QuantumPhysics.dvi

Expanding in powers of the exponential, we find,

Tr

(

U(T)

)

=

∑∞

n=0

e−iTEn/ ̄h En= ̄hω

(

n+

1

2

)

(15.16)

and recover the energy levels of the harmonic oscillator, each one with multiplicity 1.

15.2 The Aharonov-Bohm Effect

In classical electrodynamics, the motion of charged particles is completely determined by the electric
Eand magneticBfield strengths. This is because in classical physics, the equations of motion
completely determine the dynamics, and for charged particles, the force is completely specified by
the Lorentz formula,

F=eE+ev×B (15.17)

Here,Fis the force exerted on a particle of chargeeand velocityv=dx/dt. The gauge potentials
(Φ,A) are usually introduced only as auxiliary variables, when deriving the field equations from a
Lagrangian or from a Hamiltonian. Indeed, these quantitiesare given by

L =

1

2

mv^2 +eA(t,x)·v−eΦ(t,x)

H =

1

2 m

(p−eA(t,x))^2 +eΦ(t,x) (15.18)

We have local gauge invariance, under Φ→Φ−∂Λ/∂tandA→A+∇~Λ. For example, the
Lagrangian transforms as follows,L→L+edΛ/dt.

Figure 17: The Aharonov-Bohm set-up (figure from Sakurai)

In quantum mechanics, the dynamics of charged particles involves the vector potential in a
fundamental way, either in the Hamiltonian operator formulation or in the Lagrangian path integral
formulation. The Aharonov-Bohm effect is one of the most striking examples of the fundamental
role played in quantum mechanics by the vector potential.

The set-up is as in Fig 9. Consider an infinite cylinder along thez-axis, of which only a cross-
section in thex−yplane in shown in Fig 9. The cylinder is impenetrable, so the particle we study