Interacting Charged Particles in Uniform Magnetic Fields

  • Hanns Ruder
  • Günter Wunner
  • Heinz Herold
  • Florian Geyer
Part of the Astronomy and Astrophysics Library book series (AAL)

Abstract

Let us consider a system of N charged particles with charges e i and masses m i (i = 1,..., N) in a uniform magnetic field B. The value of the vector potential A at the position r i of particle i is abbreviated by A i = A(r i ). In this section no special gauge is adopted. The Hamiltonian of the system is given by
$$H = \sum\limits_{i = 1}^N {\frac{1} {{2{m_i}}}{{({p_i} - {e_i}{A_i})}^2} + V({r_1},...,{r_N})}$$
(2.1)
where the potential energy V of the interaction is assumed to be momentum independent and translationally invariant. Generally, the conserved quantity that is associated with the translational invariance of a system, the generalized total momentum, is defined as the generator of infinitesimal translations. In the presence of a magnetic field it must be taken into account that a displacement of the origin changes the vector potential. Therefore, an additional gauge transformation is necessary to preserve the translational invariance of H. Thus we arrive at the following form of the generalized momentum operator:
$${P_{0\mu }} = \sum\limits_{i = 1}^N {\frac{h} {i}{\kern 1pt} \frac{\partial } {{\partial x{i_\mu }}} - {e_i}} \int_{}^{{r_i}} {\frac{{\partial A}} {{\partial {x_\mu }}}.dr} $$
(2.2)
(μ = 1, 2, 3 denotes the Cartesian components).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Avron, J.E., Herbst, I.W., and Simon B. (1978): Separation of center of mass in homogeneous magnetic fields. Ann. Phys. (N.Y.) 114, 431.CrossRefMATHADSMathSciNetGoogle Scholar
  2. Blumberg, W.A.M., Itano, W.M., and Larson, D.J. (1979): Theory of the photodetachment of negative ions in magnetic field. Phys. Rev. D 19, 139.ADSGoogle Scholar
  3. Canuto, V., and Ventura, J. (1977): Quantizing magnetic fields in astrophysics. Fundam. Cosmic Phys. 2, 203.ADSGoogle Scholar
  4. Herold, H., Ruder, H., and Wunner, G. (1981): The two-body problem in the presence of a homogeneous magnetic field. J. Phys. B: At. Mol. Phys. 14, 751.CrossRefADSGoogle Scholar
  5. Johnson, B.R., Hirschfelder, J.O., and Yang, K.-H. (1983): Interaction of atoms, molecules, and ions with constant electric and magnetic fields. Rev. Mod. Phys. 55, 109.CrossRefADSMathSciNetGoogle Scholar
  6. O’Connell, R.F. (1979): Effect of the proton mass on the spectrum of the hydrogen atom in a strong magnetic field. Phys. Lett. 70A, 389.ADSGoogle Scholar
  7. Pavlov-Verevkin, V.B., and Zhilinskii, B.I. (1980a): The hydrogen atom in a superstrong magnetic field. Phys. Lett. 75A, 279.ADSGoogle Scholar
  8. Pavlov-Verevkin, V.B., and Zhilinskii, B.I. (1980b): Neutral hydrogen-like system in a magnetic field. Phys. Lett. 78A, 244.ADSGoogle Scholar
  9. Surmelian, G.L., and O’Connell, R.F. (1974): Energy spectrum of hydrogen-like atoms in a strong magnetic field. Astrophys. J. 190, 741.CrossRefADSGoogle Scholar
  10. Wunner, G., Ruder, H., and Herold, H. (1980): Comment on the effect of the proton mass on the spectrum of the hydrogen atom in a very strong magnetic field. Phys. Lett. 79A, 159.ADSGoogle Scholar
  11. Wunner, G., Ruder, H., and Herold, H. (1981a): Energy levels, electromagnetic transitions and annihilation of positronium in strong magnetic fields. J. Phys. B: At. Mol. Phys. 14, 765.CrossRefADSGoogle Scholar
  12. Wunner, G., Ruder, H., and Herold, H. (1981b): Energy levels and oscillator strengths for the two-body problem in magnetic fields. Astrophys. J. 247, 374.CrossRefADSGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Hanns Ruder
    • 1
  • Günter Wunner
    • 2
  • Heinz Herold
    • 1
  • Florian Geyer
    • 1
  1. 1.Theoretische AstrophysikUniversität TübingenTübingenGermany
  2. 2.Theoretische Physik, Lehrstuhl 1Ruhr-Universität BochumBochumGermany

Personalised recommendations