Gauge Invariance: Finding Energy Spectrum in 1D Ring

In summary, the conversation was about a problem in the Merzbacher 3rd edition textbook involving a particle in a 1-d ring in the presence of a magnetic field. The person was trying to find the corresponding energy spectrum and attempted to change gauges to a new one. However, a colleague pointed out that the energy spectrum found was different from the one in a linked article, and the reason for this was because the chosen function for the gauge transform did not preserve the magnetic field. The person realized their mistake and thanked anyone who read and thought about the issue.
  • #1
tbthomps
2
0

Homework Statement



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.

Homework Equations



None (read next section)

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 I am 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 into 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 I am simply asking where mine broke down)
 
Physics news on Phys.org
  • #2
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.
 

1. What is gauge invariance?

Gauge invariance is a fundamental principle in physics that refers to the idea that the physical laws and equations of a system should remain unchanged under certain transformations known as gauge transformations. These transformations do not alter the physical properties or behavior of the system, but rather represent different mathematical descriptions of the same physical state.

2. How does gauge invariance apply to finding energy spectrum in a 1D ring?

In the context of a 1D ring, gauge invariance means that the energy spectrum of the system should remain unchanged even when we change the gauge, or the mathematical description, of the system. This allows us to use different gauge choices to simplify the calculations while still obtaining the same physical results.

3. What is the significance of finding the energy spectrum in a 1D ring?

The energy spectrum in a 1D ring provides important information about the allowed energy levels and states of the system. This can help us understand the behavior and properties of the system, and can also be used to predict and explain experimental results.

4. How is the energy spectrum in a 1D ring calculated?

The energy spectrum in a 1D ring is typically calculated using mathematical techniques such as Fourier analysis or the Schrödinger equation. These methods take into account the physical properties and boundary conditions of the system to determine the allowed energy levels and states.

5. Can gauge invariance be applied to other systems besides a 1D ring?

Yes, gauge invariance is a fundamental principle that can be applied to many different physical systems. It is commonly used in quantum field theory, electromagnetism, and other areas of physics to ensure that the laws and equations of a system remain consistent under different mathematical descriptions.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Special and General Relativity
Replies
4
Views
924
  • Advanced Physics Homework Help
Replies
1
Views
968
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
7
Views
1K
  • Advanced Physics Homework Help
Replies
10
Views
324
  • Advanced Physics Homework Help
Replies
3
Views
287
Replies
5
Views
834
Replies
2
Views
1K
Replies
1
Views
3K
Back
Top