Register to reply 
Gauge Invariance 
Share this thread: 
#1
Feb1910, 05:55 PM

P: 2

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 1d ring constrained to the xy 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 gaugeinvariant 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 1d 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 measurezero set in 2space) 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) 


#2
Feb1910, 10:13 PM

P: 2

Found my problem; the issue at hand was I was being too mathematiciany :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. 


Register to reply 
Related Discussions  
What is Gauge Invariance in QFT?  High Energy, Nuclear, Particle Physics  15  
Gauge invariance  Quantum Physics  4  
Gauge invariance and it's relation to gauge bosons  High Energy, Nuclear, Particle Physics  6 