Statistical Mechanics: Calculating Pressure on a 3D Box Wall

[vladittude0583]

Homework Statement

Consider a particle confined within a box in the shape of a cube of edges L_x=L_y=L_z. The possible energy levels of this particle are then given by the quantized energy for a particle in a 3D box.

Calculate explicitly the force per unit area (or pressure) on this wall. By averaging over all possible states, find an expression for the mean pressure on this wall. (Exploit the property that the average values of the quantum numbers must all be equal by symmetry). Show that this mean pressure can be very simply expressed in terms of the mean energy of the particle and the volume of the box.

Homework Equations

In part (a), it was said that the particle exerts a force on a wall perpendicular to the x-axis such that F_x= - dE/dL_x (these are partial derivatives).

The Attempt at a Solution

I don't want the final answer, however, I just need guidance on how to get there

1) This is through a quasistatic process such that dW=(mean pressurve)*dV
2) What does it mean when "exploit the property that the average values of the quantum numbers are all equa by symmetry?"
3) I know that the length along the x-axis changing by an amount dL_x whereas the area of the wall is A=L_yL_z=constant. How do I related these values together?
4) I understand how to solve it using Kinetic Theory of Gases, however, my professr prefers using the method of pressure, etc.
5) Just some guidance is greatly appreciated. Thanks!

 

[torquil]

Regarding your question 3, it could mean that you will be able to determine a quantity that is given by the sum of the three average values, but not the average values individually. In that case you could exploit a symmetry to say that each of the average values would be 1/3 of the total.

Torquil