Gauge Invariance

tbthomps

1. The problem statement, all variables and given/known data

So I was doing a problem out of Merzbacher 3rd edition (end of chapter 4 problem 3); the homework set has already been turned in but I wanted to run this by you all and see what you thought.

I am essentially working with a particle in a 1-d ring constrained to the x-y plane in 3space in the presence of a magnetic field given by

A = C ( -y / x^2+y^2 , x / x^2+y^2 , 0) where C is constant and phi (in the electric field) is assumed to be identically zero.

I was asked to find the corresponding energy spectrum En.


2. Relevant equations

None (read next section)


3. The attempt at a solution

The approach I took was to first change gauges; you can see that A is simply the negative gradient of arctan(x/y) times a constant so letting f(x,y,z) = C*arctan(x/y) we get that our new gauge A2 = A1 + grad(f) = 0

Since an energy spectrum is gauge-invariant it should suffice to find the spectrum in this new gauge; the Hamiltonian corresponding to the new gauge is simply the canonical energy Hamiltonian.

So Im thinking "awesome"; the solutions to this particular problem are well understood (in the presence of no magnetic field) and can be seen at

http://en.wikipedia.org/wiki/Particle_in_a_ring

My Question

According to a colleague (who read a solution out of a book somewhere) the energy spectrum is NOT the same as the one in the the linked article above (the 1-d box with no magnetic field); the spectrum he found was the canonical one + value depending on C.

So my question is.. what did I do wrong? It is easily proven that Gauge transformations leave the energy operator invariant ( That is, measuring the energy of a solution in one gauge and the corresponding solution in to second gauge give the same result)

The only thing I can think of is that my scalar function f(x,y,z) and its gradient had singularities and therefore are not admissible (even though both the function and the gradient exist except on a measure-zero set in 2-space)

Can anyone shed any light on the failing of my procedure? ( I saw the textbook my friend referenced and i understand the procedure it took; so im simply asking where mine broke down)

tbthomps

Found my problem; the issue at hand was I was being too mathematician-y :P

The problem is that this particular choice of function, f(x,y,z) (above), is not a valid gauge transform for this system. Gauge transforms (as it clearly says in merzbacher) must PRESERVE the resulting magnetic field.

Simply put, Curl ( grad (f) ) is not zero (for this choice of f) and thus the resulting magnetic field from my gauge transform is

B2 = curl (A2) = curl ( A + grad(f) ) = curl (0) = 0

whereas the original magnetic field was B1 = curl(A) was nonzero

The text said 'where f(x,y,z,t) is any function' and I took the word 'any' too literally (being a math guy) and lost sight of the underlying physics of gauge transforms.

Thanks to anyone who read and thought about this.