QuantumPhysics.dvi

7 Charged particle in an electro-magnetic field

Of fundamental importance is the problem of an electrically chargedparticle in the presence

of electro-magnetic fields. To construct the Schr ̈odinger equation for this system, we use

the correspondence principle. We start from the classical Lagrangian for a non-relativistic

particle with massm, and electric chargee, in the presence of a general electro-magnetic

field, given in terms of the electric potential Φ and the vector potentialA,

L(r,r ̇,t) =

1

2

mr ̇^2 −eΦ(r,t) +eA(r,t)·r ̇ (7.1)

We allow the electro-magnetic fields to be time-dependent as this would be required, for

example, when dealing with a charged particle in the presence of an electro-magnetic wave.

The momentumpcanonically conjugate toris given by,

p=

∂L

∂r ̇

=mr ̇+eA(r,t) (7.2)

The Hamiltonian is then obtained by eliminatingr ̇ in favor ofpis the expression,

H=p·r ̇−L(r,r ̇,t) (7.3)

This problem is algebraic; it is easily carried out explicitly, and we find,

H(r,p,t) =

1

2 m

(

p−eA(r,t)

) 2

+eΦ(r,t) (7.4)

By the correspondence principle, we promote the classical variablesrandp to operators

obeying canonical commutation relations. The time-dependent Schr ̈odinger equation in the

position realization is then given by,

i ̄h

∂ψ(r,t)

∂t

=

1

2 m

(

−i ̄h∇r−eA(r,t)

) 2

ψ(r,t) +eΦ(r,t)ψ(r,t) (7.5)

In the remainder of this chapter, we proceed to exhibiting the gauge invariance of this

equation, and then studying applications to constant fields, Landau levels, the Aharonov-

Bohm effect, and the quantization conditions for Dirac magnetic monopoles.

7.1 Gauge transformations and gauge invariance

Recall that the electric and magnetic fields are given in terms of Φ andAby

B=∇×A E=−

∂A

∂t

−∇Φ (7.6)