2153:
2173:
trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation (although the physical situation is different). Zitterbewegung-like oscillations of ultracold atoms in optical lattices were predicted in 2008. In 2013, zitterbewegung was simulated in a
1821:
2159:
The original massive Dirac particle can then be viewed as being composed of two massless components, each of which continually converts itself to the other. Since the components are massless they move at the speed of light, and their spin is constrained to be about the direction of motion, but each
2172:
Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence. It has also been simulated in atomic systems that provide analogues of a free Dirac particle. The first such example, in 2010, placed a
1862:
More recently, it has been noted that in the case of free particles it could just be an artifact of the simplified theory. Zitterbewegung appear as due to the "small components" of the Dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic
1312:
2015:
2746:
Suprano, Alessia; Zia, Danilo; Polino, Emanuele; Poderini, Davide; Carvacho, Gonzalo; Sciarrino, Fabio; Lugli, Matteo; Bisio, Alessandro; Perinotti, Paolo. "Photonic cellular automaton simulation of relativistic quantum fields: observation of
Zitterbewegung".
1575:
1839:
In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves. The standard relativistic velocity can be recovered by taking a
242:
1114:
1587:
997:
866:
1844:, when the positive and negative components are decoupled. Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.
1126:
424:
2148:{\displaystyle \left\{{\begin{matrix}\sigma ^{\mu }\partial _{\mu }\psi _{\rm {R}}=m\psi _{\rm {L}}\\{\overline {\sigma }}^{\mu }\partial _{\mu }\psi _{\rm {L}}=m\psi _{\rm {R}}\end{matrix}}\right.}
1976:
1398:
748:
663:
2699:
Leblanc, L. J.; Beeler, M. C.; Jimenez-Garcia, K.; Perry, A. R.; Sugawa, S.; Williams, R. A.; Spielman, I.B. (2013). "Direct observation of zitterbewegung in a Bose–Einstein condensate".
1423:
307:
613:
580:
550:
161:
523:
266:
1914:
1816:{\displaystyle x_{k}(t)=x_{k}(0)+c^{2}p_{k}H^{-1}t+{\tfrac {1}{2}}i\hbar cH^{-1}\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)\left(e^{-{\frac {2iHt}{\hbar }}}-1\right)}
1019:
499:
2007:
913:
468:
448:
122:
This apparent oscillatory motion is often interpreted as an artifact of using the Dirac equation in a single particle description and disappears when using
2495:
Catillon, P.; Cue, N.; Gaillard, M. J.; et al. (2008-07-01). "A Search for the de
Broglie Particle Internal Clock by Means of Electron Channeling".
1855:
states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron
1827:
The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the reduced
2886:
Dóra, Balász; Cayssol, Jérôme; Simon, Ference; Moessner, Roderich (2012). "Optically engineering the topological properties of a spin Hall insulator".
763:
2180:
An optical analogue of zitterbewegung was demonstrated in a quantum cellular automaton implemented with orbital angular momentum states of light
1986:
components), where each is the source term in the other's equation of motion, with a coupling constant proportional to the original particle's
2423:
2314:
2456:
1841:
1307:{\displaystyle \hbar {\frac {d\alpha _{k}(t)}{dt}}=i\left=2\left(i\gamma _{k}m-\sigma _{kl}p^{l}\right)=2i\left(p_{k}-\alpha _{k}H\right)}
1414:
are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.
343:
3024:
2597:
Gerritsma, R.; Kirchmair, G.; Zähringer, F.; Solano, E.; Blatt, R.; Roos, C. F. (2010). "Quantum simulation of the Dirac equation".
2479:
2160:
has opposite helicity: and since the spin remains constant, the direction of the velocity reverses, leading to the characteristic
3040:
2833:
330:
77:
1925:
1324:
1875:. Nevertheless, it is an interesting way to understand certain QED effects heuristically from the single particle picture.
3055:
2174:
2768:
Schliemann, John (2005). "Zitterbewegung of
Electronic Wave Packets in III-V Zinc-Blende Semiconductor Quantum Wells".
1917:
84:
produces an apparent fluctuation (up to the speed of light) of the position of an electron around the median, with an
53:
682:
618:
2446:
Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese
Physics B, 17(11), 4170.
1570:{\displaystyle \alpha _{k}(t)=\left(\alpha _{k}(0)-cp_{k}H^{-1}\right)e^{-{\frac {2iHt}{\hbar }}}+cp_{k}H^{-1}}
32:
2949:
Shi, Likun; Zhang, Shoucheng; Cheng, Kai (2013). "Anomalous
Electron Trajectory in Topological Insulators".
2888:
2770:
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269:
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123:
61:
49:
2547:
585:
1864:
237:{\displaystyle H\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {x} ,t)}
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2475:
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2347:
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2277:
2259:
754:
502:
85:
2831:
Katsnelson, M. I. (2006). "Zitterbewegung, chirality, and minimal conductivity in graphene".
2978:
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2302:
2267:
2249:
508:
251:
2405:
2296:
1899:
1893:, by observing that the Dirac equation can be reformulated by splitting the four-component
1109:{\displaystyle \alpha _{k}(t)=e^{\frac {iHt}{\hbar }}\alpha _{k}e^{-{\frac {iHt}{\hbar }}}}
1856:
37:
992:{\displaystyle \left\langle \left({\frac {dx_{k}(t)}{dt}}\right)^{2}\right\rangle =c^{2}}
2974:
2911:
2856:
2793:
2724:
2622:
2565:
2508:
2245:
3009:
2660:
Vaishnav, J. Y.; Clark, C. W. (2008). "Observing
Zitterbewegung with Ultracold Atoms".
2272:
2229:
2200:
1992:
553:
477:
471:
152:
135:
131:
69:
1003:
as if the "root mean square speed" in every direction of space is the speed of light.
3049:
2990:
2872:
2817:
2732:
2532:
2433:
2324:
2225:
2183:
Other proposals for condensed-matter analogues include semiconductor nanostructures,
1920:
1890:
1889:
An alternative perspective of the physical meaning of zitterbewegung was provided by
334:
310:
57:
2935:
2385:
2919:
2685:
2646:
1894:
453:
433:
81:
2801:
2864:
65:
2982:
48: 'motion') is the theoretical prediction of a rapid oscillatory motion of
2516:
2415:
2306:
2205:
1868:
2524:
2351:
2263:
861:{\displaystyle {\frac {dx_{k}(t)}{dt}}={\frac {i}{\hbar }}\left=c\alpha _{k}}
1987:
2927:
2809:
2638:
2583:
2457:"Dirac equation - is Zitterbewegung an artefact of single-particle theory?"
2281:
2254:
2847:
2784:
2184:
1852:
314:
73:
2630:
318:
20:
2340:Über die kräftefreie Bewegung in der relativistischen Quantenmechanik
2574:
2753:
2965:
2902:
2715:
2676:
2613:
419:{\displaystyle H=\beta mc^{2}+\sum _{j=1}^{3}\alpha _{j}p_{j}c}
2474:(Sixth Printing ed.). Alfred A. Knopf. pp. 628–632.
2142:
1831:. That oscillation term is the so-called zitterbewegung.
1120:
The time-dependence of the velocity operator is given by
1971:{\displaystyle \psi =(\psi _{\rm {L}},\psi _{\rm {R}})}
1393:{\displaystyle \sigma _{kl}\equiv {\frac {i}{2}}\left.}
2344:
On the free movement in relativistic quantum mechanics
2024:
1675:
621:
588:
561:
480:
456:
436:
2230:"An Interpretation of Dirac's Theory of the Electron"
2018:
1995:
1928:
1902:
1590:
1426:
1327:
1129:
1022:
916:
766:
685:
531:
511:
346:
278:
254:
164:
1013:, one implements the Heisenberg picture, which says
2177:of 50,000 atoms of Rb confined in an optical trap.
2147:
2001:
1970:
1908:
1815:
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1392:
1306:
1108:
991:
860:
742:
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607:
574:
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517:
493:
462:
442:
418:
301:
260:
236:
1851:(QED) the negative-energy states are replaced by
672:, the time dependence of an arbitrary observable
134:, a small correction of the energy level of the
2234:Proceedings of the National Academy of Sciences
743:{\displaystyle -i\hbar {\frac {dQ}{dt}}=\left.}
658:{\textstyle \alpha _{j}=\gamma _{0}\gamma _{j}}
8:
2548:"Quantum physics: Trapped ion set to quiver"
893:The above equation shows that the operator
753:In particular, the time-dependence of the
2964:
2901:
2846:
2783:
2752:
2714:
2675:
2612:
2573:
2271:
2253:
2131:
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2052:
2051:
2041:
2031:
2023:
2017:
1994:
1958:
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1943:
1942:
1927:
1901:
1863:motion. It doesn't appear in the correct
1779:
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1752:
1742:
1717:
1699:
1674:
1659:
1649:
1639:
1617:
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1128:
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163:
56:. This prediction was first discussed by
904:-th component of a "velocity operator".
2217:
1795:
1689:
1531:
1130:
1099:
1062:
811:
692:
255:
194:
130:, the zitterbewegung is related to the
64:in 1930 as a result of analysis of the
44: 'to tremble, jitter' and
2346:] (in German). pp. 418–428.
1867:, or rather, it is resolved by using
302:{\displaystyle \psi (\mathbf {x} ,t)}
31:
7:
3019:. Vol. II. pp. 950–952.
2373:] (in German). pp. 63–72.
2132:
2114:
2099:
2071:
2053:
2038:
1959:
1944:
208:
200:
14:
2387:Applications of Quantum Mechanics
886:is the position operator at time
2371:Quantum Dynamics of the Electron
2367:Zur Quantendynamik des Elektrons
286:
221:
175:
3041:Zitterbewegung in New Scientist
2834:The European Physical Journal B
1842:Foldy–Wouthuysen transformation
608:{\textstyle \beta =\gamma _{0}}
2920:10.1103/PhysRevLett.108.056602
2686:10.1103/PhysRevLett.100.153002
2407:Relativistic Quantum Mechanics
2298:Relativistic Quantum Mechanics
1965:
1935:
1729:
1723:
1629:
1623:
1607:
1601:
1470:
1464:
1443:
1437:
1155:
1149:
1039:
1033:
948:
942:
789:
783:
296:
282:
231:
217:
185:
171:
80:between positive and negative
1:
2802:10.1103/PhysRevLett.94.206801
2725:10.1088/1367-2630/15/7/073011
2546:Wunderlich, Christof (2010).
907:Note that this implies that
450:is the mass of the particle,
2087:
1918:left-handed and right-handed
552:are matrices related to the
575:{\textstyle \gamma _{\mu }}
545:{\displaystyle \alpha _{j}}
76:in free space, in which an
54:relativistic wave equations
3072:
2983:10.1103/PhysRevB.87.161115
2865:10.1140/epjb/e2006-00203-1
2393:. University of Cambridge.
2164:or zitterbewegung motion.
1882:
1879:Zigzag picture of fermions
1006:To add time-dependence to
900:can be interpreted as the
33:[ˈtsɪtɐ.bəˌveːɡʊŋ]
2517:10.1007/s10701-008-9225-1
2416:10.1007/978-3-642-88082-7
2307:10.1007/978-3-642-88082-7
2404:Greiner, Walter (1995).
2365:Schrödinger, E. (1931).
2338:Schrödinger, E. (1930).
2295:Greiner, Walter (1995).
2175:Bose–Einstein condensate
1916:into a pair of massless
2889:Physical Review Letters
2771:Physical Review Letters
2663:Physical Review Letters
2470:Penrose, Roger (2004).
2168:Experimental simulation
1865:second quantized theory
1849:quantum electrodynamics
270:reduced Planck constant
2702:New Journal of Physics
2497:Foundations of Physics
2189:topological insulators
2149:
2003:
1972:
1910:
1817:
1571:
1394:
1308:
1110:
993:
862:
744:
659:
609:
576:
546:
519:
518:{\displaystyle \beta }
495:
464:
444:
420:
392:
303:
262:
261:{\displaystyle \hbar }
238:
2553:Nature News and Views
2255:10.1073/pnas.14.7.553
2150:
2004:
1973:
1921:two-component spinors
1911:
1909:{\displaystyle \psi }
1818:
1572:
1395:
1309:
1111:
994:
863:
745:
660:
610:
577:
547:
520:
496:
465:
445:
421:
372:
304:
263:
239:
147:Free spin-1/2 fermion
119:radians per second.
60:in 1928 and later by
29:German pronunciation:
3056:Quantum field theory
3008:Messiah, A. (1962).
2384:Tong, David (2017).
2016:
1993:
1926:
1900:
1885:Feynman checkerboard
1588:
1424:
1325:
1127:
1020:
914:
764:
683:
619:
586:
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509:
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344:
276:
252:
162:
124:quantum field theory
50:elementary particles
2975:2013PhRvB..87p1115S
2912:2012PhRvL.108e6602D
2857:2006EPJB...51..157K
2794:2005PhRvL..94t6801S
2631:10.1038/nature08688
2623:2010Natur.463...68G
2566:2010Natur.463...37W
2509:2008FoPh...38..659C
2472:The Road to Reality
2246:1928PNAS...14..553B
1869:Feynman propagators
676:obeys the equation
151:The time-dependent
111:, or approximately
2145:
2140:
1999:
1968:
1906:
1829:Compton wavelength
1813:
1684:
1567:
1403:Now, because both
1390:
1304:
1106:
989:
858:
740:
670:Heisenberg picture
655:
605:
572:
542:
515:
494:{\textstyle p_{j}}
491:
460:
440:
416:
299:
258:
234:
3017:Quantum Mechanics
2952:Physical Review B
2425:978-3-540-99535-7
2316:978-3-540-99535-7
2090:
2002:{\displaystyle m}
1798:
1683:
1534:
1352:
1167:
1102:
1065:
960:
814:
801:
755:position operator
713:
503:momentum operator
215:
86:angular frequency
72:for relativistic
68:solutions of the
62:Erwin Schrödinger
3063:
3030:
3014:
3010:"XX, Section 37"
2995:
2994:
2968:
2946:
2940:
2939:
2905:
2883:
2877:
2876:
2850:
2848:cond-mat/0512337
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2787:
2785:cond-mat/0410321
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3002:Further reading
2999:
2998:
2948:
2947:
2943:
2885:
2884:
2880:
2830:
2829:
2825:
2767:
2766:
2762:
2745:
2744:
2740:
2698:
2697:
2693:
2659:
2658:
2654:
2607:(7277): 68–71.
2596:
2595:
2591:
2575:10.1038/463037a
2560:(7277): 37–39.
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2014:
2013:
1991:
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1938:
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1897:
1887:
1881:
1837:
1781:
1771:
1770:
1766:
1748:
1738:
1713:
1712:
1708:
1695:
1655:
1645:
1635:
1613:
1591:
1586:
1585:
1554:
1544:
1517:
1507:
1489:
1479:
1454:
1453:
1449:
1427:
1422:
1421:
1411:
1408:
1404:
1372:
1359:
1358:
1354:
1328:
1323:
1322:
1286:
1273:
1272:
1268:
1244:
1231:
1215:
1211:
1207:
1186:
1179:
1175:
1159:
1139:
1135:
1125:
1124:
1088:
1078:
1068:
1051:
1045:
1023:
1018:
1017:
1011:
1007:
979:
952:
932:
928:
922:
921:
917:
912:
911:
901:
898:
894:
887:
878:
873:
848:
827:
820:
816:
793:
773:
769:
762:
761:
722:
718:
705:
697:
681:
680:
673:
645:
635:
622:
617:
616:
595:
584:
583:
562:
557:
556:
532:
527:
526:
507:
506:
481:
476:
475:
452:
451:
432:
431:
403:
393:
359:
342:
341:
326:
274:
273:
250:
249:
207:
199:
160:
159:
149:
144:
114:
112:
102:
99:
93:
92:
90:
89:
28:
17:
16:Particle effect
12:
11:
5:
3069:
3067:
3059:
3058:
3048:
3047:
3044:
3043:
3036:
3035:External links
3033:
3032:
3031:
3025:
3003:
3000:
2997:
2996:
2959:(16): 161115.
2941:
2878:
2841:(2): 157–160.
2823:
2778:(20): 206801.
2760:
2738:
2691:
2652:
2589:
2538:
2503:(7): 659–664.
2487:
2480:
2462:
2448:
2439:
2424:
2396:
2376:
2357:
2330:
2315:
2287:
2240:(7): 553–559.
2226:Breit, Gregory
2216:
2215:
2213:
2210:
2209:
2208:
2203:
2201:Casimir effect
2196:
2193:
2169:
2166:
2157:
2156:
2143:
2134:
2129:
2125:
2122:
2116:
2111:
2105:
2101:
2095:
2089:
2086:
2080:
2079:
2073:
2068:
2064:
2061:
2055:
2050:
2044:
2040:
2034:
2030:
2026:
2025:
2022:
1998:
1967:
1961:
1956:
1952:
1946:
1941:
1937:
1934:
1931:
1905:
1880:
1877:
1836:
1835:Interpretation
1833:
1825:
1824:
1811:
1807:
1804:
1797:
1793:
1790:
1787:
1784:
1778:
1774:
1769:
1764:
1758:
1755:
1751:
1745:
1741:
1737:
1734:
1731:
1728:
1725:
1720:
1716:
1711:
1705:
1702:
1698:
1694:
1691:
1688:
1682:
1679:
1673:
1670:
1665:
1662:
1658:
1652:
1648:
1642:
1638:
1634:
1631:
1628:
1625:
1620:
1616:
1612:
1609:
1606:
1603:
1598:
1594:
1579:
1578:
1564:
1561:
1557:
1551:
1547:
1543:
1540:
1533:
1529:
1526:
1523:
1520:
1514:
1510:
1505:
1499:
1496:
1492:
1486:
1482:
1478:
1475:
1472:
1469:
1466:
1461:
1457:
1452:
1448:
1445:
1442:
1439:
1434:
1430:
1406:
1401:
1400:
1389:
1385:
1379:
1375:
1371:
1366:
1362:
1357:
1351:
1348:
1343:
1338:
1335:
1331:
1316:
1315:
1302:
1298:
1293:
1289:
1285:
1280:
1276:
1271:
1267:
1264:
1261:
1257:
1251:
1247:
1241:
1238:
1234:
1230:
1227:
1222:
1218:
1214:
1210:
1206:
1203:
1199:
1193:
1189:
1185:
1182:
1178:
1174:
1171:
1165:
1162:
1157:
1154:
1151:
1146:
1142:
1138:
1132:
1118:
1117:
1101:
1097:
1094:
1091:
1085:
1081:
1075:
1071:
1064:
1060:
1057:
1054:
1048:
1044:
1041:
1038:
1035:
1030:
1026:
1009:
1001:
1000:
986:
982:
978:
974:
969:
964:
958:
955:
950:
947:
944:
939:
935:
931:
925:
920:
896:
876:
870:
869:
855:
851:
847:
844:
840:
834:
830:
826:
823:
819:
813:
810:
805:
799:
796:
791:
788:
785:
780:
776:
772:
751:
750:
739:
735:
731:
728:
725:
721:
717:
711:
708:
703:
700:
694:
691:
688:
652:
648:
642:
638:
634:
629:
625:
602:
598:
594:
591:
569:
565:
554:Gamma matrices
539:
535:
514:
488:
484:
472:speed of light
463:{\textstyle c}
459:
443:{\textstyle m}
439:
428:
427:
415:
410:
406:
400:
396:
390:
385:
382:
379:
375:
371:
366:
362:
358:
355:
352:
349:
298:
295:
292:
288:
284:
281:
257:
246:
245:
233:
230:
227:
223:
219:
213:
210:
205:
202:
196:
193:
190:
187:
184:
181:
177:
173:
170:
167:
155:is written as
153:Dirac equation
148:
145:
143:
140:
70:Dirac equation
25:zitterbewegung
15:
13:
10:
9:
6:
4:
3:
2:
3068:
3057:
3054:
3053:
3051:
3042:
3039:
3038:
3034:
3028:
3026:9780471597681
3022:
3018:
3011:
3006:
3005:
3001:
2992:
2988:
2984:
2980:
2976:
2972:
2967:
2962:
2958:
2954:
2953:
2945:
2942:
2937:
2933:
2929:
2925:
2921:
2917:
2913:
2909:
2904:
2899:
2896:(5): 056602.
2895:
2891:
2890:
2882:
2879:
2874:
2870:
2866:
2862:
2858:
2854:
2849:
2844:
2840:
2836:
2835:
2827:
2824:
2819:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2786:
2781:
2777:
2773:
2772:
2764:
2761:
2755:
2750:
2742:
2739:
2734:
2730:
2726:
2722:
2717:
2712:
2709:(7): 073011.
2708:
2704:
2703:
2695:
2692:
2687:
2683:
2678:
2673:
2669:
2665:
2664:
2656:
2653:
2648:
2644:
2640:
2636:
2632:
2628:
2624:
2620:
2615:
2610:
2606:
2602:
2601:
2593:
2590:
2585:
2581:
2576:
2571:
2567:
2563:
2559:
2555:
2554:
2549:
2542:
2539:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2491:
2488:
2483:
2481:0-224-04447-8
2477:
2473:
2466:
2463:
2458:
2452:
2449:
2443:
2440:
2435:
2431:
2427:
2421:
2417:
2413:
2409:
2408:
2400:
2397:
2389:
2388:
2380:
2377:
2372:
2368:
2361:
2358:
2353:
2349:
2345:
2341:
2334:
2331:
2326:
2322:
2318:
2312:
2308:
2304:
2300:
2299:
2291:
2288:
2283:
2279:
2274:
2269:
2265:
2261:
2256:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2221:
2218:
2211:
2207:
2204:
2202:
2199:
2198:
2194:
2192:
2190:
2186:
2181:
2178:
2176:
2167:
2165:
2163:
2127:
2123:
2120:
2109:
2103:
2093:
2084:
2066:
2062:
2059:
2048:
2042:
2032:
2028:
2020:
2012:
2011:
2010:
1996:
1989:
1985:
1981:
1954:
1950:
1939:
1932:
1929:
1922:
1919:
1903:
1896:
1892:
1891:Roger Penrose
1886:
1878:
1876:
1874:
1870:
1866:
1860:
1858:
1854:
1850:
1845:
1843:
1834:
1832:
1830:
1809:
1805:
1802:
1791:
1788:
1785:
1782:
1776:
1772:
1767:
1762:
1756:
1753:
1749:
1743:
1739:
1735:
1732:
1726:
1718:
1714:
1709:
1703:
1700:
1696:
1692:
1686:
1680:
1677:
1671:
1668:
1663:
1660:
1656:
1650:
1646:
1640:
1636:
1632:
1626:
1618:
1614:
1610:
1604:
1596:
1592:
1584:
1583:
1582:
1562:
1559:
1555:
1549:
1545:
1541:
1538:
1527:
1524:
1521:
1518:
1512:
1508:
1503:
1497:
1494:
1490:
1484:
1480:
1476:
1473:
1467:
1459:
1455:
1450:
1446:
1440:
1432:
1428:
1420:
1419:
1418:
1415:
1387:
1383:
1377:
1373:
1369:
1364:
1360:
1355:
1349:
1346:
1341:
1336:
1333:
1329:
1321:
1320:
1319:
1300:
1296:
1291:
1287:
1283:
1278:
1274:
1269:
1265:
1262:
1259:
1255:
1249:
1245:
1239:
1236:
1232:
1228:
1225:
1220:
1216:
1212:
1208:
1204:
1201:
1197:
1191:
1187:
1183:
1180:
1176:
1172:
1169:
1163:
1160:
1152:
1144:
1140:
1136:
1123:
1122:
1121:
1095:
1092:
1089:
1083:
1079:
1073:
1069:
1058:
1055:
1052:
1046:
1042:
1036:
1028:
1024:
1016:
1015:
1014:
1004:
984:
980:
976:
972:
967:
962:
956:
953:
945:
937:
933:
929:
923:
918:
910:
909:
908:
905:
891:
883:
879:
853:
849:
845:
842:
838:
832:
828:
824:
821:
817:
808:
803:
797:
794:
786:
778:
774:
770:
760:
759:
758:
756:
737:
733:
729:
726:
723:
719:
715:
709:
706:
701:
698:
689:
686:
679:
678:
677:
671:
666:
650:
646:
640:
636:
632:
627:
623:
600:
596:
592:
589:
567:
563:
555:
537:
533:
512:
504:
486:
482:
473:
457:
437:
413:
408:
404:
398:
394:
388:
383:
380:
377:
373:
369:
364:
360:
356:
353:
350:
347:
340:
339:
338:
336:
335:free particle
332:
329:is the Dirac
324:
320:
316:
312:
311:wave function
293:
290:
279:
271:
228:
225:
211:
203:
191:
188:
182:
179:
168:
165:
158:
157:
156:
154:
146:
141:
139:
137:
133:
129:
128:hydrogen atom
125:
120:
105:
97:
87:
83:
82:energy states
79:
75:
71:
67:
63:
59:
58:Gregory Breit
55:
51:
47:
43:
39:
34:
26:
22:
3016:
2956:
2950:
2944:
2893:
2887:
2881:
2838:
2832:
2826:
2775:
2769:
2763:
2741:
2706:
2700:
2694:
2667:
2661:
2655:
2604:
2598:
2592:
2557:
2551:
2541:
2500:
2496:
2490:
2471:
2465:
2451:
2442:
2406:
2399:
2386:
2379:
2370:
2366:
2360:
2343:
2339:
2333:
2297:
2290:
2237:
2233:
2220:
2182:
2179:
2171:
2161:
2158:
1983:
1979:
1895:Dirac spinor
1888:
1861:
1846:
1838:
1826:
1581:and finally
1580:
1416:
1402:
1317:
1119:
1005:
1002:
906:
892:
881:
874:
871:
757:is given by
752:
667:
429:
247:
150:
121:
103:
95:
78:interference
45:
41:
36:, from
24:
18:
331:Hamiltonian
132:Darwin term
66:wave packet
2754:2402.07672
2670:: 153002.
2212:References
2206:Lamb shift
1883:See also:
1871:and doing
136:s-orbitals
126:. For the
52:that obey
2991:118446413
2966:1109.4771
2903:1105.5963
2873:119353065
2818:118979437
2733:119190847
2716:1303.0914
2677:0711.3270
2614:0909.0674
2533:121875694
2525:1572-9516
2434:124404090
2352:881393652
2325:124404090
2264:0027-8424
2128:ψ
2110:ψ
2104:μ
2100:∂
2094:μ
2088:¯
2085:σ
2067:ψ
2049:ψ
2043:μ
2039:∂
2033:μ
2029:σ
1988:rest mass
1955:ψ
1940:ψ
1930:ψ
1904:ψ
1803:−
1796:ℏ
1777:−
1754:−
1733:−
1715:α
1701:−
1690:ℏ
1661:−
1560:−
1532:ℏ
1513:−
1495:−
1474:−
1456:α
1429:α
1374:γ
1361:γ
1342:≡
1330:σ
1288:α
1284:−
1233:σ
1229:−
1217:γ
1188:α
1141:α
1131:ℏ
1100:ℏ
1084:−
1070:α
1063:ℏ
1025:α
850:α
812:ℏ
693:ℏ
687:−
647:γ
637:γ
624:α
597:γ
590:β
568:μ
564:γ
534:α
513:β
395:α
374:∑
354:β
321:particle
319:fermionic
280:ψ
256:ℏ
209:∂
204:ψ
201:∂
195:ℏ
169:ψ
74:electrons
3050:Category
2936:15507388
2928:22400947
2810:16090266
2639:20054392
2584:20054385
2282:16587362
2228:(1928).
2195:See also
2185:graphene
1853:positron
973:⟩
919:⟨
323:spin-1/2
315:bispinor
46:Bewegung
2971:Bibcode
2908:Bibcode
2853:Bibcode
2790:Bibcode
2647:4322378
2619:Bibcode
2562:Bibcode
2505:Bibcode
2273:1085609
2242:Bibcode
1417:First:
668:In the
501:is the
470:is the
317:) of a
309:is the
268:is the
108:
91:
42:zittern
21:physics
3023:
2989:
2934:
2926:
2871:
2816:
2808:
2731:
2645:
2637:
2600:Nature
2582:
2531:
2523:
2478:
2432:
2422:
2350:
2323:
2313:
2280:
2270:
2262:
2162:zigzag
1318:where
872:where
505:, and
430:where
325:, and
248:where
142:Theory
38:German
23:, the
3013:(pdf)
2987:S2CID
2961:arXiv
2932:S2CID
2898:arXiv
2869:S2CID
2843:arXiv
2814:S2CID
2780:arXiv
2749:arXiv
2729:S2CID
2711:arXiv
2672:arXiv
2643:S2CID
2609:arXiv
2529:S2CID
2430:S2CID
2391:(PDF)
2369:[
2342:[
2321:S2CID
2009:, as
1857:pairs
582:, as
333:of a
40:
3021:ISBN
2924:PMID
2806:PMID
2635:PMID
2580:PMID
2521:ISSN
2476:ISBN
2420:ISBN
2348:OCLC
2311:ISBN
2278:PMID
2260:ISSN
2187:and
1982:and
1978:(or
1410:and
615:and
525:and
2979:doi
2916:doi
2894:108
2861:doi
2798:doi
2721:doi
2682:doi
2668:100
2627:doi
2605:463
2570:doi
2558:463
2513:doi
2412:doi
2303:doi
2268:PMC
2250:doi
1984:zag
1980:zig
1873:QED
1847:In
113:1.6
88:of
19:In
3052::
3015:.
2985:.
2977:.
2969:.
2957:87
2955:.
2930:.
2922:.
2914:.
2906:.
2892:.
2867:.
2859:.
2851:.
2839:51
2837:.
2812:.
2804:.
2796:.
2788:.
2776:94
2774:.
2727:.
2719:.
2707:15
2705:.
2680:.
2666:.
2641:.
2633:.
2625:.
2617:.
2603:.
2578:.
2568:.
2556:.
2550:.
2527:.
2519:.
2511:.
2501:38
2499:.
2428:.
2418:.
2410:.
2319:.
2309:.
2301:.
2276:.
2266:.
2258:.
2248:.
2238:14
2236:.
2232:.
2191:.
1859:.
890:.
665:.
474:,
337::
272:,
138:.
117:10
96:mc
3029:.
2993:.
2981::
2973::
2963::
2938:.
2918::
2910::
2900::
2875:.
2863::
2855::
2845::
2820:.
2800::
2792::
2782::
2757:.
2751::
2735:.
2723::
2713::
2688:.
2684::
2674::
2649:.
2629::
2621::
2611::
2586:.
2572::
2564::
2535:.
2515::
2507::
2484:.
2459:.
2436:.
2414::
2354:.
2327:.
2305::
2284:.
2252::
2244::
2155:.
2133:R
2124:m
2121:=
2115:L
2072:L
2063:m
2060:=
2054:R
2021:{
1997:m
1966:)
1960:R
1951:,
1945:L
1936:(
1933:=
1823:.
1810:)
1806:1
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