Expectation value of a dynamical variable problem.

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Homework Help Overview

The discussion revolves around the cancellation of an extra phase factor in the context of expectation values of dynamical variables in quantum mechanics. Participants are exploring the implications of the phase factor's independence from spatial variables on the behavior of wave functions.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning why the extra phase factor cancels out and whether its independence from x is significant. There is discussion about the effects of the phase factor when multiplied by its conjugate and how it interacts with operators acting on wave functions.

Discussion Status

The conversation is progressing with participants clarifying their understanding of the role of the phase factor. Some have noted that if the phase factor were dependent on x, it would require different treatment when applying operators, indicating a productive exploration of the topic.

Contextual Notes

There is an emphasis on the mathematical properties of wave functions and operators, particularly regarding differentiation and the product rule, which are relevant to the discussion of the phase factor's impact on expectation values.

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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:

Prob18.jpg


Homework Equations



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The Attempt at a Solution



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Normalization said:

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:

Prob18.jpg


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.
 
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.
 
Normalization said:
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.
 
Cool, thanks :D
 

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