Momentum of constant wave function

  • Thread starter J.L.A.N.
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  • #1
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1. The scenario
If we have a small cuboid volume embedded in a larger dito with periodic boundary conditions, and a wave function that is constant inside the former, while zero everywhere else; what can we then know about the momentum?

Homework Equations


I. Âψ = Aψ (A being the measured eigenvalue corresponding to the Hermitian operator Â)
II. p = -iħ
III. ψx=0 = ψx=L (Periodic b.c. for each of the three pairs of opposite sides of the larger cuboid)
IV. [xi,pj] = iħδi,j

3. The attempt
I'm thinking about the standard approach (I.) of measurement by acting on the wave function with the Hermitian operator corresponding to the quantity of interest - in this case the momentum, the operator of which is written just above (II.). Acting on a constant wave function the result is obviously zero (since we are differentiating a constant function), but I am not sure of what this really tells us about the momentum in general, for the above scenario. Is there any other approach that could produce a non-zero momentum (perhaps by utilizing (III.))? What about the discontinuity at the boundary of the inner volume?
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 
  • #3
BvU
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Dear JLAN,

No responses so far, so I'll put in a suggestion.
Your ##-I\hbar\nabla## operator is awkward with such a wave function.
If you are already familiar with the particle in a box, perhaps you can make an inroad by considering the given wave form as a superposition of eigenfunctions for that potential (A set of plane waves) at t=0 and then look at the time development of the fourier transform ?
 

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