Motion in an Electromagnetic Field

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))