# 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).

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## References

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© 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