973:
2608:
104:
20:
701:
1433:
2277:
2855:
275:) of the 2s and 2p states where the electron tends to be to the left, which will acquire a lower energy, and other hybrid orbitals where the electron tends to be to the right, which will acquire a higher energy. Therefore, the formerly degenerate energy levels will split into slightly lower and slightly higher energy levels.
1219:
1585:
3032:
of finite width. These resonances may decay in finite time via field ionization. For low lying states and not too strong fields the decay times are so long, however, that for all practical purposes the system can be regarded as bound. For highly excited states and/or very strong fields ionization may
240:
An electric field pointing from left to right, for example, tends to pull nuclei to the right and electrons to the left. In another way of viewing it, if an electronic state has its electron disproportionately to the left, its energy is lowered, while if it has the electron disproportionately to the
219:
discussed at length the Stark effect in his third paper on quantum theory (in which he introduced his perturbation theory), once in the manner of the 1916 work of
Epstein (but generalized from the old to the new quantum theory) and once by his (first-order) perturbation approach. Finally, Epstein
968:{\displaystyle V_{\mathrm {int} }\approx \phi (\mathbf {0} )\int _{\mathcal {V}}\rho (\mathbf {r} )d^{3}r-\sum _{i=1}^{3}F_{i}\int _{\mathcal {V}}\rho (\mathbf {r} )r_{i}d^{3}r\equiv q\phi (\mathbf {0} )-\sum _{i=1}^{3}\mu _{i}F_{i}=q\phi (\mathbf {0} )-{\boldsymbol {\mu }}\cdot \mathbf {F} ,}
175:
performed classical mechanical calculations of quasi-elastically bound electrons in an electric field. By using experimental indices of refraction he gave an estimate of the Stark splittings. This estimate was a few orders of magnitude too low. Not deterred by this prediction, Stark undertook
2272:
1074:
Turning now to quantum mechanics an atom or a molecule can be thought of as a collection of point charges (electrons and nuclei), so that the second definition of the dipole applies. The interaction of atom or molecule with a uniform external field is described by the operator
2617:
2603:{\displaystyle E_{k}^{(2)}=\sum _{k'\neq k}{\frac {\langle \psi _{k}^{0}|V_{\mathrm {int} }|\psi _{k^{\prime }}^{0}\rangle \langle \psi _{k'}^{0}|V_{\mathrm {int} }|\psi _{k}^{0}\rangle }{E_{k}^{(0)}-E_{k'}^{(0)}}}\equiv -{\frac {1}{2}}\sum _{i,j=1}^{3}\alpha _{ij}F_{i}F_{j}}
1646:
have opposite parity (obtain plus and minus under inversion), because only functions of opposite parity give non-vanishing matrix elements. Degenerate zeroth-order states of opposite parity occur for excited hydrogen-like (one-electron) atoms or
Rydberg states. Neglecting
2980:
481:
2068:. The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent
1464:
668:
1870:
220:
reconsidered the linear and quadratic Stark effect from the point of view of the new quantum theory. He derived equations for the line intensities which were a decided improvement over
Kramers's results obtained by the old quantum theory.
2045:
584:
2086:
1428:{\displaystyle (\mathbf {V} _{\mathrm {int} })_{kl}=\langle \psi _{k}^{0}|V_{\mathrm {int} }|\psi _{l}^{0}\rangle =-\mathbf {F} \cdot \langle \psi _{k}^{0}|{\boldsymbol {\mu }}|\psi _{l}^{0}\rangle ,\qquad k,l=1,\ldots ,g.}
1130:
3212:
Intensities of
Spectral Lines. On the Application of the Quantum Theory to the Problem of Relative Intensities of the Components of the Fine Structure and of the Stark Effect of the Lines of the Hydrogen
1058:
this is no longer true. Nevertheless, it is often justified to omit it in this case, too. Indeed, the Stark effect is observed in spectral lines, which are emitted when an electron "jumps" between two
2871:
389:
85:. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the
1735:
1202:
2850:{\displaystyle \alpha _{ij}=-2\sum _{k'\neq k}{\frac {\langle \psi _{k}^{0}|\mu _{i}|\psi _{k'}^{0}\rangle \langle \psi _{k'}^{0}|\mu _{j}|\psi _{k}^{0}\rangle }{E_{k}^{(0)}-E_{k'}^{(0)}}}.}
1784:
1459:
384:
326:
1930:
589:
494:
1015:
696:
350:
1644:
3010:
227:
theory of the atom, higher-order corrections are not. Measurements of the Stark effect under high field strengths confirmed the correctness of the new quantum theory.
3088:
203:, with corrections for relativistic kinetic energy and coupling between electron spin and orbital motion. The first quantum mechanical treatment (in the framework of
1439:= 1 (as is often the case for electronic states of molecules) the first-order energy becomes proportional to the expectation (average) value of the dipole operator
1910:
1890:
1779:
1755:
1580:{\displaystyle E^{(1)}=-\mathbf {F} \cdot \langle \psi _{1}^{0}|{\boldsymbol {\mu }}|\psi _{1}^{0}\rangle =-\mathbf {F} \cdot \langle {\boldsymbol {\mu }}\rangle .}
3024:
The perturbative treatment of the Stark effect has some problems. In the presence of an electric field, states of atoms and molecules that were previously bound (
1052:
1078:
993:
3346:
3108:
Courtney, Michael; Neal
Spellmeyer; Hong Jiao; Daniel Kleppner (1995). "Classical, semiclassical, and quantum dynamics of lithium in an electric field".
1054:), so the first, monopole, term in the expression above is identically zero. This is also the case for a neutral atom or molecule. However, for an
93:. For most spectral lines, the Stark effect is either linear (proportional to the applied electric field) or quadratic with a high accuracy.
1662:
2267:{\displaystyle H^{(0)}\psi _{k}^{0}=E_{k}^{(0)}\psi _{k}^{0},\quad k=0,1,\ldots ,\quad E_{0}^{(0)}<E_{1}^{(0)}\leq E_{2}^{(0)},\dots }
1604:. Atoms and molecules possessing inversion symmetry do not have a (permanent) dipole moment and hence do not show a linear Stark effect.
1066:
of the radiator but not its charge, the effects of the monopole interaction on the initial and final states exactly cancel each other.
3288:
223:
While the first-order-perturbation (linear) Stark effect in hydrogen is in agreement with both the old BohrâSommerfeld model and the
2868:(which is often justified â unless extremely weak electric fields are considered), the polarizability tensor of atoms is isotropic,
1920:
491:
converges fast, so only a few first terms give an accurate approximation. Namely, keeping only the zero- and first-order terms,
1923:(but not for linear and asymmetric molecules). In first approximation a molecule may be seen as a rigid rotor. A symmetric top
3455:
3428:
3406:
3380:
3356:
156:. The discovery of this effect contributed importantly to the development of quantum theory and Stark was awarded with the
1151:
3171:(Observations of the effect of the electric field on spectral lines I. Transverse effect), Annalen der Physik, vol.
54:
2975:{\displaystyle \alpha _{ij}\equiv \alpha _{0}\delta _{ij}\Longrightarrow E^{(2)}=-{\frac {1}{2}}\alpha _{0}F^{2}.}
476:{\displaystyle V_{\mathrm {int} }=\int _{\mathcal {V}}\rho (\mathbf {r} )\phi (\mathbf {r} )\,d^{3}\mathbf {r} .}
3219:(On the influence of an electric field on the fine structure of hydrogen lines), Zeitschrift fĂŒr Physik, vol.
119:= 0. Note how a complicated pattern of the energy levels emerges as the electric field increases, not unlike
3236:(On the hydrogen spectrum from the point of view of the new quantum mechanics). Zeitschrift fĂŒr Physik, vol.
3068:
96:
The Stark effect can be observed both for emission and absorption lines. The latter is sometimes called the
2081:
663:{\textstyle F_{i}\equiv -\left.\left({\frac {\partial \phi }{\partial r_{i}}}\right)\right|_{\mathbf {0} }}
1442:
3460:
2080:
As stated, the quadratic Stark effect is described by second-order perturbation theory. The zeroth-order
358:
300:
3289:"Locally Excited State-Charge Transfer State Coupled Dyes as Optically Responsive Neuron Firing Probes"
3078:
248:, because the electron is more distant from the nucleus, so it travels farther left and farther right.
148:, who discovered it in 1913. It was independently discovered in the same year by the Italian physicist
3287:
Sirbu, Dumitru; Butcher, John B.; Waddell, Paul G.; Andras, Peter; Benniston, Andrew C. (2017-09-18).
998:
353:
1865:{\displaystyle 16=1+3+5+7\;\;\Longrightarrow \;\;n=4\;{\text{contains}}\;s\oplus p\oplus d\oplus f.}
677:
331:
3175:, pp. 965â983 (1914). Published earlier (1913) in Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss.
3063:
1617:
35:
3450:
2988:
487:
and quantum-mechanically alike. If the potential varies weakly over the charge distribution, the
252:
196:
86:
46:
28:
2069:
157:
199:
derived formulas for intensities of spectral transitions. Kramers also included the effect of
3465:
3046:
272:
268:
1614:
for systems with an inversion center it is necessary that some of the unperturbed functions
3117:
1895:
1875:
1764:
1740:
1208:= 1). According to perturbation theory the first-order energies are the eigenvalues of the
176:
measurements on excited states of the hydrogen atom and succeeded in observing splittings.
2040:{\displaystyle |JKM\rangle =(D_{MK}^{J})^{*}\quad {\text{with}}\quad M,K=-J,-J+1,\dots ,J}
216:
8:
2865:
1133:
1031:
579:{\displaystyle \phi (\mathbf {r} )\approx \phi (\mathbf {0} )-\sum _{i=1}^{3}r_{i}F_{i},}
488:
284:
191:
were independently able to derive equations for the linear and quadratic Stark effect in
184:
3121:
3395:
3370:
3083:
3073:
978:
188:
180:
120:
3201:
K. Schwarzschild, Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss. April 1916, p. 548
3169:
Beobachtungen ĂŒber den Effekt des elektrischen Feldes auf
Spektrallinien I. Quereffekt
3424:
3402:
3376:
3352:
3319:
3311:
3217:Ăber den EinfluĂ eines elektrischen Feldes auf die Feinstruktur der Wasserstofflinien
3133:
3025:
1063:
484:
224:
204:
149:
58:
81:, where a spectral line is split into several components due to the presence of the
23:
Computed energy level spectrum of hydrogen as a function of the electric field near
3303:
3125:
1601:
1018:
208:
128:
124:
90:
3341:
2065:
290:
168:
244:
Other things being equal, the effect of the electric field is greater for outer
3107:
2611:
1648:
294:
264:
260:
245:
212:
200:
172:
145:
82:
74:
50:
1757:
is the azimuthal (angular momentum) quantum number. For instance, the excited
3444:
3315:
3129:
3058:
164:
78:
70:
3323:
3307:
3154:(On the electric analogue of the Zeeman effect), Annalen der Physik, vol.
3034:
1125:{\displaystyle V_{\mathrm {int} }=-\mathbf {F} \cdot {\boldsymbol {\mu }}.}
132:
108:
3137:
1924:
1059:
3271:
The Stark Effect from the Point of View of
Schroedinger's Quantum Theory
2982:
For some molecules this expression is a reasonable approximation, too.
256:
3234:Ăber dass Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik
3029:
1593:
of the first rank), the diagonal elements of the perturbation matrix
1148:-fold degenerate state with orthonormal zeroth-order state functions
3045:
The Stark effect is at the basis of the spectral shift measured for
1028:
Classical macroscopic objects are usually neutral or quasi-neutral (
1132:
This operator is used as a perturbation in first- and second-order
192:
1919:
The first-order Stark effect occurs in rotational transitions of
1590:
1022:
3016:
positive, i.e., the quadratic Stark shift is always negative.
297:. The interaction energy of a continuous charge distribution
89:(Stark broadening) of spectral lines by charged particles in
3388:(Chapter 17 provides a comprehensive treatment, as of 1935.)
100:, but this term is no longer used in the modern literature.
611:
289:
The Stark effect originates from the interaction between a
3368:
103:
73:
of atoms and molecules due to the presence of an external
19:
3286:
1136:
to account for the first- and second-order Stark effect.
1055:
111:-level spectrum as a function of the electric field near
1651:
effects, such a state with the principal quantum number
3340:
3033:
have to be accounted for. (See also the article on the
2274:
is assumed to be solved. The perturbation theory gives
259:, an electron has the same energy whether it is in the
1730:{\displaystyle n^{2}=\sum _{\ell =0}^{n-1}(2\ell +1),}
592:
53:
breaks the degeneracy. Energy levels can cross due to
2991:
2874:
2620:
2280:
2089:
1933:
1912:
are odd under parity. Hence hydrogen-like atoms with
1898:
1878:
1787:
1767:
1743:
1665:
1620:
1467:
1445:
1222:
1154:
1081:
1034:
1001:
981:
704:
680:
497:
392:
361:
334:
303:
267:
states. However, in an electric field, there will be
3049:
used for imaging of the firing activity of neurons.
3347:
A History of the
Theories of Aether and Electricity
1197:{\displaystyle \psi _{1}^{0},\ldots ,\psi _{g}^{0}}
1062:. Since such a transition only alters the internal
3394:
3004:
2974:
2849:
2602:
2266:
2039:
1904:
1884:
1864:
1773:
1749:
1729:
1638:
1579:
1453:
1427:
1196:
1124:
1046:
1009:
987:
967:
690:
662:
578:
475:
378:
344:
320:
3152:Ueber das Elektrische Analogon des Zeemaneffectes
3442:
2051:+1)-fold degenerate energy for |K| > 0 and (2
1589:Because the electric dipole moment is a vector (
293:distribution (atom or molecule) and an external
3265:
3263:
152:, and in Italy it is thus sometimes called the
144:The effect is named after the German physicist
1144:Let the unperturbed atom or molecule be in a
3392:
3260:
2786:
2728:
2725:
2667:
2466:
2400:
2397:
2329:
1948:
1892:are even under parity, while those with odd
1571:
1563:
1546:
1498:
1388:
1340:
1323:
1262:
3418:
3101:
3028:), become formally (non-square-integrable)
77:. It is the electric-field analogue of the
16:Spectral line splitting in electrical field
3369:E. U. Condon & G. H. Shortley (1935).
1837:
1831:
1821:
1820:
1816:
1815:
1017:are, respectively, the total charge (zero
251:The Stark effect can lead to splitting of
2055:+1)-fold degenerate energy for K=0. Here
454:
102:
18:
1567:
1522:
1447:
1364:
1115:
950:
586:where we introduced the electric field
3443:
1204:. (Non-degeneracy is the special case
1069:
278:
3436:(Stark effect for rotating molecules)
3349:. II. The Modern Theories (1800-1950)
1916:>1 show first-order Stark effect.
1607:In order to obtain a non-zero matrix
1600:vanish between states with a certain
698:. Therefore, the interaction becomes
3210:H. A. Kramers, Roy. Danish Academy,
3089:Stark effect in semiconductor optics
1454:{\displaystyle {\boldsymbol {\mu }}}
3364:(Early history of the Stark effect)
379:{\displaystyle \phi (\mathbf {r} )}
321:{\displaystyle \rho (\mathbf {r} )}
13:
3334:
3251:Quantisierung als Eigenwertproblem
2861:gives the quadratic Stark effect.
2440:
2437:
2434:
2385:
2364:
2361:
2358:
1872:The one-electron states with even
1297:
1294:
1291:
1240:
1237:
1234:
1094:
1091:
1088:
819:
746:
717:
714:
711:
683:
628:
620:
420:
405:
402:
399:
337:
328:, confined within a finite volume
179:By the use of the BohrâSommerfeld
14:
3477:
3351:. American Institute of Physics.
1761:= 4 state contains the following
69:is the shifting and splitting of
1927:has the unperturbed eigenstates
1556:
1491:
1333:
1228:
1107:
958:
939:
875:
832:
759:
733:
654:
522:
505:
466:
447:
433:
369:
311:
3280:
3040:
2188:
2163:
2075:
2072:of the symmetric top molecule.
1991:
1985:
1394:
1010:{\displaystyle \mathbf {\mu } }
3375:. Cambridge University Press.
3296:Chemistry - A European Journal
3243:
3226:
3204:
3195:
3178:
3161:
3144:
2928:
2922:
2914:
2836:
2830:
2807:
2801:
2767:
2752:
2701:
2686:
2516:
2510:
2487:
2481:
2447:
2424:
2371:
2348:
2297:
2291:
2253:
2247:
2229:
2223:
2205:
2199:
2140:
2134:
2101:
2095:
1976:
1954:
1935:
1817:
1721:
1706:
1527:
1517:
1479:
1473:
1369:
1359:
1304:
1281:
1247:
1223:
1139:
943:
935:
879:
871:
836:
828:
763:
755:
737:
729:
691:{\displaystyle {\mathcal {V}}}
526:
518:
509:
501:
451:
443:
437:
429:
373:
365:
345:{\displaystyle {\mathcal {V}}}
315:
307:
1:
3186:Zur Theorie des Starkeffektes
3094:
3069:Quantum-confined Stark effect
1639:{\displaystyle \psi _{i}^{0}}
1216:matrix with general element
241:right, its energy is raised.
3456:Foundational quantum physics
3372:The Theory of Atomic Spectra
1025:of the charge distribution.
230:
7:
3401:. Springer-Verlag, Berlin.
3253:, Annalen der Physik, vol.
3188:, Annalen der Physik, vol.
3052:
3019:
3005:{\displaystyle \alpha _{0}}
2610:with the components of the
235:
10:
3482:
3421:Molecular Rotation Spectra
3397:Theoretical Atomic Physics
282:
139:
483:This expression is valid
163:Inspired by the magnetic
3414:(Stark effect for atoms)
3257:Issue 13, 437â490 (1926)
3130:10.1103/PhysRevA.51.3604
253:degenerate energy levels
3342:Edmond Taylor Whittaker
3273:, Physical Review, vol
1921:symmetric top molecules
674:to be somewhere within
670:and assumed the origin
354:electrostatic potential
29:magnetic quantum number
3308:10.1002/chem.201703366
3079:InglisâTeller equation
3047:voltage-sensitive dyes
3006:
2976:
2851:
2604:
2566:
2268:
2070:electric dipole moment
2041:
1906:
1886:
1866:
1775:
1751:
1731:
1705:
1640:
1581:
1455:
1429:
1198:
1126:
1048:
1011:
989:
969:
905:
802:
692:
664:
580:
552:
477:
380:
346:
322:
273:quantum superpositions
255:. For example, in the
181:("old") quantum theory
171:'s explanation of it,
158:Nobel Prize in Physics
136:
62:
3393:H. Friedrich (1990).
3223:, pp. 199â223 (1920)
3158:, pp. 197â208 (1901).
3007:
2985:For the ground state
2977:
2852:
2612:polarizability tensor
2605:
2540:
2269:
2064:is an element of the
2042:
1907:
1905:{\displaystyle \ell }
1887:
1885:{\displaystyle \ell }
1867:
1776:
1774:{\displaystyle \ell }
1752:
1750:{\displaystyle \ell }
1732:
1679:
1659:-fold degenerate and
1641:
1582:
1456:
1430:
1199:
1127:
1049:
1012:
990:
970:
885:
782:
693:
665:
581:
532:
478:
381:
347:
323:
154:StarkâLo Surdo effect
106:
55:underlying symmetries
22:
3419:H. W. Kroto (1992).
3277:, pp. 695â710 (1926)
3192:, pp. 489â520 (1916)
3064:AutlerâTownes effect
2989:
2872:
2618:
2278:
2087:
1931:
1896:
1876:
1785:
1765:
1741:
1663:
1618:
1465:
1443:
1220:
1152:
1079:
1032:
999:
979:
702:
678:
590:
495:
390:
359:
332:
301:
263:state or any of the
195:. Four years later,
167:, and especially by
98:inverse Stark effect
49:; application of an
47:degenerate sublevels
3423:. Dover, New York.
3302:(58): 14639â14649.
3122:1995PhRvA..51.3604C
2866:hyperfine structure
2840:
2811:
2785:
2750:
2724:
2684:
2520:
2491:
2465:
2422:
2396:
2346:
2301:
2257:
2233:
2209:
2159:
2144:
2120:
1974:
1635:
1545:
1515:
1387:
1357:
1322:
1279:
1193:
1169:
1134:perturbation theory
1070:Perturbation theory
1047:{\displaystyle q=0}
489:multipole expansion
352:, with an external
285:Multipole expansion
279:Multipole expansion
87:pressure broadening
3461:Physical phenomena
3084:Electric field NMR
3074:Stark spectroscopy
3002:
2972:
2847:
2815:
2791:
2771:
2731:
2705:
2670:
2663:
2600:
2495:
2471:
2451:
2403:
2375:
2332:
2325:
2281:
2264:
2237:
2213:
2189:
2145:
2124:
2106:
2037:
1957:
1902:
1882:
1862:
1771:
1747:
1727:
1636:
1621:
1577:
1531:
1501:
1451:
1425:
1373:
1343:
1308:
1265:
1194:
1179:
1155:
1122:
1064:degrees of freedom
1044:
1007:
985:
965:
688:
660:
576:
473:
376:
342:
318:
225:quantum-mechanical
189:Karl Schwarzschild
160:in the year 1919.
137:
63:
3430:978-0-486-67259-5
3408:978-0-387-54179-2
3382:978-0-521-09209-8
3358:978-0-88318-523-0
3110:Physical Review A
3026:square-integrable
2947:
2842:
2643:
2538:
2522:
2305:
1989:
1835:
988:{\displaystyle q}
642:
217:Erwin Schrödinger
205:Werner Heisenberg
150:Antonino Lo Surdo
129:dynamical systems
59:Coulomb potential
57:of motion in the
3473:
3434:
3412:
3400:
3386:
3362:
3328:
3327:
3293:
3284:
3278:
3267:
3258:
3249:E. Schrödinger,
3247:
3241:
3230:
3224:
3215:, p. 287 (1919);
3208:
3202:
3199:
3193:
3182:
3176:
3165:
3159:
3148:
3142:
3141:
3116:(5): 3604â3620.
3105:
3011:
3009:
3008:
3003:
3001:
3000:
2981:
2979:
2978:
2973:
2968:
2967:
2958:
2957:
2948:
2940:
2932:
2931:
2913:
2912:
2900:
2899:
2887:
2886:
2856:
2854:
2853:
2848:
2843:
2841:
2839:
2828:
2827:
2810:
2799:
2789:
2784:
2779:
2770:
2765:
2764:
2755:
2749:
2744:
2743:
2723:
2718:
2717:
2704:
2699:
2698:
2689:
2683:
2678:
2665:
2662:
2655:
2633:
2632:
2609:
2607:
2606:
2601:
2599:
2598:
2589:
2588:
2579:
2578:
2565:
2560:
2539:
2531:
2523:
2521:
2519:
2508:
2507:
2490:
2479:
2469:
2464:
2459:
2450:
2445:
2444:
2443:
2427:
2421:
2416:
2415:
2395:
2390:
2389:
2388:
2374:
2369:
2368:
2367:
2351:
2345:
2340:
2327:
2324:
2317:
2300:
2289:
2273:
2271:
2270:
2265:
2256:
2245:
2232:
2221:
2208:
2197:
2158:
2153:
2143:
2132:
2119:
2114:
2105:
2104:
2046:
2044:
2043:
2038:
1990:
1987:
1984:
1983:
1973:
1968:
1938:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1871:
1869:
1868:
1863:
1836:
1833:
1780:
1778:
1777:
1772:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1704:
1693:
1675:
1674:
1645:
1643:
1642:
1637:
1634:
1629:
1586:
1584:
1583:
1578:
1570:
1559:
1544:
1539:
1530:
1525:
1520:
1514:
1509:
1494:
1483:
1482:
1460:
1458:
1457:
1452:
1450:
1434:
1432:
1431:
1426:
1386:
1381:
1372:
1367:
1362:
1356:
1351:
1336:
1321:
1316:
1307:
1302:
1301:
1300:
1284:
1278:
1273:
1258:
1257:
1245:
1244:
1243:
1231:
1203:
1201:
1200:
1195:
1192:
1187:
1168:
1163:
1131:
1129:
1128:
1123:
1118:
1110:
1099:
1098:
1097:
1053:
1051:
1050:
1045:
1016:
1014:
1013:
1008:
1006:
994:
992:
991:
986:
974:
972:
971:
966:
961:
953:
942:
925:
924:
915:
914:
904:
899:
878:
858:
857:
848:
847:
835:
824:
823:
822:
812:
811:
801:
796:
775:
774:
762:
751:
750:
749:
736:
722:
721:
720:
697:
695:
694:
689:
687:
686:
669:
667:
666:
661:
659:
658:
657:
651:
647:
643:
641:
640:
639:
626:
618:
602:
601:
585:
583:
582:
577:
572:
571:
562:
561:
551:
546:
525:
508:
482:
480:
479:
474:
469:
464:
463:
450:
436:
425:
424:
423:
410:
409:
408:
385:
383:
382:
377:
372:
351:
349:
348:
343:
341:
340:
327:
325:
324:
319:
314:
209:matrix mechanics
3481:
3480:
3476:
3475:
3474:
3472:
3471:
3470:
3441:
3440:
3431:
3409:
3383:
3359:
3337:
3335:Further reading
3332:
3331:
3291:
3285:
3281:
3269:P. S. Epstein,
3268:
3261:
3248:
3244:
3231:
3227:
3209:
3205:
3200:
3196:
3184:P. S. Epstein,
3183:
3179:
3166:
3162:
3149:
3145:
3106:
3102:
3097:
3055:
3043:
3022:
2996:
2992:
2990:
2987:
2986:
2963:
2959:
2953:
2949:
2939:
2921:
2917:
2905:
2901:
2895:
2891:
2879:
2875:
2873:
2870:
2869:
2864:Neglecting the
2829:
2820:
2819:
2800:
2795:
2790:
2780:
2775:
2766:
2760:
2756:
2751:
2745:
2736:
2735:
2719:
2710:
2709:
2700:
2694:
2690:
2685:
2679:
2674:
2666:
2664:
2648:
2647:
2625:
2621:
2619:
2616:
2615:
2594:
2590:
2584:
2580:
2571:
2567:
2561:
2544:
2530:
2509:
2500:
2499:
2480:
2475:
2470:
2460:
2455:
2446:
2433:
2432:
2428:
2423:
2417:
2408:
2407:
2391:
2384:
2380:
2379:
2370:
2357:
2356:
2352:
2347:
2341:
2336:
2328:
2326:
2310:
2309:
2290:
2285:
2279:
2276:
2275:
2246:
2241:
2222:
2217:
2198:
2193:
2154:
2149:
2133:
2128:
2115:
2110:
2094:
2090:
2088:
2085:
2084:
2078:
2066:Wigner D-matrix
2063:
1986:
1979:
1975:
1969:
1961:
1934:
1932:
1929:
1928:
1897:
1894:
1893:
1877:
1874:
1873:
1832:
1786:
1783:
1782:
1766:
1763:
1762:
1742:
1739:
1738:
1694:
1683:
1670:
1666:
1664:
1661:
1660:
1630:
1625:
1619:
1616:
1615:
1613:
1599:
1566:
1555:
1540:
1535:
1526:
1521:
1516:
1510:
1505:
1490:
1472:
1468:
1466:
1463:
1462:
1446:
1444:
1441:
1440:
1382:
1377:
1368:
1363:
1358:
1352:
1347:
1332:
1317:
1312:
1303:
1290:
1289:
1285:
1280:
1274:
1269:
1250:
1246:
1233:
1232:
1227:
1226:
1221:
1218:
1217:
1188:
1183:
1164:
1159:
1153:
1150:
1149:
1142:
1114:
1106:
1087:
1086:
1082:
1080:
1077:
1076:
1072:
1033:
1030:
1029:
1002:
1000:
997:
996:
980:
977:
976:
957:
949:
938:
920:
916:
910:
906:
900:
889:
874:
853:
849:
843:
839:
831:
818:
817:
813:
807:
803:
797:
786:
770:
766:
758:
745:
744:
740:
732:
710:
709:
705:
703:
700:
699:
682:
681:
679:
676:
675:
653:
652:
635:
631:
627:
619:
617:
613:
610:
609:
597:
593:
591:
588:
587:
567:
563:
557:
553:
547:
536:
521:
504:
496:
493:
492:
465:
459:
455:
446:
432:
419:
418:
414:
398:
397:
393:
391:
388:
387:
368:
360:
357:
356:
336:
335:
333:
330:
329:
310:
302:
299:
298:
287:
281:
269:hybrid orbitals
246:electron shells
238:
233:
197:Hendrik Kramers
169:Hendrik Lorentz
142:
17:
12:
11:
5:
3479:
3469:
3468:
3463:
3458:
3453:
3451:Atomic physics
3439:
3438:
3429:
3416:
3407:
3390:
3381:
3366:
3357:
3336:
3333:
3330:
3329:
3279:
3259:
3242:
3225:
3203:
3194:
3177:
3160:
3143:
3099:
3098:
3096:
3093:
3092:
3091:
3086:
3081:
3076:
3071:
3066:
3061:
3054:
3051:
3042:
3039:
3021:
3018:
2999:
2995:
2971:
2966:
2962:
2956:
2952:
2946:
2943:
2938:
2935:
2930:
2927:
2924:
2920:
2916:
2911:
2908:
2904:
2898:
2894:
2890:
2885:
2882:
2878:
2846:
2838:
2835:
2832:
2826:
2823:
2818:
2814:
2809:
2806:
2803:
2798:
2794:
2788:
2783:
2778:
2774:
2769:
2763:
2759:
2754:
2748:
2742:
2739:
2734:
2730:
2727:
2722:
2716:
2713:
2708:
2703:
2697:
2693:
2688:
2682:
2677:
2673:
2669:
2661:
2658:
2654:
2651:
2646:
2642:
2639:
2636:
2631:
2628:
2624:
2597:
2593:
2587:
2583:
2577:
2574:
2570:
2564:
2559:
2556:
2553:
2550:
2547:
2543:
2537:
2534:
2529:
2526:
2518:
2515:
2512:
2506:
2503:
2498:
2494:
2489:
2486:
2483:
2478:
2474:
2468:
2463:
2458:
2454:
2449:
2442:
2439:
2436:
2431:
2426:
2420:
2414:
2411:
2406:
2402:
2399:
2394:
2387:
2383:
2378:
2373:
2366:
2363:
2360:
2355:
2350:
2344:
2339:
2335:
2331:
2323:
2320:
2316:
2313:
2308:
2304:
2299:
2296:
2293:
2288:
2284:
2263:
2260:
2255:
2252:
2249:
2244:
2240:
2236:
2231:
2228:
2225:
2220:
2216:
2212:
2207:
2204:
2201:
2196:
2192:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2162:
2157:
2152:
2148:
2142:
2139:
2136:
2131:
2127:
2123:
2118:
2113:
2109:
2103:
2100:
2097:
2093:
2077:
2074:
2059:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1982:
1978:
1972:
1967:
1964:
1960:
1956:
1953:
1950:
1947:
1944:
1941:
1937:
1901:
1881:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1830:
1827:
1824:
1819:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1770:
1746:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1703:
1700:
1697:
1692:
1689:
1686:
1682:
1678:
1673:
1669:
1649:fine-structure
1633:
1628:
1624:
1611:
1597:
1576:
1573:
1569:
1565:
1562:
1558:
1554:
1551:
1548:
1543:
1538:
1534:
1529:
1524:
1519:
1513:
1508:
1504:
1500:
1497:
1493:
1489:
1486:
1481:
1478:
1475:
1471:
1449:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1393:
1390:
1385:
1380:
1376:
1371:
1366:
1361:
1355:
1350:
1346:
1342:
1339:
1335:
1331:
1328:
1325:
1320:
1315:
1311:
1306:
1299:
1296:
1293:
1288:
1283:
1277:
1272:
1268:
1264:
1261:
1256:
1253:
1249:
1242:
1239:
1236:
1230:
1225:
1191:
1186:
1182:
1178:
1175:
1172:
1167:
1162:
1158:
1141:
1138:
1121:
1117:
1113:
1109:
1105:
1102:
1096:
1093:
1090:
1085:
1071:
1068:
1043:
1040:
1037:
1005:
984:
964:
960:
956:
952:
948:
945:
941:
937:
934:
931:
928:
923:
919:
913:
909:
903:
898:
895:
892:
888:
884:
881:
877:
873:
870:
867:
864:
861:
856:
852:
846:
842:
838:
834:
830:
827:
821:
816:
810:
806:
800:
795:
792:
789:
785:
781:
778:
773:
769:
765:
761:
757:
754:
748:
743:
739:
735:
731:
728:
725:
719:
716:
713:
708:
685:
656:
650:
646:
638:
634:
630:
625:
622:
616:
612:
608:
605:
600:
596:
575:
570:
566:
560:
556:
550:
545:
542:
539:
535:
531:
528:
524:
520:
517:
514:
511:
507:
503:
500:
472:
468:
462:
458:
453:
449:
445:
442:
439:
435:
431:
428:
422:
417:
413:
407:
404:
401:
396:
375:
371:
367:
364:
339:
317:
313:
309:
306:
295:electric field
283:Main article:
280:
277:
237:
234:
232:
229:
213:Wolfgang Pauli
201:fine structure
173:Woldemar Voigt
146:Johannes Stark
141:
138:
83:magnetic field
75:electric field
71:spectral lines
51:electric field
15:
9:
6:
4:
3:
2:
3478:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3448:
3446:
3437:
3432:
3426:
3422:
3417:
3415:
3410:
3404:
3399:
3398:
3391:
3389:
3384:
3378:
3374:
3373:
3367:
3365:
3360:
3354:
3350:
3348:
3343:
3339:
3338:
3325:
3321:
3317:
3313:
3309:
3305:
3301:
3297:
3290:
3283:
3276:
3272:
3266:
3264:
3256:
3252:
3246:
3240:p. 336 (1926)
3239:
3235:
3229:
3222:
3218:
3214:
3207:
3198:
3191:
3187:
3181:
3174:
3170:
3164:
3157:
3153:
3147:
3139:
3135:
3131:
3127:
3123:
3119:
3115:
3111:
3104:
3100:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3062:
3060:
3059:Zeeman effect
3057:
3056:
3050:
3048:
3038:
3036:
3031:
3027:
3017:
3015:
2997:
2993:
2983:
2969:
2964:
2960:
2954:
2950:
2944:
2941:
2936:
2933:
2925:
2918:
2909:
2906:
2902:
2896:
2892:
2888:
2883:
2880:
2876:
2867:
2862:
2860:
2844:
2833:
2824:
2821:
2816:
2812:
2804:
2796:
2792:
2781:
2776:
2772:
2761:
2757:
2746:
2740:
2737:
2732:
2720:
2714:
2711:
2706:
2695:
2691:
2680:
2675:
2671:
2659:
2656:
2652:
2649:
2644:
2640:
2637:
2634:
2629:
2626:
2622:
2614:α defined by
2613:
2595:
2591:
2585:
2581:
2575:
2572:
2568:
2562:
2557:
2554:
2551:
2548:
2545:
2541:
2535:
2532:
2527:
2524:
2513:
2504:
2501:
2496:
2492:
2484:
2476:
2472:
2461:
2456:
2452:
2429:
2418:
2412:
2409:
2404:
2392:
2381:
2376:
2353:
2342:
2337:
2333:
2321:
2318:
2314:
2311:
2306:
2302:
2294:
2286:
2282:
2261:
2258:
2250:
2242:
2238:
2234:
2226:
2218:
2214:
2210:
2202:
2194:
2190:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2160:
2155:
2150:
2146:
2137:
2129:
2125:
2121:
2116:
2111:
2107:
2098:
2091:
2083:
2073:
2071:
2067:
2062:
2058:
2054:
2050:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1980:
1970:
1965:
1962:
1958:
1951:
1945:
1942:
1939:
1926:
1922:
1917:
1915:
1899:
1879:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1828:
1825:
1822:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1768:
1760:
1744:
1724:
1718:
1715:
1712:
1709:
1701:
1698:
1695:
1690:
1687:
1684:
1680:
1676:
1671:
1667:
1658:
1654:
1650:
1631:
1626:
1622:
1610:
1605:
1603:
1596:
1592:
1587:
1574:
1560:
1552:
1549:
1541:
1536:
1532:
1511:
1506:
1502:
1495:
1487:
1484:
1476:
1469:
1438:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1391:
1383:
1378:
1374:
1353:
1348:
1344:
1337:
1329:
1326:
1318:
1313:
1309:
1286:
1275:
1270:
1266:
1259:
1254:
1251:
1215:
1211:
1207:
1189:
1184:
1180:
1176:
1173:
1170:
1165:
1160:
1156:
1147:
1137:
1135:
1119:
1111:
1103:
1100:
1083:
1067:
1065:
1061:
1057:
1041:
1038:
1035:
1026:
1024:
1023:dipole moment
1020:
1003:
982:
962:
954:
946:
932:
929:
926:
921:
917:
911:
907:
901:
896:
893:
890:
886:
882:
868:
865:
862:
859:
854:
850:
844:
840:
825:
814:
808:
804:
798:
793:
790:
787:
783:
779:
776:
771:
767:
752:
741:
726:
723:
706:
673:
648:
644:
636:
632:
623:
614:
606:
603:
598:
594:
573:
568:
564:
558:
554:
548:
543:
540:
537:
533:
529:
515:
512:
498:
490:
486:
470:
460:
456:
440:
426:
415:
411:
394:
362:
355:
304:
296:
292:
286:
276:
274:
271:(also called
270:
266:
262:
258:
254:
249:
247:
242:
228:
226:
221:
218:
214:
210:
206:
202:
198:
194:
190:
186:
182:
177:
174:
170:
166:
165:Zeeman effect
161:
159:
155:
151:
147:
134:
130:
127:in classical
126:
125:closed orbits
122:
118:
114:
110:
105:
101:
99:
94:
92:
88:
84:
80:
79:Zeeman effect
76:
72:
68:
60:
56:
52:
48:
44:
40:
38:
33:
30:
26:
21:
3466:Spectroscopy
3435:
3420:
3413:
3396:
3387:
3371:
3363:
3345:
3299:
3295:
3282:
3274:
3270:
3254:
3250:
3245:
3237:
3233:
3228:
3220:
3216:
3211:
3206:
3197:
3189:
3185:
3180:
3172:
3168:
3163:
3155:
3151:
3146:
3113:
3109:
3103:
3044:
3041:Applications
3035:Rydberg atom
3023:
3013:
2984:
2863:
2858:
2082:eigenproblem
2079:
2076:Second order
2060:
2056:
2052:
2048:
1918:
1913:
1758:
1656:
1652:
1608:
1606:
1594:
1588:
1436:
1213:
1209:
1205:
1145:
1143:
1073:
1060:bound states
1027:
671:
288:
250:
243:
239:
222:
185:Paul Epstein
178:
162:
153:
143:
121:bifurcations
116:
112:
97:
95:
67:Stark effect
66:
64:
42:
41:consists of
36:
31:
24:
2857:The energy
1925:rigid rotor
1140:First order
485:classically
131:leading to
34:= 0. Each
3445:Categories
3232:W. Pauli,
3167:J. Stark,
3150:W. Voigt,
3095:References
3030:resonances
1021:) and the
257:Bohr model
3316:0947-6539
2994:α
2951:α
2937:−
2915:⟹
2903:δ
2893:α
2889:≡
2877:α
2813:−
2787:⟩
2773:ψ
2758:μ
2733:ψ
2729:⟨
2726:⟩
2707:ψ
2692:μ
2672:ψ
2668:⟨
2657:≠
2645:∑
2638:−
2623:α
2569:α
2542:∑
2528:−
2525:≡
2493:−
2467:⟩
2453:ψ
2405:ψ
2401:⟨
2398:⟩
2386:′
2377:ψ
2334:ψ
2330:⟨
2319:≠
2307:∑
2262:…
2235:≤
2183:…
2147:ψ
2108:ψ
2029:…
2014:−
2005:−
1981:∗
1949:⟩
1900:ℓ
1880:ℓ
1854:⊕
1848:⊕
1842:⊕
1818:⟹
1769:ℓ
1745:ℓ
1713:ℓ
1699:−
1685:ℓ
1681:∑
1623:ψ
1572:⟩
1568:μ
1564:⟨
1561:⋅
1553:−
1547:⟩
1533:ψ
1523:μ
1503:ψ
1499:⟨
1496:⋅
1488:−
1448:μ
1414:…
1389:⟩
1375:ψ
1365:μ
1345:ψ
1341:⟨
1338:⋅
1330:−
1324:⟩
1310:ψ
1267:ψ
1263:⟨
1181:ψ
1174:…
1157:ψ
1116:μ
1112:⋅
1104:−
1004:μ
955:⋅
951:μ
947:−
933:ϕ
908:μ
887:∑
883:−
869:ϕ
863:≡
826:ρ
815:∫
784:∑
780:−
753:ρ
742:∫
727:ϕ
724:≈
629:∂
624:ϕ
621:∂
607:−
604:≡
534:∑
530:−
516:ϕ
513:≈
499:ϕ
441:ϕ
427:ρ
416:∫
363:ϕ
305:ρ
231:Mechanism
211:) was by
115:= 15 for
27:= 15 for
3344:(1987).
3324:28833695
3213:Spectrum
3053:See also
3020:Problems
2825:′
2741:′
2715:′
2653:′
2505:′
2413:′
2315:′
2047:with 2(2
1834:contains
1781:states,
236:Overview
193:hydrogen
107:Lithium
3138:9912027
3118:Bibcode
140:History
109:Rydberg
91:plasmas
3427:
3405:
3379:
3355:
3322:
3314:
3136:
3014:always
1737:where
1602:parity
1591:tensor
1019:moment
975:where
291:charge
3292:(PDF)
133:chaos
39:level
3425:ISBN
3403:ISBN
3377:ISBN
3353:ISBN
3320:PMID
3312:ISSN
3134:PMID
2211:<
1988:with
995:and
386:is
187:and
65:The
45:â 1
3304:doi
3255:385
3156:309
3126:doi
3037:).
3012:is
1655:is
1612:int
1598:int
1435:If
1056:ion
207:'s
123:of
3447::
3318:.
3310:.
3300:23
3298:.
3294:.
3275:28
3262:^
3238:36
3190:50
3173:43
3132:.
3124:.
3114:51
3112:.
2061:MK
1789:16
1461:,
1212:Ă
265:2p
261:2s
215:.
183:,
135:.
3433:.
3411:.
3385:.
3361:.
3326:.
3306::
3221:3
3140:.
3128::
3120::
2998:0
2970:.
2965:2
2961:F
2955:0
2945:2
2942:1
2934:=
2929:)
2926:2
2923:(
2919:E
2910:j
2907:i
2897:0
2884:j
2881:i
2859:E
2845:.
2837:)
2834:0
2831:(
2822:k
2817:E
2808:)
2805:0
2802:(
2797:k
2793:E
2782:0
2777:k
2768:|
2762:j
2753:|
2747:0
2738:k
2721:0
2712:k
2702:|
2696:i
2687:|
2681:0
2676:k
2660:k
2650:k
2641:2
2635:=
2630:j
2627:i
2596:j
2592:F
2586:i
2582:F
2576:j
2573:i
2563:3
2558:1
2555:=
2552:j
2549:,
2546:i
2536:2
2533:1
2517:)
2514:0
2511:(
2502:k
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2488:)
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2482:(
2477:k
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2462:0
2457:k
2448:|
2441:t
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2419:0
2410:k
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2322:k
2312:k
2303:=
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2292:(
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2259:,
2254:)
2251:0
2248:(
2243:2
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2206:)
2203:0
2200:(
2195:0
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2186:,
2180:,
2177:1
2174:,
2171:0
2168:=
2165:k
2161:,
2156:0
2151:k
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2138:0
2135:(
2130:k
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2122:=
2117:0
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2099:0
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2092:H
2057:D
2053:J
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2035:J
2032:,
2026:,
2023:1
2020:+
2017:J
2011:,
2008:J
2002:=
1999:K
1996:,
1993:M
1977:)
1971:J
1966:K
1963:M
1959:D
1955:(
1952:=
1946:M
1943:K
1940:J
1936:|
1914:n
1860:.
1857:f
1851:d
1845:p
1839:s
1829:4
1826:=
1823:n
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1810:+
1807:5
1804:+
1801:3
1798:+
1795:1
1792:=
1759:n
1725:,
1722:)
1719:1
1716:+
1710:2
1707:(
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1691:0
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1677:=
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1632:0
1627:i
1609:V
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1575:.
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1550:=
1542:0
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1477:1
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1420:g
1417:,
1411:,
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1399:,
1396:k
1392:,
1384:0
1379:l
1370:|
1360:|
1354:0
1349:k
1334:F
1327:=
1319:0
1314:l
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1298:t
1295:n
1292:i
1287:V
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1276:0
1271:k
1260:=
1255:l
1252:k
1248:)
1241:t
1238:n
1235:i
1229:V
1224:(
1214:g
1210:g
1206:g
1190:0
1185:g
1177:,
1171:,
1166:0
1161:1
1146:g
1120:.
1108:F
1101:=
1095:t
1092:n
1089:i
1084:V
1042:0
1039:=
1036:q
983:q
963:,
959:F
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940:0
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930:q
927:=
922:i
918:F
912:i
902:3
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894:=
891:i
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876:0
872:(
866:q
860:r
855:3
851:d
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833:r
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820:V
809:i
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799:3
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791:=
788:i
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772:3
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760:r
756:(
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730:(
718:t
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538:i
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523:0
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506:r
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471:.
467:r
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444:(
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412:=
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338:V
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312:r
308:(
117:m
113:n
61:.
43:n
37:n
32:m
25:n
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