Momentum of constant wave function

[J.L.A.N.]

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. ψ_x=0 = ψ_x=L (Periodic b.c. for each of the three pairs of opposite sides of the larger cuboid)
IV. [x_i,p_j] = iħδ_i,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?

 

[BvU]

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 ?