The results obtained by Pauli in his 1926 article on the hydrogen atom made essential use of the dynamical so(4) symmetry of the bound states. Pauli used this symmetry to calculate the perturbed energy levels of an hydrogen atom in a uniform electric field (Stark effect) and in uniform electric and magnetic fields. Although the Stark effect in hydrogen has been studied experimentally, Pauli’s results in mixed fields have been studied only for Rydberg states of rubidium atoms in crossed fields and for lithium atoms in parallel fields.
REFERENCES
1.
E.
Schrödinger
, “The non-relativistic equation of the de Broglie waves
,” Ann. Phys. (N.Y.)
79
, 361
–376
(1926
).2.
W.
Pauli
, “On the hydrogen spectrum from the standpoint of the new quantum mechanics
,” Z. Phys.
36
, 336
–363
(1926
). This article is reprinted in Ref. 3.3.
B. L. Van der Waerden, Sources of Quantum Mechanics (Dover, New York, 1968).
4.
L. D. Landau and E. M. Lifschitz, Quantum Mechanics (Pergamon, New York, 1977).
5.
H.
Goldstein
, “More on the prehistory of the Laplace or Runge-Lenz vector
,” Am. J. Phys.
44
, 1123
–1124
(1976
).6.
As is well known, this analysis is equivalent to solving the Schrödinger equation in parabolic coordinates.
7.
L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968).
8.
W. Greiner and B. Muller, Quantum Mechanics (Springer-Verlag, New York, 1994).
9.
D. Oliver, The Shaggy Steed of Physics (Springer-Verlag, New York, 1994).
10.
This is reminiscent of the calculation of the one-dimensional harmonic oscillator spectrum using creation and annihilation operators instead of solving Schrödinger equation by power series.
11.
H. G.
Becker
and K.
Bleuler
, “O(4)-Symmetrie und Stark-effekt des H Atoms
,” Z. Naturforsch. A
31A
, 517
–523
(1976
).12.
G.
Bonneau
, J.
Faraut
, and G.
Valent
, “Self-adjoint extensions of operators and the teaching of quantum mechanics
,” Am. J. Phys.
69
, 322
–331
(2001
).13.
We point out that is non-Hermitian because we have
14.
We should not use this relation for because in this case we have and this operator is already diagonal. Similarly, for we have
15.
I.
Waller
, “Der Starkeffekt zweiter Ordnung bei Wasserstoff und die Rydbergkorrektion der Spektra von He und
” Z. Phys.
38
, 635
–646
(1926
).16.
G.
Wentzel
, “Eine Verallgemeinerung der Qunatenbedingungen für die Zwecke der Wellenmechanik
,” Z. Phys.
38
, 518
–529
(1926
).17.
P. S.
Epstein
, “The Stark effect from the point of view of Schrödinger’s quantum theory
,” Phys. Rev.
28
, 695
–710
(1926
).18.
E.
Luc-Koenig
and A.
Bachelier
, “Systematic theoretic study of the Stark spectrum of atomic hydrogen I: density of continuum states
,” J. Phys. D
13
, 1743
–1767
(1980
).19.
F.
Penent
, D.
Delande
, F.
Biraben
, and J. C.
Gay
, “Rydberg atoms in crossed electric and magnetic fields. Experimental evidence for Pauli quantization
,” Opt. Commun.
49
, 184
–188
(1984
).20.
P.
Cacciani
, E.
Luc-Koenig
, J.
Pinard
, C.
Thomas
, and S.
Liberman
, “Experimental study and analysis of the lithium atom in presence of parallel electric and magnetic fields
,” Phys. Rev. Lett.
56
, 1467
–1470
(1986
).21.
G.
Raithel
, M.
Fauth
, and H.
Walther
, “Quasi-Landau resonances in the spectra of rubidium Rydberg atoms in crossed electric and magnetic fields
,” Phys. Rev. A
44
, 1898
–1909
(1991
).22.
E. A.
Solov’ev
, “Second-order perturbation theory for a hydrogen atom in crossed electric and magnetic fields
,” Sov. Phys. JETP
58
, 63
–66
(1983
).23.
J.
Main
, M.
Schwacke
, and G.
Wunner
, “Hydrogen atom in combined electric and magnetic fields with arbitrary mutual orientations
,” Phys. Rev. A
57
, 1149
–1157
(1998
).24.
J. L. Basdevant and J. Dalibard, The Quantum Mechanics Solver (Springer-Verlag, New York, 2000).
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