Expectation value of a dynamical variable problem.

[Normalization]

Homework Statement

Why does the extra phase factor cancel out? Is it because you are multiplying the wave-function with the extra phase factor by its conjugate and if so, why should it matter that the extra phase factor is independent of x?

All relevant information, the solution and equation referred to in the solution is given below:

Homework Equations

Given in picture

The Attempt at a Solution

See problem statement

 

[Dick]

Homework Statement

Why does the extra phase factor cancel out? Is it because you are multiplying the wave-function with the extra phase factor by its conjugate and if so, why should it matter that the extra phase factor is independent of x?

All relevant information, the solution and equation referred to in the solution is given below:

Homework Equations

Given in picture

The Attempt at a Solution

See problem statement

Yes, the phase get multiplied by its conjugate. If the phase had had an x dependence then when the operator Q (which could contain d/dx) acts on ψ it could also change the phase factor.

 

[Normalization]

Oh I see :) Because the phase factor is independent of x it acts as a constant on ψ so taking the derivative of one of the ψ's wouldn't affect the value of the phase factor, but if it was dependent of x one would need to use the product rule to take the partial of one of the psi's.

 

[Dick]

Oh I see :) Because the phase factor is independent of x it acts as a constant on ψ so taking the derivative of one of the ψ's wouldn't affect the value of the phase factor, but if it was dependent of x one would need to use the product rule to take the partial of one of the psi's.

Right! The phase has to commute with the operator Q if it's not going to change expectation values.

 

[Normalization]

Cool, thanks :D