Expectation Values of x & p for Wavefunction u(x,0)

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Homework Help Overview

The discussion revolves around the expectation values of position and momentum

for a given wavefunction of a particle at time t=0. The original poster seeks clarification on why the relationship

= m d/dt does not hold in this scenario.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definitions of momentum and its expectation value, questioning the applicability of the Ehrenfest theorem in this context. There is a focus on the distinction between the definitions of

    and p.

Discussion Status

Participants are actively questioning the assumptions behind the relationship between

and the time derivative of . Some guidance has been offered regarding the definitions involved, but no consensus has been reached on the implications of the Ehrenfest theorem in this case.

Contextual Notes

The discussion highlights the specific definitions used for momentum and expectation values, as well as the constraints imposed by the wavefunction's form. There is an emphasis on the mathematical definitions rather than physical interpretations.

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Homework Statement


A particle is represented(at t=0) by the wavefunction

u(x,0) = A(a^2 - x^2) if -a<x<a
= 0 otherwise
Determine <x> & <p>.
It is given in the book that in this case <p> \neq m*d/dt<x>. Could someone please tell me the reason for this?

Homework Equations





The Attempt at a Solution

 
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because if you use <p> = m d<x>/dt, you would determine the expectation value of the momentum via that of x.
But <p> is given only by ∫u p u* dx - it's a definition.
 
Last edited:
Rick88 said:
because if you use <p> = m d<x>/dt, you would determine the expectation value of the momentum via that of x.
But <p> is given only by ∫u p u* dx - it's a definition.

What I don't understand is that according to Ehrenfest theorem,
<p> = m* d/dt<x>
Why is this not true in this case?
 
How does your book DEFINE p mathematically?
 
The integral is just the definition of <p>.
p and <p> are not the same thing, so you can't use the formula for p to find <p>.
 

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