Momentum of constant wave function

In summary, the conversation discusses the scenario of a small cuboid volume embedded in a larger ditto with periodic boundary conditions and a constant wave function inside the smaller cuboid. The question is posed about what we can know about the momentum in this scenario. The conversation also mentions the standard approach of measurement using a Hermitian operator and discusses the possibility of utilizing the periodic boundary conditions to produce a non-zero momentum. The suggestion is made to consider the given wave function as a superposition of eigenfunctions for the potential and analyze the time development of the Fourier transform.
  • #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?
 
  • #3
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 ?
 

1. What is the definition of momentum in a constant wave function?

Momentum is a measure of the amount of motion an object has and is defined as the product of an object's mass and its velocity.

2. How is momentum related to a constant wave function?

In a constant wave function, momentum is directly proportional to the wavelength of the wave. This means that as the wavelength increases, the momentum of the wave also increases.

3. What is the uncertainty principle in relation to the momentum of a constant wave function?

The uncertainty principle states that it is impossible to simultaneously determine the exact momentum and position of a particle. This means that the momentum of a constant wave function can only be known within a certain range of values.

4. How is the momentum of a constant wave function calculated?

The momentum of a constant wave function can be calculated using the equation p=ħk, where p is the momentum, ħ is the reduced Planck's constant, and k is the wave number (2π/λ, where λ is the wavelength).

5. What is the significance of momentum in a constant wave function?

Momentum plays a crucial role in understanding the behavior of particles and waves in quantum mechanics. It helps us understand the movement and interactions of particles, and is a fundamental quantity in many physical theories and laws.

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