195:
particles approaching the barrier have less kinetic energy than the step: the particles are reflected. If the particles have more energy than the step, some are transmitted past the step, while some are reflected. The ratio of reflection to transmission depends on the energy difference. Relativity adds a third solution: very steep potential steps appear to create particles and antiparticles that then change the calculated ratio of transmission and reflection. The theoretical tools called quantum mechanics cannot handle the creation of particles, making any analysis of the relativistic case suspect. Before antiparticles where discovered and quantum field theory developed, this third solution was not understood. The puzzle came to be called the Klein paradox.
1758:
25:
487:. Planck showed that the blackbody oscillators must be restricted to quantum transitions. In 1927, Dirac published his first work on quantum electrodynamics by using quantum field theory. With this foundation, Heisenberg, Jordan, and Pauli incorporated relativistic invariance in quantized Maxwell's equations in 1928 and 1929.
445:, the only elementary particle other than the electron known at the time. Dirac pointed out Klein's negative electrons could not convert themselves to positive protons and suggested that the extra states were all filled with electrons already. Then a proton would amount to a missing electron in these lower states.
1736:
One interpretation of the paradox is that a potential step cannot reverse the direction of the group velocity of a massless relativistic particle. This explanation best suits the single particle solution cited above. Other, more complex interpretations are suggested in literature, in the context of
1053:
286:
published the paper describing what later came to be called the Klein paradox in 1929, just as physicists were grappling with two problems: how to combine the theories of relativity and quantum mechanics and how to understand the coupling of matter and light known as electrodynamics. The paradox
194:
The Klein paradox is an unexpected consequence of relativity on the interaction of quantum particles with electrostatic potentials. The quantum mechanical problem of free particles striking an electrostatic step potential has two solutions when relativity is ignored. One solution applies when the
494:
showed that, under the conditions of the paradox, two currents of opposite charge are spontaneously generated at the step. In modern terminology pairs of electrons and positrons are spontaneously created the step potential. These results were confirmed in 1981 by Hansen and
Ravndal using a more
335:
Klein found that these extra states caused absurd results from models for electrons striking a large, sharp change in electrostatic potential: a negative current appeared beyond the barrier. Significantly Dirac's theory only predicted single-particle states. Creation or annihilation of particles
182:. The paradox presented a quantum mechanical objection to the notion of an electron confined within a nucleus. This clear and precise paradox suggested that an electron could not be confined within a nucleus by any potential well. The meaning of this paradox was intensely debated by
880:
1585:
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1171:
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are complex numbers. Both the incoming and transmitted wave functions are associated with positive group velocity (Blue lines in Fig.1), whereas the reflected wave function is associated with negative group velocity. (Green lines in Fig.1)
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For the massive case, the calculations are similar to the above. The results are as surprising as in the massless case. The transmission coefficient is always larger than zero, and approaches 1 as the potential step goes to infinity.
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of the atom published in 1913 assumed electrons in motion around a compact positive nucleus. An atomic electron obeying classical mechanics in the presence of a positive charged nucleus experiences a
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developed a new mechanics for the electron, a quantum mechanics that reproduced Bohr's results. Schrodinger and other physicists knew this mechanics was incomplete: it did not include effects of
1878:
These results were expanded to higher dimensions, and to other types of potentials, such as a linear step, a square barrier, a smooth potential, etc. Many experiments in electron transport in
313:: they should radiate energy and accelerate in to the atomic core. The success of the Bohr model in predicting atomic spectra suggested that the classical mechanics could not be correct.
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For massive particles, the electric field strength required to observe the effect is enormous. The electric potential energy change similar to the rest energy of the incoming particle,
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Resolution of the paradox would require quantum field theory which developed alongside quantum mechanics but at a slower pace due its many complexities. The concept goes back to
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developed for electrodynamics to resolve the paradox. Thus the background of the paradox has two threads: the development of quantum mechanics and of quantum electrodynamics.
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Assuming the particle is propagating from the left, we obtain two solutions — one before the step, in region (1) and one under the potential, in region (2):
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showed that this would make atoms unstable. Finally in 1931 Dirac concluded that these states must correspond to a new "anti-electron" particle. In 1932
1048:{\displaystyle \psi _{1}=Ae^{ipx}\left({\begin{matrix}1\\1\end{matrix}}\right)+A'e^{-ipx}\left({\begin{matrix}-1\\1\end{matrix}}\right),\quad p=E_{0}\,}
171:), the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.
328:
solved the first issue in 1928 with his relativistic quantum theory of the electron. The combination was more accurate and also predicted electron
1238:
287:
raised questions about how relativity was added to quantum mechanics in Dirac's first attempt. It would take the development of the new
2301:
Hund, Fritz. "Materieerzeugung im anschaulichen und im gequantelten
Wellenbild der Materie." Zeitschrift fĂĽr Physik 117.1 (1941): 1-17.
1335:
1580:{\displaystyle R={\frac {\left|A'\right|^{2}}{\left|A\right|^{2}}},\quad T={\frac {\left|B\right|^{2}}{\left|A\right|^{2}}}\,}
760:{\displaystyle \left(\sigma _{x}p+V\right)\psi =E_{0}\psi ,\quad V={\begin{cases}0,&x<0\\V_{0},&x>0\end{cases}}}
351:
to investigate sloped steps. Sauter was able to confirm Bohr's conjecture: the paradoxical result only appeared for a step of
332:. However, it also included twice as many states as expected, all with lower energy than the ones involved in atomic physics.
2226:
1626:
73:) is a quantum phenomenon related to particles encountering high-energy potential barriers. It is named after physicist
1914:
Klein, O. (1929). "Die
Reflexion von Elektronen an einem Potentialsprung nach der relativistischen Dynamik von Dirac".
437:
Throughout 1929 and 1930, a series of papers by different physicists attempted to understand Dirac's extra states.
1166:{\displaystyle \psi _{2}=Be^{ikx}\left({\begin{matrix}1\\1\end{matrix}}\right),\quad \left|k\right|=V_{0}-E_{0}\,}
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However it took another 10 years before the theory could be applied to the problem of the Klein paradox. In 1941
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1979:
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41:
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263:, which works out to 10 V/cm for electrons. For electrons, such extreme fields might only be relevant in
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1996:
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2008:
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The traditional resolution uses particle–anti-particle pair production in the context of
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who discovered in 1929. Originally, Klein obtained a paradoxical result by applying the
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97:. However, Klein's result showed that if the potential is at least of the order of the
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867:{\displaystyle \sigma _{x}=\left({\begin{matrix}0&1\\1&0\end{matrix}}\right)}
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The definition of the probability current associated with the Dirac equation is:
503:
Consider a massless relativistic particle approaching a potential step of height
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491:
329:
272:
168:
78:
2449:
Dombey, N; Calogeracos, A. (July 1999). "Seventy years of the Klein paradox".
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where the unrestrained tunnelling is shown to occur due to the existence of
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model for neutral particles within the nucleus, before the discovery of the
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800:
438:
348:
2161:
Das Sarma, S.; Adam, Shaffique; Hwang, E. H.; Rossi, Enrico (2011-05-16).
2263:
2122:
2060:
283:
268:
74:
2410:"Introductory review on quantum field theory with space–time resolution"
1757:
1315:{\displaystyle J_{i}=\psi _{i}^{\dagger }\sigma _{x}\psi _{i},\ i=1,2\,}
339:
The Klein result was widely discussed immediately after it publication.
1935:
1194:
We now want to calculate the transmission and reflection coefficients,
476:
340:
325:
306:
183:
2499:
2361:
Pendry, J. B. (2007). "PHYSICS: Negative
Refraction for Electrons?".
2139:
1733:
And so the transmission coefficient is 1 and there is no reflection.
458:
450:
2020:
1858:, then partial reflection rather than total reflection will result.
275:
at graphene p-n junctions the effect can be studied experimentally.
2104:
Katsnelson, M. I.; Novoselov, K. S.; Geim, A. K. (September 2006).
1879:
343:
thought the result was related to the abrupt step and as a result
2179:
442:
179:
38:
The diagrams and interpretation presented here need confirmation.
336:
could not be correctly analyzed in the single particle theory.
1451:{\displaystyle J_{1}=2\left,\quad J_{2}=2\left|B\right|^{2}\,}
879:
174:
The immediate application of the paradox was to
Rutherford's
753:
2478:
Robinson, T. R. (2012). "On Klein tunneling in graphene".
483:
so successful in many applications, fails to predict the
2219:
Inward bound: of matter and forces in the physical world
597:
2240:
2238:
2160:
2103:
1673:{\displaystyle \left|A\right|^{2}=\left|B\right|^{2}\,}
1100:
1000:
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833:
267:>170 nuclei or evaporation at the event horizon of
2106:"Chiral tunnelling and the Klein paradox in graphene"
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883:Fig. 1 A depiction of the dispersion relation, the
441:suggested they corresponded to recently discovered
2163:"Electronic transport in two-dimensional graphene"
1961:
1882:rely on the Klein paradox for massless particles.
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1463:The transmission and reflection coefficients are:
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457:experimentally observed these particles, renamed
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2245:Calogeracos, A.; Dombey, N. (September 1999).
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2042:Calogeracos, A.; Dombey, N. (September 1999).
1797:If the energy of the particle is in the range
324:nor the interaction of matter and radiation.
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2304:
2221:(Reprint ed.). Oxford: Clarendon Press
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464:
93:into a barrier is observed, with exponential
2407:
1988:
1966:. In French, A. P.; Kennedy, P. J. (eds.).
1909:
1907:
294:
2408:Cheng, T.; Su, Q.; Grobe, R. (July 2010).
2314:Hansen, Alex; Ravndal, Finn (1981-06-01).
2247:"History and physics of the Klein paradox"
2044:"History and physics of the Klein paradox"
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393:over a distance similar to the electrons
34:needs attention from an expert in physics
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1994:
1904:
1851:{\displaystyle mc^{2}<E<Ve-mc^{2}}
878:
89:. In nonrelativistic quantum mechanics,
1959:
1721:{\displaystyle \left|A'\right|^{2}=0\,}
2509:
2360:
1950:
44:may be able to help recruit an expert.
1970:. Harvard University Press. pp.
1913:
498:
2316:"Klein's Paradox and Its Resolution"
2216:
1752:
887:-axis represents momentum while the
18:
1592:Continuity of the wave function at
481:Maxwell's classical electrodynamics
386:{\displaystyle \Delta V>2mc^{2}}
189:
13:
2401:
358:
14:
2528:
1995:Holstein, Barry R. (1998-06-01).
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1963:"Niels Bohr and Nuclear Physics"
1862:Resolutions for the massive case
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132:{\displaystyle Ve\approx mc^{2}}
23:
2354:
1748:
1529:
1412:
1124:
1027:
690:
623:, follows the time-independent
566:{\displaystyle E_{0}<V_{0}e}
471:History of quantum field theory
228:, would need to occur over the
2295:
2035:
1968:Niels Bohr: A Centenary Volume
1873:
603:The particle's wave function,
63:relativistic quantum mechanics
1:
2471:10.1016/S0370-1573(99)00023-X
1897:
427:{\displaystyle \lambda =h/mc}
301:History of quantum mechanics
7:
2480:American Journal of Physics
2001:American Journal of Physics
1885:
1743:particle–antiparticle pairs
792:{\displaystyle \sigma _{x}}
81:to the familiar problem of
36:. The specific problem is:
10:
2533:
2340:10.1088/0031-8949/23/6/002
1960:Stuewer, Roger H. (1985).
1223:They are derived from the
468:
465:Positron-electron creation
298:
278:
2434:10.1080/00107510903450559
2197:10.1103/RevModPhys.83.407
2167:Reviews of Modern Physics
256:{\displaystyle \hbar m/c}
891:-axis represents energy.
295:Dirac equation mysteries
186:and others at the time.
2375:10.1126/science.1140178
2281:10.1080/001075199181387
2078:10.1080/001075199181387
1178:where the coefficients
2217:Pais, Abraham (2002).
1916:Zeitschrift fĂĽr Physik
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479:'s demonstration that
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221:{\displaystyle mc^{2}}
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616:{\displaystyle \psi }
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523:{\displaystyle V_{0}}
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2414:Contemporary Physics
2251:Contemporary Physics
2048:Contemporary Physics
1868:quantum field theory
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1739:quantum field theory
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1216:{\displaystyle T,R.}
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289:quantum field theory
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2492:2012AmJPh..80..141R
2463:1999PhR...315...41D
2426:2010ConPh..51..315C
2332:1981PhyS...23.1036H
2273:1999ConPh..40..313C
2189:2011RvMP...83..407D
2132:2006NatPh...2..620K
2070:1999ConPh..40..313C
2013:1998AmJPh..66..507H
1928:1929ZPhy...53..157K
1611:{\displaystyle x=0}
1269:
495:general treatment.
83:electron scattering
42:WikiProject Physics
2517:Physical paradoxes
1936:10.1007/BF01339716
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1768:. You can help by
1745:at the potential.
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447:Robert Oppenheimer
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395:Compton wavelength
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322:special relativity
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230:Compton wavelength
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145:electric potential
129:
91:electron tunneling
2500:10.1119/1.3658629
2228:978-0-19-851997-3
1997:"Klein's paradox"
1892:List of paradoxes
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586:{\displaystyle p}
530:with energy
434:, about 2 x 10m.
345:Arnold Sommerfeld
318:Edwin Schrodinger
232:of the particle,
153:elementary charge
87:potential barrier
59:
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16:Quantum phenomena
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2486:(2): 141–147.
2475:
2457:(1–3): 41–58.
2446:
2420:(4): 315–330.
2403:
2400:
2397:
2396:
2353:
2303:
2294:
2257:(5): 313–321.
2234:
2227:
2202:
2173:(2): 407–470.
2153:
2116:(9): 620–625.
2110:Nature Physics
2091:
2054:(5): 313–321.
2034:
2007:(6): 507–512.
1987:
1980:
1949:
1902:
1901:
1899:
1896:
1895:
1894:
1887:
1884:
1875:
1872:
1863:
1860:
1845:
1841:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1814:
1810:
1806:
1794:
1793:The Klein zone
1791:
1784:
1783:
1763:
1761:
1750:
1747:
1731:
1730:
1729:
1728:
1716:
1713:
1708:
1703:
1699:
1696:
1692:
1680:
1666:
1661:
1658:
1655:
1650:
1645:
1640:
1637:
1634:
1607:
1604:
1601:
1590:
1589:
1588:
1587:
1571:
1566:
1563:
1560:
1553:
1548:
1545:
1542:
1535:
1532:
1528:
1521:
1516:
1513:
1510:
1503:
1498:
1494:
1491:
1487:
1480:
1477:
1461:
1460:
1459:
1458:
1444:
1439:
1436:
1433:
1428:
1425:
1420:
1416:
1411:
1407:
1401:
1396:
1392:
1389:
1385:
1380:
1375:
1370:
1367:
1364:
1358:
1354:
1351:
1346:
1342:
1327:In this case:
1325:
1324:
1323:
1322:
1310:
1307:
1304:
1301:
1298:
1292:
1287:
1283:
1277:
1273:
1267:
1262:
1258:
1254:
1249:
1245:
1212:
1209:
1206:
1203:
1176:
1175:
1174:
1173:
1159:
1155:
1151:
1146:
1142:
1138:
1134:
1131:
1128:
1123:
1119:
1112:
1109:
1108:
1105:
1102:
1101:
1098:
1092:
1089:
1086:
1082:
1078:
1075:
1070:
1066:
1055:
1041:
1037:
1033:
1030:
1026:
1022:
1015:
1012:
1011:
1008:
1005:
1002:
1001:
998:
992:
989:
986:
983:
979:
974:
971:
967:
963:
956:
953:
952:
949:
946:
945:
942:
936:
933:
930:
926:
922:
919:
914:
910:
877:
876:
875:
874:
862:
855:
852:
850:
847:
846:
843:
840:
838:
835:
834:
831:
827:
822:
818:
786:
782:
770:
769:
768:
767:
754:
749:
746:
743:
740:
738:
733:
729:
725:
724:
721:
718:
715:
712:
710:
707:
704:
703:
701:
696:
693:
689:
686:
681:
677:
673:
670:
666:
662:
659:
656:
651:
647:
642:
625:Dirac equation
612:
582:
562:
557:
553:
549:
544:
540:
517:
513:
500:
497:
492:Friedrich Hund
469:Main article:
466:
463:
423:
420:
416:
412:
409:
406:
380:
376:
372:
369:
366:
363:
360:
299:Main article:
296:
293:
280:
277:
273:quasiparticles
271:, but for 2-D
252:
248:
244:
241:
215:
211:
207:
191:
188:
169:speed of light
126:
122:
118:
115:
112:
109:
79:Dirac equation
57:
56:
31:
29:
22:
15:
9:
6:
4:
3:
2:
2529:
2518:
2515:
2514:
2512:
2501:
2497:
2493:
2489:
2485:
2481:
2476:
2472:
2468:
2464:
2460:
2456:
2452:
2447:
2443:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2411:
2406:
2405:
2392:
2388:
2384:
2380:
2376:
2372:
2368:
2364:
2357:
2349:
2345:
2341:
2337:
2333:
2329:
2325:
2321:
2317:
2310:
2308:
2298:
2290:
2286:
2282:
2278:
2274:
2270:
2265:
2260:
2256:
2252:
2248:
2241:
2239:
2230:
2224:
2220:
2213:
2211:
2209:
2207:
2198:
2194:
2190:
2186:
2181:
2176:
2172:
2168:
2164:
2157:
2149:
2145:
2141:
2137:
2133:
2129:
2124:
2119:
2115:
2111:
2107:
2100:
2098:
2096:
2087:
2083:
2079:
2075:
2071:
2067:
2062:
2057:
2053:
2049:
2045:
2038:
2030:
2026:
2022:
2018:
2014:
2010:
2006:
2002:
1998:
1991:
1983:
1977:
1973:
1969:
1964:
1956:
1954:
1945:
1941:
1937:
1933:
1929:
1925:
1921:
1917:
1910:
1908:
1903:
1893:
1890:
1889:
1883:
1881:
1871:
1869:
1859:
1843:
1839:
1835:
1832:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1790:
1780:
1771:
1767:
1764:This section
1762:
1759:
1755:
1754:
1746:
1744:
1740:
1734:
1714:
1711:
1706:
1701:
1697:
1694:
1690:
1681:
1664:
1659:
1656:
1653:
1648:
1643:
1638:
1635:
1632:
1623:
1622:
1621:
1620:
1619:
1605:
1602:
1599:
1569:
1564:
1561:
1558:
1551:
1546:
1543:
1540:
1533:
1530:
1526:
1519:
1514:
1511:
1508:
1501:
1496:
1492:
1489:
1485:
1478:
1475:
1468:
1467:
1466:
1465:
1464:
1442:
1437:
1434:
1431:
1426:
1423:
1418:
1414:
1409:
1405:
1399:
1394:
1390:
1387:
1383:
1378:
1373:
1368:
1365:
1362:
1356:
1352:
1349:
1344:
1340:
1332:
1331:
1330:
1329:
1328:
1308:
1305:
1302:
1299:
1296:
1290:
1285:
1281:
1275:
1271:
1265:
1260:
1256:
1252:
1247:
1243:
1235:
1234:
1233:
1232:
1231:
1228:
1226:
1210:
1207:
1204:
1201:
1192:
1157:
1153:
1149:
1144:
1140:
1136:
1132:
1129:
1126:
1121:
1117:
1110:
1103:
1096:
1090:
1087:
1084:
1080:
1076:
1073:
1068:
1064:
1056:
1039:
1035:
1031:
1028:
1024:
1020:
1013:
1006:
1003:
996:
990:
987:
984:
981:
977:
972:
969:
965:
961:
954:
947:
940:
934:
931:
928:
924:
920:
917:
912:
908:
900:
899:
898:
897:
896:
890:
886:
881:
860:
853:
848:
841:
836:
829:
825:
820:
816:
808:
807:
806:
805:
804:
802:
784:
780:
747:
744:
741:
736:
731:
727:
719:
716:
713:
708:
705:
699:
694:
691:
687:
684:
679:
675:
671:
668:
664:
660:
657:
654:
649:
645:
640:
632:
631:
630:
629:
628:
626:
610:
601:
599:
594:
580:
560:
555:
551:
547:
542:
538:
515:
511:
496:
493:
488:
486:
482:
478:
472:
462:
460:
456:
455:Carl Anderson
452:
448:
444:
440:
435:
421:
418:
414:
410:
407:
404:
396:
378:
374:
370:
367:
364:
361:
350:
346:
342:
337:
333:
331:
327:
323:
319:
314:
312:
311:Lorentz force
308:
302:
292:
290:
285:
276:
274:
270:
266:
250:
246:
242:
239:
231:
213:
209:
205:
196:
187:
185:
181:
177:
172:
170:
166:
162:
161:electron mass
158:
154:
150:
146:
142:
124:
120:
116:
113:
110:
107:
100:
99:electron mass
96:
92:
88:
84:
80:
76:
72:
68:
67:Klein paradox
64:
53:
43:
39:
35:
32:This article
30:
21:
20:
2483:
2479:
2454:
2450:
2417:
2413:
2366:
2362:
2356:
2323:
2319:
2297:
2254:
2250:
2218:
2170:
2166:
2156:
2113:
2109:
2051:
2047:
2037:
2004:
2000:
1990:
1967:
1919:
1915:
1877:
1865:
1796:
1787:
1774:
1770:adding to it
1765:
1749:Massive case
1735:
1732:
1591:
1462:
1326:
1229:
1193:
1177:
894:
888:
884:
801:Pauli matrix
771:
602:
595:
502:
489:
474:
439:Hermann Weyl
436:
349:Fritz Sauter
338:
334:
315:
304:
282:
264:
197:
193:
173:
164:
156:
148:
140:
70:
66:
60:
50:October 2019
47:
37:
33:
1874:Other cases
284:Oscar Klein
269:black holes
75:Oskar Klein
1981:0674624165
1898:References
1618:, yields:
1227:currents.
477:Max Planck
341:Niels Bohr
326:Paul Dirac
307:Bohr model
184:Niels Bohr
2442:0010-7514
2391:122548440
2348:0031-8949
2289:0010-7514
2180:1003.4731
2148:1745-2481
2086:0010-7514
2029:0002-9505
1944:121771000
1833:−
1379:−
1282:ψ
1272:σ
1266:†
1257:ψ
1150:−
1065:ψ
1004:−
982:−
909:ψ
817:σ
781:σ
685:ψ
669:ψ
646:σ
611:ψ
459:positrons
451:Igor Tamm
405:λ
359:Δ
240:ℏ
114:≈
2511:Category
2383:17332397
1886:See also
1880:graphene
1777:May 2018
1698:′
1493:′
1391:′
973:′
316:In 1926
2488:Bibcode
2459:Bibcode
2422:Bibcode
2363:Science
2328:Bibcode
2269:Bibcode
2185:Bibcode
2128:Bibcode
2066:Bibcode
2009:Bibcode
1972:197–220
1924:Bibcode
799:is the
443:protons
279:History
180:neutron
167:is the
159:is the
151:is the
143:is the
139:(where
95:damping
85:from a
2440:
2389:
2381:
2346:
2287:
2225:
2146:
2084:
2027:
1978:
1942:
1294:
347:asked
65:, the
2387:S2CID
2259:arXiv
2175:arXiv
2118:arXiv
2056:arXiv
1940:S2CID
2438:ISSN
2379:PMID
2344:ISSN
2285:ISSN
2223:ISBN
2144:ISSN
2082:ISSN
2025:ISSN
1976:ISBN
1824:<
1818:<
1186:and
772:And
745:>
717:<
548:<
365:>
330:spin
305:The
163:and
2496:doi
2467:doi
2455:315
2430:doi
2371:doi
2367:315
2336:doi
2277:doi
2193:doi
2136:doi
2074:doi
2017:doi
1932:doi
1772:.
61:In
2513::
2494:.
2484:80
2482:.
2465:.
2453:.
2436:.
2428:.
2418:51
2416:.
2412:.
2385:.
2377:.
2365:.
2342:.
2334:.
2324:23
2322:.
2318:.
2306:^
2283:.
2275:.
2267:.
2255:40
2253:.
2249:.
2237:^
2205:^
2191:.
2183:.
2171:83
2169:.
2165:.
2142:.
2134:.
2126:.
2112:.
2108:.
2094:^
2080:.
2072:.
2064:.
2052:40
2050:.
2046:.
2023:.
2015:.
2005:66
2003:.
1999:.
1974:.
1952:^
1938:.
1930:.
1920:53
1918:.
1906:^
1870:.
1184:A′
1182:,
803::
627::
593:.
461:.
397:,
155:,
147:,
2502:.
2498::
2490::
2473:.
2469::
2461::
2444:.
2432::
2424::
2393:.
2373::
2350:.
2338::
2330::
2291:.
2279::
2271::
2261::
2231:.
2199:.
2195::
2187::
2177::
2150:.
2138::
2130::
2120::
2114:2
2088:.
2076::
2068::
2058::
2031:.
2019::
2011::
1984:.
1946:.
1934::
1926::
1844:2
1840:c
1836:m
1830:e
1827:V
1821:E
1813:2
1809:c
1805:m
1779:)
1775:(
1715:0
1712:=
1707:2
1702:|
1695:A
1691:|
1665:2
1660:|
1657:B
1654:|
1649:=
1644:2
1639:|
1636:A
1633:|
1606:0
1603:=
1600:x
1570:2
1565:|
1562:A
1559:|
1552:2
1547:|
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1541:|
1534:=
1531:T
1527:,
1520:2
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1512:A
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1502:2
1497:|
1490:A
1486:|
1479:=
1476:R
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1435:B
1432:|
1427:2
1424:=
1419:2
1415:J
1410:,
1406:]
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1388:A
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1111:1
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1097:(
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991:x
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720:0
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695:=
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52:)
48:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.