Gauge Transformations

Assume the Hamiltonian of a system is


and the state vector of the system at t=t0 is |y(t0)>.  Then the state vector a time t is

The zero of the potential energy can be chosen arbitrarily.  Assume we add a constant V0 to the potential energy.  Then the state vector a time t is

We have

This is a very simple example of a class of transformations, called gauge transformations.  There are no observable effects if every state vector in the world is multiplied by a common factor .  All matrix elements and expectation values remain unchanged.  Only phase differences are observable.  Our adding V0 to the potential energy amounts to changing the total energy of the system, but only energy differences are observable.

Consider a particle in a uniform gravitational field.  The gravitational potential energy is mfG, and the classical equation of motion is


The mass term drops out of the equation of motion.  In quantum mechanics the mass term no longer drops out. It appears in the combination Image4384.gif (901 bytes).  We have

Does this lead to observable effects?
Ehrenfest’s theorem gives us the equation for the trajectory of the center of the wave packet describing the particle,


The combination does not appear here.  The appearance of in the Schroedinger equation does not lead to observable changes in the trajectory of a particle moving under the influence of gravity.  However, purely quantum mechanical effects are often observed through interference effects.


Assume a beam of thermal neutrons can move from point A to point B along two different path in the apparatus shown in the figure.  Assume the apparatus is tilted such that the normal to the plane shown makes an angle q with the z-axis.  Assume Find the intensity of the neutron beam at B as a function of the tilt angle q.

Gauge transformations in electromagnetism

The Hamiltonian of a particle with charge e and mass m moving in an electromagnetic field is (in Gaussian units)

Here p is the canonical momentum of the particle, The mechanical momentum is  The electric and magnetic fields


are invariant under the gauge transformation

where f(r,t) is some scalar field.  Under a gauge transformation the Schroedinger equation,




If  then



and the original equation is restored.

For the Schroedinger equation to remain unchanged, the wave function y(r,t) must change into This corresponds to a phase change, which varies from one point to another, not a global phase factor.  For the state vector we have

In order for the behavior of an observable A to be invariant under a gauge transformation we need

A true physical quantity is an observable for which this is true.  The observable r is a true physical quantity.  For the observable p we have

GTpG = GTpG - GTGp +p = GT[p,G] + p


We used  

The canonical momentum p is gauge dependent and thus is not a true physical quantity.  However the mechanical momentum is gauge invariant. 

The Aharonov-Bohm effect

Suppose the magnetic field is in a cylindrical region of radius R and zero everywhere else.

The vector potential A is not zero outside the cylindrical region. 

Assume the cylindrical region of radius R is inaccessible to a charged particle of mass m and charge e.  It never penetrates the region with the magnetic field.  Is the motion of the particle affected by the presence of the magnetic field?

If there is no field present, the Hamiltonian of the particle is


while in the presence of the magnetic field it is


If we write then the wave function y in the presence of the field is given in terms of the no field wave function y by


Here S denotes the path that the charged particle follows.  By following a path through a region where the field is zero but the vector potential is not zero the charged particle’s wave function acquires an additional phase


Again this phase change can be observed through interference effects.

Suppose a beam of electrons can reach a detector along two different path in a double slit experiment

The additional difference in the phases of y1 and y2 is



we have


Here F is the magnetic flux.  The additional phase difference displaces the interference pattern that is observed as a function of position on the detector.  This Aharonov-Bohm effect has been observed.  It is a purely quantum mechanical effect.  Classically the motion of the particle is determined by Newton's laws and the Lorentz force is zero in regions where B is zero.  In quantum mechanics the phase of the wave function depends on the vector potential A.  However the observable effects depend on phase differences, which in turn depend on the flux of the magnetic field B.  Observable effects have to be gauge invariant.