On the spectrum of the hydrogen atom in an ultrastrong magnetic field
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On the Spectrum of the Hydrogen Atom in an Ultrastrong Magnetic Field V. S. Popova and B. M. Karnakovb a
Institute of Theoretical and Experimental Physics, Moscow, 117218 Russia National Research Nuclear University “MEPhI,” Moscow, 115409 Russia email: [email protected]
b
Received April 27, 2011
Abstract—Various approaches to computing the energies of the ground state and excited levels of the hydro gen atom in an ultrastrong magnetic field B that considerably exceeds the field Ba = m e e3c/3 ~ 109 G are considered. The effects of polarization of vacuum and anomalous magnetic moment of the electron on the position of the atomic levels are discussed. The vacuum polarization effects are negligibly weak for B < 1015 G but become significant in fields B 1016 G, in which these effects qualitatively modify the atomic spectrum in this range. The difference in the behaviors of the even and odd energy levels for B Ba is analyzed and the formulas for the energies of odd levels as a function of field B are refined. DOI: 10.1134/S1063776111160060 2
1. INTRODUCTION The quantummechanical problem of the spec trum of the hydrogen atom in a magnetic field B Ba,
lar momentum of the electron on the direction of the 2 3
magnetic field), Ba = m e e = 2.35 × 109 G is the atomic unit of the magnetic field strength, Ᏼ = B/Ba is the dimensionless reduced field, Bcr = α–2Ba = 4.41 × 1013 G is the critical (or Schwinger) field in quantum electrodynamics [21–23], aB = 1/mee2 is the Bohr radius, aH = (eB)–1/2 is the magnetic length (or the Landau radius), ωL = eB/me is the Larmor or cyclotron frequency, and α = e2 = 1/137 (in Gaussian system of units). In addition, we can write the following rela tions:
2
where Ba = m e e3c/3 ~ 109 G, is of considerable inter est for the astrophysics of neutron stars, viz., pulsars with fields B ~ 1011–1013 G [1, 2] and magnetars, which form a special class of neutron stars in which magnetic fields may reach recordhigh values up to 1015 G [3]. This problem has been considered by many authors [4–17] beginning with the pioneering work by Schiff and Snyder [4], who introduced the adiabatic approximation that was subsequently employed by all authors. Since the variables in the Schrödinger equa tions cannot be separated, various numerical methods were used (see [6–13] and the literature cited therein); analytic formulas for energy levels are also available [5, 13–17]. The question arises of the accuracy of various approximations and the field of their application, which has been elucidated insufficiently and which forms the subject of this work. We will also consider the effect of polarization of the vacuum and anomalous magnetic moment of the electron on the position of the atomic levels, as well as the recently discovered effect of “freezing” of energies of even energy levels in ultrastrong fields B 1016 G [18–20]. More exact asymptotic expansions for the energies of odd levels in the hydrogen atom will be obtained and the qualitative difference in the behaviors of
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