1021:
1329:) can be applied to Landau–Zener systems which are coupled to baths composed of infinite many oscillators and/or spin baths (dissipative Landau–Zener transitions). They provide exact expressions for transition probabilities averaged over final bath states if the evolution begins from the ground state at zero temperature, see in Ref. for oscillator baths and for universal results including spin baths in Ref.
75:
728:
25:
1016:{\displaystyle {\begin{aligned}P_{\rm {D}}&=e^{-2\pi \Gamma }\\\Gamma &={a^{2}/\hbar \over \left|{\frac {\partial }{\partial t}}(E_{2}-E_{1})\right|}={a^{2}/\hbar \over \left|{\frac {dq}{dt}}{\frac {\partial }{\partial q}}(E_{2}-E_{1})\right|}\\&={a^{2} \over \hbar |\alpha |}\end{aligned}}}
1303:
Landau–Zener transitions in infinite linear chains. This class contains the systems with formally infinite number of interacting states. Although most known their instances can be obtained as limits of the finite size models (such as the Tavis–Cummings model), there are also cases that do not belong
1182:
Demkov–Osherov model that describes a single level that crosses a band of parallel levels. A surprising fact about the solution of this model is coincidence of the exactly obtained transition probability matrix with its form obtained with a simple semiclassical independent crossing approximation.
181:
The first simplification makes this a semi-classical treatment. In the case of an atom in a magnetic field, the field strength becomes a classical variable which can be precisely measured during the transition. This requirement is quite restrictive as a linear change will not, in general, be the
1313:
Applications of the Landau–Zener solution to the problems of quantum state preparation and manipulation with discrete degrees of freedom stimulated the study of noise and decoherence effects on the transition probability in a driven two-state system. Several compact analytical results have been
1209:
with a bosonic mode in a linearly time-dependent magnetic field. This is the richest known solved system. It has combinatorial complexity: the dimension of its state vector space is growing exponentially with the number of spins N. The transition probabilities in this model are described by the
140:
If the system starts, in the infinite past, in the lower energy eigenstate, we wish to calculate the probability of finding the system in the upper energy eigenstate in the infinite future (a so-called Landau–Zener transition). For infinitely slow variation of the energy difference (that is, a
1165:
matrices with time-independent elements. The goal of the multistate Landau–Zener theory is to determine elements of the scattering matrix and the transition probabilities between states of this model after evolution with such a
Hamiltonian from negative infinite to positive infinite time. The
595:
1317:
Using the
Schwinger–Keldysh Green's function, a rather complete and comprehensive study on the effect of quantum noise in all parameter regimes were performed by Ao and Rammer in late 1980s, from weak to strong coupling, low to high temperature, slow to fast passage, etc. Concise analytical
1169:
There are exact formulas, called hierarchy constraints, that provide analytical expressions for special elements of the scattering matrix in any multi-state Landau–Zener model. Special cases of these relations are known as the
Brundobler–Elser (BE) formula,), and the
1299:
are constant parameters. This is the earliest known solvable system, which was discussed by
Majorana in 1932. Among the other examples there are models of a pair of degenerate level crossing, and the 1D quantum Ising chain in a linearly changing magnetic
145:
tells us that no such transition will take place, as the system will always be in an instantaneous eigenstate of the
Hamiltonian at that moment in time. At non-zero velocities, transitions occur with probability as described by the Landau–Zener formula.
1318:
expressions were obtained in various limits, showing the rich behaviors of such problem. The effects of nuclear spin bath and heat bath coupling on the Landau–Zener process was explored by
Sinitsyn and Prokof'ev and Pokrovsky and Sun, respectively.
1217:
Reducible (or composite) multistate Landau–Zener models. This class consists of systems that can be decoupled to subsets of other solvable and simpler models by a symmetry transformation. The notable example is an arbitrary spin
Hamiltonian
1177:
There are also integrability conditions that, when they are satisfied, lead to exact expressions for the entire scattering matrices in multistate Landau–Zener models. Numerous such completely solvable models have been identified, including:
1213:
Spin clusters interacting with time-dependent magnetic fields. This class of models shows relatively complex behavior of the transition probabilities due to the path interference effects in the semiclassical independent crossing
417:
The details of Zener's solution are somewhat opaque, relying on a set of substitutions to put the equation of motion into the form of the Weber equation and using the known solution. A more transparent solution is provided by
435:
166:
fields. In order that the equations of motion for the system might be solved analytically, a set of simplifications are made, known collectively as the Landau–Zener approximation. The simplifications are as follows:
266:
1314:
derived to describe these effects, including the
Kayanuma formula for a strong diagonal noise, and Pokrovsky–Sinitsyn formula for the coupling to a fast colored noise with off-diagonal components.
733:
1186:
Generalized bow-tie model. The model describes coupling of two (or one in the degenerate case limit) levels to a set of otherwise noninteracting diabatic states that cross at a single point.
43:
1271:
688:
1088:
720:
374:
The final simplification requires that the time–dependent perturbation does not couple the diabatic states; rather, the coupling must be due to a static deviation from a
1143:
341:
305:
1050:
of the two-level system's
Hamiltonian coupling the bases, and as such it is half the distance between the two unperturbed eigenenergies at the avoided crossing, when
365:
656:
629:
400:
1044:
1401:
2174:
R. Malla; V. Y. Chernyak; C. Sun; N. A. Sinitsyn (2022). "Coherent
Reaction between Molecular and Atomic Bose-Einstein Condensates: Integrable Model".
590:{\displaystyle v_{\rm {LZ}}={{\frac {\partial }{\partial t}}|E_{2}-E_{1}| \over {\frac {\partial }{\partial q}}|E_{2}-E_{1}|}\approx {\frac {dq}{dt}},}
3079:
K. Saito; M. Wubs; S. Kohler; Y. Kayanuma; P. Hanggi (2007). "Dissipative Landau-Zener transitions of a qubit: Bath-specific and universal behavior".
1990:
Yu. N. Demkov; V. I. Osherov (1968). "Stationary and nonstationary problems in quantum mechanics that can be solved by means of contour integration".
2695:
154:
Such transitions occur between states of the entire system; hence any description of the system must include all external influences, including
1543:
1338:
2995:
D. Sun; A. Abanov; V. L. Pokrovsky (2009). "Static and Dynamic properties of a Fermi-gas of cooled atoms near a wide Feshbach resonance".
1183:
With some generalizations, this property appears in almost all solvable Landau–Zener systems with a finite number of interacting states.
3016:
M. Wubs; K. Saito; S. Kohler; P. Hanggi; Y. Kayanuma (2006). "Gauging a quantum heat bath with dissipative Landau-Zener transitions".
2019:
Yu. N. Demkov; V. N. Ostrovsky (2001). "The exact solution of the multistate Landau–Zener type model: the generalized bow-tie model".
61:
191:
113:
varying such that the energy separation of the two states is a linear function of time. The formula, giving the probability of a
2940:
D. Sun; A. Abanov; V. L. Pokrovsky (2008). "Molecular production at a broad Feshbach resonance in a Fermi gas of cooled atoms".
1688:
B. Dobrescu; N. A. Sinitsyn (2006). "Comment on 'Exact results for survival probability in the multistate Landau–Zener model'".
1561:
1380:
604:
is the perturbation variable (electric or magnetic field, molecular bond-length, or any other perturbation to the system), and
367:
is a constant. For the case of an atom in a magnetic field this corresponds to a linear change in magnetic field. For a linear
110:
423:
1780:
N. A. Sinitsyn (2004). "Counterintuitive transitions in the multistate Landau–Zener problem with linear level crossings".
1304:
to this classification. For example, there are solvable infinite chains with nonzero couplings between non-nearest states.
1174:,. Discrete symmetries often lead to constraints that reduce the number of independent elements of the scattering matrix.
1596:
N. A. Sinitsyn; J. Lin; V. Y. Chernyak (2017). "Constraints on scattering amplitudes in multistate Landau-Zener theory".
2693:
Y. Kayanuma (1984). "Nonadiabatic Transitions in Level Crossing with Energy Fluctuation. I. Analytical Investigations".
1210:
q-deformed binomial statistics. This solution has found practical applications in physics of Bose-Einstein condensates.
86:(which may be vary in time). The dashed lines represent the energies of the diabatic states, which cross each other at
1835:
M. V. Volkov; V. N. Ostrovsky (2005). "No-go theorem for bands of potential curves in multistate Landau–Zener model".
1098:
The simplest generalization of the two-state Landau–Zener model is a multistate system with a Hamiltonian of the form
1743:
M. V. Volkov; V. N. Ostrovsky (2004). "Exact results for survival probability in the multistate Landau–Zener model".
3140:
2119:
C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution".
1358:
106:
2840:
N. A. Sinitsyn; N. Prokof'ev (2003). "Nuclear spin bath effects on Landau–Zener transitions in nanomagnets".
3018:
2514:
2176:
114:
2512:
J. Dziarmaga (2005). "Dynamics of a Quantum Phase Transition: Exact Solution of the Quantum Ising Model".
1454:
2402:
G. S. Vasilev; S. S. Ivanov; N. V. Vitanov (2007). "Degenerate Landau-Zener model: Analytical solution".
2457:
R. W. Cherng; L. S. Levitov (2006). "Entropy and correlation functions of a driven quantum spin chain".
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2140:
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N. A. Sinitsyn; F. Li (2016). "Solvable multistate model of Landau-Zener transitions in cavity QED".
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2001:
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1047:
1880:
N. A. Sinitsyn (2015). "Exact results for models of multichannel quantum nonadiabatic transitions".
93:, and the full lines represent the energy of the adiabatic states (eigenvalues of the Hamiltonian).
3145:
2347:
A. Patra; E. A. Yuzbashyan (2015). "Quantum integrability in the multistate Landau–Zener problem".
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P. Ao; J. Rammer (1991). "Quantum Dynamics of a Two-State System in a Dissipative Environment".
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V. Y. Chernyak; N. A. Sinitsyn; C. Sun (2019). "Dynamic spin localization and gamma-magnets".
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N. A. Sinitsyn (2002). "Multiparticle Landau–Zener problem: Application to quantum dots".
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F. Li; N. A. Sinitsyn (2016). "Dynamic Symmetries and Quantum Nonadiabatic Transitions".
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transition probabilities are the absolute value squared of scattering matrix elements.
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406:
163:
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126:
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V. L. Pokrovsky; N. A. Sinitsyn (2002). "Landau–Zener transitions in a linear chain".
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V. L. Pokrovsky; D. Sun (2007). "Fast quantum noise in the Landau–Zener transition".
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1968:
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E. C. G. Stueckelberg (1932). "Theorie der unelastischen Stösse zwischen Atomen".
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V. L. Pokrovsky; N. A. Sinitsyn (2004). "Fast noise in the Landau–Zener theory".
1651:
S. Brundobler; V. Elser (1993). "S-matrix for generalized Landau–Zener problem".
171:
The perturbation parameter in the Hamiltonian is a known, linear function of time
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419:
16:
Formula for the probability that a system will change between two energy states.
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121:) transition between the two energy states, was published separately by
1504:
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The key figure of merit in this approach is the Landau–Zener velocity:
1574:
177:
The coupling in the diabatic Hamiltonian matrix is independent of time
174:
The energy separation of the diabatic states varies linearly with time
1430:
1481:
E. Majorana (1932). "Atomi orientati in campo magnetico variabile".
82:. The graph represents the energies of the system along a parameter
2253:
2190:
2135:
2080:
1951:
1612:
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results in a large diabatic transition probability and vice versa.
3001:
2956:
2591:
2418:
2363:
1896:
105:
to the equations of motion governing the transition dynamics of a
73:
658:
are the energies of the two diabatic (crossing) states. A large
347:, given by the diagonal elements of the Hamiltonian matrix, and
182:
optimal profile to achieve the desired transition probability.
18:
2575:
N. A. Sinitsyn (2013). "Landau-Zener Transitions in Chains".
185:
The second simplification allows us to make the substitution
1399:
C. Zener (1932). "Non-Adiabatic Crossing of Energy Levels".
1378:
L. Landau (1932). "Zur Theorie der Energieubertragung. II".
261:{\displaystyle \Delta E=E_{2}(t)-E_{1}(t)\equiv \alpha t,\,}
39:
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Driven Tavis–Cummings model describes interaction of
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may be too technical for most readers to understand
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1321:Exact results in multistate Landau–Zener theory (
693:Using the Landau–Zener formula the probability,
1559:C. Wittig (2005). "The Landau–Zener Formula".
8:
1402:Proceedings of the Royal Society of London A
1534:(9 ed.). Dover Publications. pp.
343:are the energies of the two states at time
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62:Learn how and when to remove this message
46:, without removing the technical details.
2696:Journal of the Physical Society of Japan
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722:, of a diabatic transition is given by
1526:Abramowitz, M.; I. A. Stegun (1976).
44:make it understandable to non-experts
7:
1339:Nonadiabatic transition state theory
371:this follows directly from point 1.
141:Landau–Zener velocity of zero), the
1530:Handbook of Mathematical Functions
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1969:10.1016/j.chemphys.2016.05.029
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3050:10.1103/PhysRevLett.97.200404
2546:10.1103/PhysRevLett.95.245701
1266:{\textstyle H=gS_{x}+btS_{z}}
683:{\displaystyle v_{\rm {LZ}}}
150:Conditions and approximation
2271:10.1103/PhysRevB.100.224304
2043:10.1088/0953-4075/34/12/309
1814:10.1088/0305-4470/37/44/016
1759:10.1088/0953-4075/37/20/003
1083:{\displaystyle E_{1}=E_{2}}
715:{\displaystyle P_{\rm {D}}}
3162:
3113:10.1103/PhysRevB.75.214308
2974:10.1209/0295-5075/83/16003
2911:10.1103/PhysRevB.76.024310
2864:10.1103/PhysRevB.67.134403
2765:10.1103/PhysRevB.67.144303
2664:10.1103/PhysRevB.65.153105
2609:10.1103/PhysRevA.87.032701
2491:10.1103/PhysRevA.73.043614
2436:10.1103/PhysRevA.75.013417
2326:10.1103/PhysRevB.66.205303
2153:10.1103/PhysRevA.94.033808
2098:10.1103/PhysRevA.93.063859
1914:10.1103/PhysRevA.90.062509
1859:10.1088/0953-4075/38/7/011
1722:10.1088/0953-4075/39/5/N01
1675:10.1088/0305-4470/26/5/037
1630:10.1103/PhysRevA.95.012140
405:, commonly described by a
1138:{\displaystyle H(t)=A+Bt}
2819:10.1103/PhysRevB.43.5397
1359:Froissart-Stora equation
1287:are spin operators, and
336:{\displaystyle E_{2}(t)}
300:{\displaystyle E_{1}(t)}
109:, with a time-dependent
107:two-state quantum system
3019:Physical Review Letters
2515:Physical Review Letters
2177:Physical Review Letters
360:{\displaystyle \alpha }
1455:Helvetica Physica Acta
1423:10.1098/rspa.1932.0165
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651:{\displaystyle E_{2}}
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624:{\displaystyle E_{1}}
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2350:Journal of Physics A
2022:Journal of Physics B
1838:Journal of Physics B
1783:Journal of Physics A
1746:Journal of Physics B
1691:Journal of Physics B
1654:Journal of Physics A
1468:10.5169/seals-110177
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1048:off-diagonal element
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99:Landau–Zener formula
3105:2007PhRvB..75u4308S
3042:2006PhRvL..97t0404W
2966:2008EL.....8316003S
2903:2007PhRvB..76b4310P
2856:2003PhRvB..67m4403S
2811:1991PhRvB..43.5397A
2757:2003PhRvB..67n4303P
2717:10.1143/JPSJ.53.108
2709:1984JPSJ...53..108K
2656:2002PhRvB..65o3105P
2601:2013PhRvA..87c2701S
2538:2005PhRvL..95x5701D
2483:2006PhRvA..73d3614C
2428:2007PhRvA..75a3417V
2373:2015JPhA...48x5303P
2318:2002PhRvB..66t5303S
2263:2019PhRvB.100v4304C
2200:2022PhRvL.129c3201M
2145:2016PhRvA..94c3808S
2090:2016PhRvA..93f3859S
2035:2001JPhB...34.2419D
2006:1968JETP...26..916D
1993:Soviet Physics JETP
1961:2016CP....481...28L
1906:2014PhRvA..90f2509S
1851:2005JPhB...38..907V
1806:2004JPhA...3710691S
1790:(44): 10691–10697.
1714:2006JPhB...39.1253D
1667:1993JPhA...26.1211B
1622:2017PhRvA..95a2140S
1497:1932NCim....9...43M
1415:1932RSPSA.137..696Z
424:contour integration
395:{\displaystyle 1/r}
1505:10.1007/BF02960953
1263:
1135:
1094:Multistate problem
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712:
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131:Ernst Stueckelberg
95:
3141:Quantum mechanics
3082:Physical Review B
2880:Physical Review B
2843:Physical Review B
2798:Physical Review B
2734:Physical Review B
2633:Physical Review B
2578:Physical Review A
2460:Physical Review A
2405:Physical Review A
2295:Physical Review B
2240:Physical Review B
2122:Physical Review A
2067:Physical Review A
1883:Physical Review A
1599:Physical Review A
1575:10.1021/jp040627u
1569:(17): 8428–8430.
1545:978-0-486-61272-0
1344:Adiabatic theorem
1039:{\displaystyle a}
1007:
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403:Coulomb potential
143:adiabatic theorem
103:analytic solution
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2683:
2649:
2647:cond-mat/0112419
2627:
2621:
2620:
2594:
2572:
2566:
2565:
2531:
2529:cond-mat/0509490
2509:
2503:
2502:
2476:
2474:cond-mat/0512689
2454:
2448:
2447:
2421:
2399:
2393:
2392:
2366:
2344:
2338:
2337:
2311:
2309:cond-mat/0212017
2289:
2283:
2282:
2256:
2234:
2228:
2227:
2193:
2171:
2165:
2164:
2138:
2116:
2110:
2109:
2083:
2061:
2055:
2054:
2016:
2010:
2009:
1987:
1981:
1980:
1954:
1938:Chemical Physics
1932:
1926:
1925:
1899:
1877:
1871:
1870:
1832:
1826:
1825:
1799:
1797:quant-ph/0403113
1777:
1771:
1770:
1740:
1734:
1733:
1707:
1705:cond-mat/0505571
1685:
1679:
1678:
1648:
1642:
1641:
1615:
1593:
1587:
1586:
1556:
1550:
1549:
1533:
1523:
1517:
1516:
1484:Il Nuovo Cimento
1478:
1472:
1471:
1449:
1443:
1442:
1396:
1390:
1389:
1375:
1272:
1270:
1269:
1264:
1262:
1261:
1243:
1242:
1208:
1206:
1205:
1202:
1199:
1144:
1142:
1141:
1136:
1089:
1087:
1086:
1081:
1079:
1078:
1066:
1065:
1045:
1043:
1042:
1037:
1022:
1020:
1019:
1014:
1012:
1008:
1006:
1005:
997:
988:
987:
978:
970:
966:
964:
960:
956:
955:
943:
942:
930:
928:
917:
915:
913:
905:
897:
890:
886:
881:
880:
870:
865:
863:
859:
855:
854:
842:
841:
829:
827:
816:
809:
805:
800:
799:
789:
773:
772:
747:
746:
745:
721:
719:
718:
713:
711:
710:
709:
689:
687:
686:
681:
679:
678:
677:
657:
655:
654:
649:
647:
646:
630:
628:
627:
622:
620:
619:
603:
596:
594:
593:
588:
583:
581:
573:
565:
560:
558:
557:
552:
551:
539:
538:
529:
524:
522:
511:
508:
507:
502:
501:
489:
488:
479:
474:
472:
461:
458:
453:
452:
451:
401:
399:
398:
393:
388:
366:
364:
363:
358:
346:
342:
340:
339:
334:
323:
322:
306:
304:
303:
298:
287:
286:
267:
265:
264:
259:
235:
234:
213:
212:
80:avoided crossing
67:
60:
56:
53:
47:
27:
26:
19:
3161:
3160:
3156:
3155:
3154:
3152:
3151:
3150:
3131:
3130:
3129:
3128:
3078:
3077:
3073:
3015:
3014:
3010:
2994:
2993:
2989:
2939:
2938:
2934:
2876:
2875:
2871:
2839:
2838:
2834:
2794:
2792:
2788:
2730:
2728:
2724:
2692:
2691:
2687:
2629:
2628:
2624:
2574:
2573:
2569:
2511:
2510:
2506:
2456:
2455:
2451:
2401:
2400:
2396:
2346:
2345:
2341:
2291:
2290:
2286:
2236:
2235:
2231:
2173:
2172:
2168:
2118:
2117:
2113:
2063:
2062:
2058:
2018:
2017:
2013:
1989:
1988:
1984:
1934:
1933:
1929:
1879:
1878:
1874:
1834:
1833:
1829:
1779:
1778:
1774:
1742:
1741:
1737:
1687:
1686:
1682:
1650:
1649:
1645:
1595:
1594:
1590:
1558:
1557:
1553:
1546:
1525:
1524:
1520:
1480:
1479:
1475:
1451:
1450:
1446:
1398:
1397:
1393:
1377:
1376:
1372:
1367:
1335:
1311:
1285:
1278:
1253:
1234:
1220:
1219:
1203:
1200:
1197:
1196:
1194:
1103:
1102:
1096:
1070:
1057:
1052:
1051:
1028:
1027:
1010:
1009:
989:
979:
968:
967:
947:
934:
921:
906:
898:
895:
891:
872:
871:
846:
833:
820:
814:
810:
791:
790:
781:
775:
774:
755:
748:
736:
727:
726:
700:
695:
694:
665:
660:
659:
638:
633:
632:
611:
606:
605:
601:
574:
566:
543:
530:
515:
509:
493:
480:
465:
459:
439:
434:
433:
415:
376:
375:
349:
348:
344:
314:
309:
308:
278:
273:
272:
226:
204:
190:
189:
152:
135:Ettore Majorana
92:
68:
57:
51:
48:
40:help improve it
37:
28:
24:
17:
12:
11:
5:
3159:
3157:
3149:
3148:
3143:
3133:
3132:
3127:
3126:
3089:(21): 214308.
3071:
3026:(20): 200404.
3008:
2987:
2932:
2869:
2850:(13): 134403.
2832:
2786:
2741:(14): 045603.
2722:
2703:(1): 108–117.
2685:
2640:(15): 153105.
2622:
2567:
2522:(24): 245701.
2504:
2449:
2394:
2357:(24): 245303.
2339:
2302:(20): 205303.
2284:
2247:(22): 224304.
2229:
2166:
2111:
2056:
2011:
1982:
1927:
1872:
1827:
1772:
1735:
1680:
1643:
1606:(1): 0112140.
1588:
1551:
1544:
1518:
1473:
1444:
1409:(6): 696–702.
1391:
1369:
1368:
1366:
1363:
1362:
1361:
1356:
1354:Bond hardening
1351:
1349:Bond softening
1346:
1341:
1334:
1331:
1310:
1309:Study of noise
1307:
1306:
1305:
1301:
1283:
1276:
1260:
1256:
1252:
1249:
1246:
1241:
1237:
1233:
1230:
1227:
1215:
1214:approximation.
1211:
1187:
1184:
1157:are Hermitian
1147:
1146:
1134:
1131:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1095:
1092:
1077:
1073:
1069:
1064:
1060:
1035:
1024:
1023:
1004:
1000:
996:
992:
986:
982:
976:
973:
971:
969:
963:
959:
954:
950:
946:
941:
937:
933:
927:
924:
920:
912:
909:
904:
901:
894:
889:
885:
879:
875:
868:
862:
858:
853:
849:
845:
840:
836:
832:
826:
823:
819:
813:
808:
804:
798:
794:
787:
784:
782:
780:
777:
776:
771:
768:
765:
762:
758:
754:
751:
749:
744:
739:
735:
734:
708:
703:
676:
673:
668:
645:
641:
618:
614:
598:
597:
586:
580:
577:
572:
569:
563:
556:
550:
546:
542:
537:
533:
528:
521:
518:
514:
506:
500:
496:
492:
487:
483:
478:
471:
468:
464:
456:
450:
447:
442:
414:
411:
407:quantum defect
391:
387:
383:
356:
332:
329:
326:
321:
317:
296:
293:
290:
285:
281:
269:
268:
256:
253:
250:
247:
244:
241:
238:
233:
229:
225:
222:
219:
216:
211:
207:
203:
200:
197:
179:
178:
175:
172:
151:
148:
127:Clarence Zener
90:
70:
69:
52:September 2018
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3158:
3147:
3144:
3142:
3139:
3138:
3136:
3122:
3118:
3114:
3110:
3106:
3102:
3097:
3092:
3088:
3084:
3083:
3075:
3072:
3067:
3063:
3059:
3055:
3051:
3047:
3043:
3039:
3034:
3029:
3025:
3021:
3020:
3012:
3009:
3003:
2998:
2991:
2988:
2983:
2979:
2975:
2971:
2967:
2963:
2958:
2953:
2949:
2945:
2944:
2936:
2933:
2928:
2924:
2920:
2919:1969.1/127339
2916:
2912:
2908:
2904:
2900:
2895:
2890:
2887:(2): 024310.
2886:
2882:
2881:
2873:
2870:
2865:
2861:
2857:
2853:
2849:
2845:
2844:
2836:
2833:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2799:
2790:
2787:
2782:
2778:
2774:
2773:1969.1/127315
2770:
2766:
2762:
2758:
2754:
2749:
2744:
2740:
2736:
2735:
2726:
2723:
2718:
2714:
2710:
2706:
2702:
2698:
2697:
2689:
2686:
2681:
2677:
2673:
2672:1969.1/146790
2669:
2665:
2661:
2657:
2653:
2648:
2643:
2639:
2635:
2634:
2626:
2623:
2618:
2614:
2610:
2606:
2602:
2598:
2593:
2588:
2585:(3): 032701.
2584:
2580:
2579:
2571:
2568:
2563:
2559:
2555:
2551:
2547:
2543:
2539:
2535:
2530:
2525:
2521:
2517:
2516:
2508:
2505:
2500:
2496:
2492:
2488:
2484:
2480:
2475:
2470:
2467:(4): 043614.
2466:
2462:
2461:
2453:
2450:
2445:
2441:
2437:
2433:
2429:
2425:
2420:
2415:
2412:(1): 013417.
2411:
2407:
2406:
2398:
2395:
2390:
2386:
2382:
2378:
2374:
2370:
2365:
2360:
2356:
2352:
2351:
2343:
2340:
2335:
2331:
2327:
2323:
2319:
2315:
2310:
2305:
2301:
2297:
2296:
2288:
2285:
2280:
2276:
2272:
2268:
2264:
2260:
2255:
2250:
2246:
2242:
2241:
2233:
2230:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2192:
2187:
2184:(3): 033201.
2183:
2179:
2178:
2170:
2167:
2162:
2158:
2154:
2150:
2146:
2142:
2137:
2132:
2129:(3): 033808.
2128:
2124:
2123:
2115:
2112:
2107:
2103:
2099:
2095:
2091:
2087:
2082:
2077:
2074:(6): 063859.
2073:
2069:
2068:
2060:
2057:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2023:
2015:
2012:
2007:
2003:
1999:
1995:
1994:
1986:
1983:
1978:
1974:
1970:
1966:
1962:
1958:
1953:
1948:
1944:
1940:
1939:
1931:
1928:
1923:
1919:
1915:
1911:
1907:
1903:
1898:
1893:
1890:(7): 062509.
1889:
1885:
1884:
1876:
1873:
1868:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1839:
1831:
1828:
1823:
1819:
1815:
1811:
1807:
1803:
1798:
1793:
1789:
1785:
1784:
1776:
1773:
1768:
1764:
1760:
1756:
1752:
1748:
1747:
1739:
1736:
1731:
1727:
1723:
1719:
1715:
1711:
1706:
1701:
1697:
1693:
1692:
1684:
1681:
1676:
1672:
1668:
1664:
1660:
1656:
1655:
1647:
1644:
1639:
1635:
1631:
1627:
1623:
1619:
1614:
1609:
1605:
1601:
1600:
1592:
1589:
1584:
1580:
1576:
1572:
1568:
1564:
1563:
1555:
1552:
1547:
1541:
1537:
1532:
1531:
1522:
1519:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1485:
1477:
1474:
1469:
1465:
1461:
1457:
1456:
1448:
1445:
1440:
1436:
1432:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1403:
1395:
1392:
1387:
1383:
1382:
1374:
1371:
1364:
1360:
1357:
1355:
1352:
1350:
1347:
1345:
1342:
1340:
1337:
1336:
1332:
1330:
1328:
1324:
1323:no-go theorem
1319:
1315:
1308:
1302:
1298:
1294:
1290:
1286:
1279:
1258:
1254:
1250:
1247:
1244:
1239:
1235:
1231:
1228:
1225:
1216:
1212:
1192:
1188:
1185:
1181:
1180:
1179:
1175:
1173:
1172:no-go theorem
1167:
1164:
1160:
1156:
1152:
1132:
1129:
1126:
1123:
1120:
1114:
1108:
1101:
1100:
1099:
1093:
1091:
1075:
1071:
1067:
1062:
1058:
1049:
1033:
1026:The quantity
998:
984:
980:
974:
972:
961:
952:
948:
944:
939:
935:
925:
910:
907:
902:
899:
892:
883:
877:
873:
866:
860:
851:
847:
843:
838:
834:
824:
811:
802:
796:
792:
785:
783:
766:
763:
760:
756:
752:
750:
737:
725:
724:
723:
701:
691:
666:
643:
639:
616:
612:
584:
578:
575:
570:
567:
561:
548:
544:
540:
535:
531:
519:
498:
494:
490:
485:
481:
469:
454:
440:
432:
431:
430:
427:
425:
421:
412:
410:
408:
404:
389:
385:
381:
372:
370:
354:
327:
319:
315:
291:
283:
279:
254:
251:
248:
245:
239:
231:
227:
223:
217:
209:
205:
201:
198:
188:
187:
186:
183:
176:
173:
170:
169:
168:
165:
161:
158:and external
157:
149:
147:
144:
138:
136:
132:
128:
124:
120:
116:
112:
108:
104:
100:
89:
85:
81:
78:Sketch of an
76:
66:
63:
55:
45:
41:
35:
32:This article
30:
21:
20:
3086:
3080:
3074:
3023:
3017:
3011:
2990:
2950:(1): 16003.
2947:
2941:
2935:
2884:
2878:
2872:
2847:
2841:
2835:
2802:
2796:
2789:
2738:
2732:
2725:
2700:
2694:
2688:
2637:
2631:
2625:
2582:
2576:
2570:
2519:
2513:
2507:
2464:
2458:
2452:
2409:
2403:
2397:
2354:
2348:
2342:
2299:
2293:
2287:
2244:
2238:
2232:
2181:
2175:
2169:
2126:
2120:
2114:
2071:
2065:
2059:
2029:(12): 2419.
2026:
2020:
2014:
1997:
1991:
1985:
1942:
1936:
1930:
1887:
1881:
1875:
1842:
1836:
1830:
1787:
1781:
1775:
1753:(20): 4069.
1750:
1744:
1738:
1695:
1689:
1683:
1658:
1652:
1646:
1603:
1597:
1591:
1566:
1560:
1554:
1529:
1521:
1491:(2): 43–50.
1488:
1482:
1476:
1459:
1453:
1447:
1406:
1400:
1394:
1385:
1379:
1373:
1320:
1316:
1312:
1296:
1292:
1288:
1281:
1274:
1190:
1176:
1168:
1162:
1158:
1154:
1150:
1148:
1097:
1025:
692:
599:
428:
416:
373:
369:Zeeman shift
270:
184:
180:
153:
139:
98:
96:
87:
83:
58:
49:
33:
2793:Table I in
1698:(5): 1253.
1661:(5): 1211.
420:Curt Wittig
137:, in 1932.
111:Hamiltonian
3146:Lev Landau
3135:Categories
2729:Eq. 42 in
2254:1905.05287
2191:2112.12302
2136:1606.08430
2081:1602.03136
1952:1604.00106
1845:(7): 907.
1613:1609.06285
1365:References
1327:BE-formula
156:collisions
123:Lev Landau
3002:0902.2178
2957:0707.3630
2617:119321544
2592:1212.2907
2499:115915571
2419:0909.5396
2389:117049526
2364:1412.4926
2334:119101393
2279:153312716
2224:245425087
2161:119317114
2106:119331736
2051:250846731
1977:119167653
1945:: 28–33.
1922:119211541
1897:1411.4307
1867:122560197
1767:250804220
1730:118943836
1513:122738040
1439:120348552
1291:>1/2;
999:α
991:ℏ
945:−
923:∂
919:∂
888:ℏ
844:−
822:∂
818:∂
807:ℏ
779:Γ
770:Γ
767:π
761:−
562:≈
541:−
517:∂
513:∂
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