Gauge transformation which counteract wave function

[exponent137]

Gauge transformation can be written as:
##\psi(\vec{r},t)\rightarrow e^{-i \frac{e}{\hbar c}f(\vec{r},t)}\psi(\vec{r},t)##
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html
Does it have any sense that we choose such function ##f##, that all right side is constant in time. Is this possible at least approximately?

 

[Brage]

As far as i understand yes, but only if your wave function that is the inverse of the time variable of your gauge transformation. That is we must have f(r,t)=R(r) + T(t). The reason for this is that the probability amplitude of the original and the transformed functions are the same, and therefore no any observation will remain unchanged through the transformation. In other words, a gauge transformation does not affect any physical outcome but changes the structure of the equation.

 

[exponent137]

I ask, because it is strange to me, that wave functions can exist, where sinusoidal form ##e^{ikx+\omega t}##, is not necessary.Do you know any other examples, where wave functions can be described without ##e^{ikx+\omega t}##? Even, if it is described in momentum space.

As one condition for this I see that wave function contains only one frequency ##\omega##.

 

[Brage]

First off I assume you mean [itex] e^{i(kx+wt)} [\itex] as otherwise your wave function will not be normalizable. And I have the perfect example, the ground state of a quantum harmonic oscillator is a Gaussian distribution [itex] e^{-\lambda x^2} [\itex], where ## \lambda ## is a constant. The wave function you have described is that of a traveling wave, but one that is not normalizable, as its probability density is constant over space.

 

[exponent137]

1. Yes, I mean ##e^{i(kx+wt)}##.
2. Yes, harmonic oscilator is one such example. I forget. But, the factor ##e^{i\omega t}## remains. It is similarly for a Infinite Potential Well in the lowest level: http://www.physics.ox.ac.uk/Users/cowley/QuantumL12.pdf?
3. Yes, a traveling wave is not renormalizable. But, I think, that according to my question, this is not a problem. I ask only, if it is possible to choose such gauge function that it neutralizes its space and time waving. Maybe the harmonic oscilator is not a good example, because I suppose that it is not possible to neutralize its oscillation? But, the above function, ##e^{i(kx+wt)}##, is also the simplest possible.

4. One of the basic rules of QM is uncertainty principle. It is based on Fourier principle, thus it demands ##e^{i(kx+wt)}## nature inside of Gaussian nature, because of Fourier transformation. But, a traveling wave ##e^{i(kx+wt)}## has not gaussian nature, thus it is not possible to make uncertainty principle from it. Thus, its ##e^{i(kx+wt)} is not necessay. But, this neutralization seems very unnatural for me, thus I suppose that it is maybe imposible.

 

[Brage]

I think you are missing the point, a gauge transformation does not have any observable effect. For example,

"in electromagnetism the electric and magnetic fields, E and B, are observable, while the potentials V ("voltage") and A (the vector potential) are not.[3] Under a gauge transformation in which a constant is added to V, no observable change occurs in E or B."
(taken from wikipedia)

So yes, you could stop the oscillations in space and time of a function like ##e^{i(kx+wt)}## with a gauge transformation, but it would produce no observable effects. If you are still confused I recommend reading "Gauge Theories of the Strong, Weak, and Electromagnetic Interactions" by Chris Quigg. Although this book goes into much more detail than I think you are interested in, it has a very good section on gauge theory in electromagnetism and the phase invariance in quantum mechanics.

 

[exponent137]

I am aware that gauge transformation does not contain any measurable effect. (except some philosphical aspect, such as Aharonov Bohm http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html ...)

But, it is strange to me, that such example with stopping of oscillations was not mentioned anywhere. Because, it is interesting from mathematical view, (similarly as proper vectors etc. )

OK, it does not disturb uncertainty principle, so it is allowed. And it seems that you are sure that there is not any other possible problem?

Is this mentioned in book: Gauge Theories of the Strong, Weak, and Electromagnetic Interactions? I do not know that I will get this book, can you, please, find some other links which will tell mi more about gauge theory of EM, maybe even about such effect?

 

[DrDu]

Hm, this is nothing else than the change between the Schroedinger and Heisenberg picture. Should be familiar to you before delving into gauge theory.

 

[radium]

The Aharonov Bohm affect, QHE, FQHE, Dirac monopole and basically other systems with nontrivial topology are examples of constraints gauge invariance puts on a system. It is the reason the hall conductance is quantized. If it were not quantized, the wave function would not be gauge invariant since it would not be single valued. Gauge invariance like the phase only causes issues like these when you talk about evolving in a closed loop. If there is some constraint to preserve gauge invariance, that means there is some singularity present (like the Dirac monopole or Dirac string in the Aharonov Bohm effect) present which makes the system topologically nontrivial.