Question

It is usually said that gauge invariance is captured by the invariance of the electric and magnetic fields on the following transformation of the potentials :

ϕ′=ϕ−∂χ∂tA′=A+∇χ
In quantum mechanics, considering an hamiltonian on the form

H=12m(P−qA)2+qϕ,
the effect of the gauge transformation can be canceled by transforming the wave function into

ψ′=eiqℏχψ.
Am I right so far?

If I am then because the gauge transformation is not supposed to change anything, it means that every expectation can be calculated equivalently using ψ′
or ψ
, which does not seem not true.

An example could be a particle in a E=B=0
region. So I can choose ϕ=A=0
, to make the hamiltonian

H=12mP2
But one of my friends could have selected ϕ=0
, A=a
, which is supposed to be equivalent to my choice since they are linked by a gauge transformation (χ=ax
). Then, the hamiltonian would be

H=12m(P−qa)2
The gauge transformed wave function he can take to make his hamiltonian similar to mine is ψ′=eiqℏaxψ
, but this function is not equivalent.

The expectation value of momentum for me is:

⟨P⟩ψ=−iℏ∫ψ∗∂xψdx
For my friend it is:

⟨P⟩ψ′=−iℏ∫e−iqℏaxψ∗∂x(eiqℏaxψ)dx=qa∫ψ∗ψdx−iℏ∫ψ∗∂xψdx
What is it that I don't understand?

Options:
A) Gauge transformations are not fully represented by the wave function transformation given; further mathematical adjustments are required to ensure equivalence.
B) Expectation values in quantum mechanics are inherently gauge-dependent, leading to differences between gauge-transformed wave functions.
C) The equivalence between different gauge choices and their corresponding wave functions is approximate and valid only in specific scenarios.
D) The expectation values of physical quantities like momentum are inherently independent of gauge transformations, indicating an error in the calculations or understanding.

Answer 1

The wave function transformation under gauge transformations is incomplete in representing the full impact on physical quantities. The discrepancy in gauge-transformed momentum expectation values reveals the need for additional mathematical adjustments, supporting option A) Gauge transformations are not fully represented by the wave function transformation given; further mathematical adjustments are required to ensure equivalence.

Step-by-step explanation:

In quantum mechanics, the wave function transformation under gauge transformations is given by ψ' =
`e^(^i^q^ℏ^χ)`ψ. However, this transformation is not sufficient to ensure the equivalence of all physical quantities, as demonstrated in the example provided.

The expectation values of physical quantities, such as momentum, exhibit gauge dependence. In the given scenario, the difference arises due to the non-commutativity of the derivative operator with the gauge transformation. Specifically, the gauge-transformed expectation value of momentum involves an additional term (qa∫ψ∗ψdx) that does not cancel out, leading to a discrepancy between the transformed wave functions.

This discrepancy highlights that the equivalence between different gauge choices and their corresponding wave functions is not exact, necessitating further mathematical adjustments to ensure consistency. Therefore, option A is the correct choice, acknowledging the need for additional considerations beyond the basic wave function transformation to account for the effects of gauge transformations on physical observables.