:
764:
759:
754:
749:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
618:
617:
614:
613:
610:
609:
606:
605:
600:
595:
590:
588:Density matrix
585:
580:
575:
570:
565:
560:
554:
551:
550:
547:
546:
542:
541:
536:
531:
526:
521:
516:
511:
510:
509:
508:
507:
492:
487:
482:
477:
472:
466:
465:
460:
459:
456:
455:
451:
450:
445:
440:
435:
430:
424:
423:
420:
419:
416:
415:
411:
410:
405:
400:
395:
390:
385:
379:
378:
377:
371:
368:
367:
364:
363:
359:
358:
353:
348:
342:
341:
340:
339:
338:
336:Delayed-choice
331:Quantum eraser
326:
325:
320:
315:
310:
305:
300:
295:
290:
285:
280:
274:
273:
270:
269:
266:
265:
261:
260:
259:
258:
248:
243:
238:
233:
228:
223:
221:Quantum number
218:
213:
208:
203:
198:
193:
187:
186:
183:
182:
179:
178:
174:
173:
168:
162:
161:
160:
155:
150:
144:
141:
140:
137:
136:
135:
134:
129:
124:
116:
115:
110:
99:
96:
92:
85:
82:
76:
73:
70:
66:
59:
56:
52:
47:
44:
33:
32:
26:
25:
15:
9:
6:
4:
3:
2:
4253:
4242:
4239:
4237:
4234:
4232:
4231:Laser science
4229:
4227:
4224:
4223:
4221:
4212:
4209:
4207:
4203:
4200:
4197:
4196:
4190:
4182:
4175:
4166:
4158:
4154:
4150:
4146:
4141:
4136:
4132:
4128:
4124:
4117:
4109:
4105:
4101:
4097:
4093:
4089:
4085:
4078:
4070:
4066:
4061:
4056:
4052:
4048:
4044:
4037:
4023:
4019:
4012:
4004:
4000:
3996:
3992:
3988:
3985:(in German).
3984:
3980:
3973:
3965:
3963:0-521-43595-1
3959:
3955:
3954:
3946:
3938:
3936:0-8493-3786-0
3932:
3928:
3927:
3919:
3910:
3902:
3898:
3895:(1): 89–109.
3894:
3890:
3883:
3875:
3871:
3867:
3863:
3859:
3855:
3848:
3840:
3836:
3832:
3828:
3824:
3820:
3816:
3812:
3805:
3797:
3793:
3789:
3785:
3781:
3777:
3770:
3766:
3760:
3758:
3748:
3743:
3739:
3735:
3731:
3727:
3723:
3716:
3714:
3698:
3692:
3688:
3687:
3679:
3670:
3665:
3658:
3642:
3638:
3634:
3627:
3623:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3592:Laser science
3590:
3588:
3585:
3583:
3582:Spectral line
3580:
3578:
3575:
3573:
3570:
3568:
3565:
3564:
3558:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3509:
3493:
3467:
3455:
3454:
3453:
3345:
3332:
3315:
3314:
3313:
3271:
3257:
3241:
3237:
3228:
3223:
3207:
3203:
3159:
3134:
3131:
3098:
3080:
3069:
3059:
3055:
3021:
3003:
2987:
2971:
2963:
2947:
2939:
2923:
2915:
2897:
2893:
2884:
2866:
2862:
2853:
2837:
2817:
2792:
2779:
2761:
2750:
2747:
2744:
2738:
2730:
2726:
2722:
2715:
2703:
2699:
2683:
2682:
2662:
2658:
2654:
2647:
2634:
2616:
2605:
2600:
2596:
2592:
2589:
2583:
2575:
2571:
2562:
2558:
2554:
2551:
2544:
2532:
2528:
2519:
2514:
2510:
2503:
2497:
2477:
2476:
2475:
2473:
2469:
2459:
2456:
2455:Eugene Wigner
2452:
2448:
2444:
2426:
2413:
2407:
2382:
2362:
2337:
2329:
2326:
2321:
2317:
2294:
2290:
2269:
2266:
2243:
2235:
2232:
2228:
2202:
2199:
2195:
2191:
2188:
2160:
2154:
2131:
2128:
2125:
2092:
2089:
2085:
2081:
2078:
2068:
2063:
2059:
2055:
2052:
2048:
2041:
2033:
2030:
2026:
2020:
2017:
2014:
2010:
2006:
2000:
1997:
1994:
1984:
1979:
1975:
1971:
1968:
1964:
1957:
1951:
1948:
1939:
1933:
1921:
1920:
1919:
1915:
1912:
1906:
1904:
1899:
1895:
1891:
1886:
1884:
1880:
1876:
1852:
1845:
1841:
1837:
1832:
1827:
1819:
1814:
1810:
1802:
1801:
1800:
1784:
1780:
1730:
1727:
1717:
1701:
1651:
1628:
1622:
1599:
1594:
1578:
1574:
1567:
1561:
1558:
1553:
1548:
1544:
1540:
1536:
1529:
1523:
1520:
1514:
1508:
1501:
1500:
1499:
1497:
1496:
1477:
1473:
1450:
1446:
1422:
1416:
1410:
1405:
1401:
1397:
1394:
1388:
1377:
1371:
1358:
1357:
1356:
1342:
1319:
1313:
1293:
1284:
1282:
1277:
1275:
1271:
1267:
1251:
1243:
1227:
1207:
1204:
1201:
1198:
1195:
1187:
1171:
1148:
1145:
1139:
1134:
1130:
1126:
1121:
1117:
1109:
1108:
1107:
1093:
1090:
1083:
1067:
1060:
1042:
1038:
1015:
1011:
996:
994:
989:
985:
981:
977:
973:
969:
965:
961:
957:
953:
948:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
894:
890:
879:
874:
872:
867:
865:
860:
859:
857:
856:
848:
845:
843:
840:
838:
835:
833:
830:
828:
825:
823:
820:
818:
815:
813:
810:
808:
805:
803:
800:
798:
795:
793:
790:
788:
785:
783:
780:
778:
775:
773:
770:
768:
765:
763:
760:
758:
755:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
619:
612:
611:
604:
601:
599:
596:
594:
591:
589:
586:
584:
581:
579:
578:Quantum chaos
576:
574:
571:
569:
566:
564:
561:
559:
556:
555:
549:
548:
540:
537:
535:
534:Transactional
532:
530:
527:
525:
524:Quantum logic
522:
520:
517:
515:
512:
506:
503:
502:
501:
498:
497:
496:
493:
491:
488:
486:
483:
481:
478:
476:
473:
471:
468:
467:
463:
458:
457:
449:
446:
444:
441:
439:
436:
434:
431:
429:
426:
425:
418:
417:
409:
406:
404:
401:
399:
396:
394:
391:
389:
386:
384:
381:
380:
376:
373:
372:
366:
365:
357:
354:
352:
349:
347:
344:
343:
337:
334:
333:
332:
329:
328:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
291:
289:
286:
284:
281:
279:
276:
275:
268:
267:
257:
254:
253:
252:
251:Wave function
249:
247:
244:
242:
239:
237:
234:
232:
231:Superposition
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
192:
189:
188:
181:
180:
172:
169:
167:
164:
163:
159:
156:
154:
151:
149:
146:
145:
139:
138:
133:
130:
128:
125:
123:
120:
119:
118:
117:
113:
80:
74:
57:
54:
50:
42:
35:
34:
31:
28:
27:
23:
22:
19:
4188:
4174:
4165:
4130:
4126:
4116:
4091:
4087:
4077:
4050:
4046:
4036:
4026:, retrieved
4021:
4011:
3989:(1): 54–73.
3986:
3982:
3972:
3952:
3945:
3925:
3918:
3909:
3892:
3888:
3882:
3857:
3853:
3847:
3814:
3810:
3804:
3779:
3775:
3729:
3726:Proc. R. Soc
3725:
3700:. Retrieved
3685:
3678:
3657:
3645:. Retrieved
3640:
3636:
3626:
3524:
3370:
3263:
3227:quantum dots
3224:
2809:
2465:
2112:
1916:
1907:
1890:ground state
1887:
1872:
1614:
1493:
1437:
1306:is given by
1285:
1278:
1163:
1002:
999:Introduction
949:
933:fluorescence
921:luminescence
913:ground state
888:
887:
433:Klein–Gordon
369:Formulations
206:Energy level
201:Entanglement
184:Fundamentals
171:Interference
122:Introduction
18:
3860:(9): 1157.
3776:Am. J. Phys
1911:phase space
1355:decays is:
822:von Neumann
807:Schrödinger
583:EPR paradox
514:Many-worlds
448:Schrödinger
403:Schrödinger
398:Phase-space
388:Interaction
293:Double-slit
271:Experiments
246:Uncertainty
216:Nonlocality
211:Measurement
196:Decoherence
166:Hamiltonian
4220:Categories
4094:(9): 378.
4060:2001.09737
4028:2024-06-21
3782:(4): 340.
3669:1703.08176
3618:References
2472:free space
1898:eigenstate
1894:QED vacuum
817:Sommerfeld
732:Heisenberg
727:Gutzwiller
667:de Broglie
615:Scientists
529:Relational
480:Copenhagen
383:Heisenberg
241:Tunnelling
142:Background
4157:0002-9505
4140:1006.4610
4108:1094-4087
4003:0044-3328
3854:Phys. Rev
3498:Γ
3485:Γ
3474:Γ
3434:Γ
3407:Γ
3380:Γ
3350:Γ
3337:Γ
3324:Γ
3294:Γ
3238:ω
3204:ω
3102:⟩
3078:⟨
3056:μ
3025:⟩
3001:⟨
2972:α
2924:ℏ
2894:ε
2863:μ
2818:ω
2783:⟩
2759:⟨
2751:α
2736:ℏ
2727:ε
2723:π
2700:μ
2638:⟩
2614:⟨
2597:ω
2593:α
2568:ℏ
2559:ε
2555:π
2529:μ
2511:ω
2498:ω
2486:Γ
2318:ω
2291:ω
2208:⟩
2135:⟩
2098:⟩
2060:ω
2053:−
2011:∑
2004:⟩
1976:ω
1969:−
1946:⟩
1934:ψ
1842:τ
1824:Γ
1781:τ
1754:Γ
1673:Γ
1583:Γ
1579:−
1541:−
1398:−
1386:∂
1369:∂
1266:frequency
1252:ν
1208:ν
1199:ω
1196:ℏ
1188:. Note:
1172:ℏ
1146:ω
1143:ℏ
1127:−
1094:ω
1091:ℏ
1068:ω
847:Zeilinger
692:Ehrenfest
421:Equations
98:⟩
95:Ψ
84:^
72:⟩
69:Ψ
46:ℏ
4202:Archived
3767:(1984).
3561:See also
3152:, where
1220:, where
1080:and an
897:molecule
772:Millikan
697:Einstein
682:Davisson
637:Blackett
622:Aharonov
490:Ensemble
470:Bayesian
375:Overview
256:Collapse
236:Symmetry
127:Glossary
4179:(2005).
4053:: 474.
4047:Quantum
3862:Bibcode
3839:6780937
3819:Bibcode
3784:Bibcode
3734:Bibcode
3539:bandgap
3527:phonons
2984:is the
2936:is the
2912:is the
2881:is the
2850:is the
1268:. The
1240:is the
1184:is the
812:Simmons
802:Rydberg
767:Moseley
747:Kramers
737:Hilbert
722:Glauber
717:Feynman
702:Everett
672:Compton
443:Rydberg
132:History
4155:
4106:
4001:
3960:
3933:
3837:
3702:1 June
3693:
3647:1 June
3551:lasers
3371:where
2964:, and
2810:where
2113:where
1892:, the
1869:Theory
1615:where
1438:where
1164:where
1082:energy
941:Lasers
917:photon
842:Zeeman
837:Wigner
787:Planck
757:Landau
742:Jordan
393:Matrix
323:Popper
4135:arXiv
4055:arXiv
3835:S2CID
3772:(PDF)
3664:arXiv
1270:phase
972:Dirac
903:or a
899:, an
797:Raman
782:Pauli
777:Onnes
712:Fermi
687:Debye
677:Dirac
642:Bloch
632:Bethe
500:Local
438:Pauli
428:Dirac
226:State
4153:ISSN
4104:ISSN
3999:ISSN
3958:ISBN
3931:ISBN
3730:A114
3704:2022
3691:ISBN
3649:2022
3531:heat
3489:nrad
3438:nrad
3354:nrad
2453:and
2375:and
2220:and
2147:and
1244:and
966:and
901:atom
832:Wien
827:Weyl
792:Rabi
762:Laue
752:Lamb
707:Fock
662:Bose
657:Born
652:Bohr
647:Bohm
627:Bell
4145:doi
4096:doi
4065:doi
3991:doi
3897:doi
3870:doi
3827:doi
3792:doi
3742:doi
3643:(8)
3502:rad
3478:rad
3411:rad
3384:tot
3341:rad
3328:tot
3298:tot
2490:rad
1758:rad
1679:rad
1589:rad
4222::
4151:.
4143:.
4131:79
4129:.
4125:.
4102:.
4090:.
4086:.
4063:.
4049:.
4045:.
4020:,
3997:.
3987:63
3981:.
3893:51
3891:.
3868:.
3858:74
3856:.
3833:.
3825:.
3815:23
3813:.
3790:.
3780:52
3778:.
3774:.
3756:^
3740:.
3728:.
3724:.
3712:^
3641:14
3639:.
3635:.
3464:QE
3222:.
3060:12
2940:,
2916:,
2885:,
2867:12
2854:,
2704:12
2533:12
1885:.
1846:21
1828:21
1815:21
1799::
1785:21
1549:21
1478:21
1451:21
1406:21
1106::
947:.
927:,
4159:.
4147::
4137::
4110:.
4098::
4092:2
4071:.
4067::
4057::
4051:5
4005:.
3993::
3966:.
3939:.
3903:.
3899::
3876:.
3872::
3864::
3841:.
3829::
3821::
3798:.
3794::
3786::
3750:.
3744::
3736::
3706:.
3672:.
3666::
3651:.
3510:.
3494:+
3468:=
3346:+
3333:=
3272:N
3242:3
3208:3
3181:r
3160:q
3139:r
3135:q
3132:=
3128:d
3106:|
3099:2
3095:|
3090:d
3085:|
3081:1
3074:|
3070:=
3066:|
3051:|
3029:|
3022:2
3018:|
3013:r
3008:|
3004:1
2997:|
2948:c
2898:0
2838:n
2793:2
2788:|
2780:2
2776:|
2771:r
2766:|
2762:1
2755:|
2748:4
2745:=
2739:c
2731:0
2716:2
2711:|
2695:|
2663:2
2659:c
2655:3
2648:2
2643:|
2635:2
2631:|
2626:r
2621:|
2617:1
2610:|
2606:n
2601:3
2590:4
2584:=
2576:3
2572:c
2563:0
2552:3
2545:2
2540:|
2524:|
2520:n
2515:3
2504:=
2501:)
2495:(
2427:2
2422:|
2417:)
2414:t
2411:(
2408:b
2404:|
2383:s
2363:k
2342:|
2338:k
2334:|
2330:c
2327:=
2322:k
2295:0
2270:s
2267:k
2247:)
2244:t
2241:(
2236:s
2233:k
2229:b
2203:s
2200:k
2196:1
2192:;
2189:g
2185:|
2164:)
2161:t
2158:(
2155:a
2132:0
2129:;
2126:e
2122:|
2093:s
2090:k
2086:1
2082:;
2079:g
2075:|
2069:t
2064:k
2056:i
2049:e
2045:)
2042:t
2039:(
2034:s
2031:k
2027:b
2021:s
2018:,
2015:k
2007:+
2001:0
1998:;
1995:e
1991:|
1985:t
1980:0
1972:i
1965:e
1961:)
1958:t
1955:(
1952:a
1949:=
1943:)
1940:t
1937:(
1930:|
1853:.
1838:1
1833:=
1820:=
1811:A
1731:e
1728:1
1702:N
1652:t
1632:)
1629:0
1626:(
1623:N
1600:,
1595:t
1575:e
1571:)
1568:0
1565:(
1562:N
1559:=
1554:t
1545:A
1537:e
1533:)
1530:0
1527:(
1524:N
1521:=
1518:)
1515:t
1512:(
1509:N
1474:A
1447:A
1423:,
1420:)
1417:t
1414:(
1411:N
1402:A
1395:=
1389:t
1381:)
1378:t
1375:(
1372:N
1343:N
1323:)
1320:t
1317:(
1314:N
1294:t
1228:h
1205:h
1202:=
1149:,
1140:=
1135:1
1131:E
1122:2
1118:E
1043:1
1039:E
1016:2
1012:E
877:e
870:t
863:v
91:|
81:H
75:=
65:|
58:t
55:d
51:d
43:i
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