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Motion in an Electromagnetic Field

  • Franz Schwabl
Part of the Advanced Texts in Physics book series (ADTP)

Abstract

We now consider a particle of mass m and charge e in an electromagnetic field. The representation of the field by the vector potential A and the scalar potential Φ
$$ E = - \frac{1}{c}\,\frac{{\partial A}}{{\partial t}} - \nabla \Phi \;;\;B = \nabla \times A $$
(7.1)
and the classical Hamiltonian
$$ H = \frac{1}{{2m}}{\left( {p - \frac{e}{c}A\left( {x,t} \right)} \right)^2} + e\Phi \left( {x,t} \right)$$
(7.2)
is known from electrodynamics. By the correspondence principle (section 2.5.1), the replacement of p by the momentum operator turns (7.2) into the Hamiltonian operator, and the time dependent Schrödinger equation takes the form
$$ i\hbar \frac{\partial }{{\partial t}}\psi = \left[ {\frac{1}{{2m}}{{\left( {\frac{\hbar }{i}\nabla - \frac{e}{c}A} \right)}^2} + e\Phi } \right]\psi $$
(7.3)

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Franz Schwabl
    • 1
  1. 1.Physik-DepartmentTechnische Universität MünchenGarchingGermany

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