Pressure in quantum mechanics

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Homework Statement

Suppose you have a particle in a box of length L (a cube). Suppose a particle is in a given state specified by three integers n1,n2,n3. By considering how this state must change when the length of the cube is changed in one direction, show that the force exerted by the particle in this state on a wall perpendicular to the direction of change is given by [itex]-\partial E/\partial L[\latex]

Homework Equations

The Attempt at a Solution

Taking the potential to be zero within the cube and infinite at the boundaries and putting the origin at one of the corners I can get a wave function:

[itex] \psi(x,y,z)=\sqrt{\frac{8}{L^3}}\sin(w_1 x)\sin(w_2 y)\sin(w_3 z) [\latex]

where the frequency the sine functions have the usual dependence (infinite well) on the integers n1,n2,and n3. The energy is the sum of the energies from each 'component':

[itex] E=\frac{2\pi^2 \hbar^2}{mL^3}(n_1^2+n_2^2+n_3^2) [\latex]

I'm stuck conceptually. I've never thought about force in quantum. Here are my thoughts which i haven't fully explored because I don't have enough time in a day to explore all these. Hopefully one of you can provide a little nudge in the correct direction?

I want to talk about forces and so I have two ideas: Ehrenfest's theorem and the Heisenberg picture. Can I attack this problem by looking at the change in <p>? I should say that I'm assuming I approach this from the force end of things and shake out the partial of E wrt L. Or should I think about moving to the Heisenberg picture and talk about the time evolution of the momentum operator?

Thanks in advance!

The Attempt at a Solution


Answers and Replies

  • #2
damn my latex didn't render. I apologize. I can fix that if anyone let's me know what I messed up.