# Statistical Mechanics and Thermodynamics

1. Jan 23, 2010

1. The problem statement, all variables and given/known data

Consider a particle confined within a box in the shape of a cube of edges Lx=Ly=Lz.

(a) Suppose that the partice is in a given state specified by particular values of the principal quantum numbers nx, ny, nz. By considering how the energy of this state must change when the length Lx of the box is changed quasistatically by a small amount dLx, show that the force exerted by the particle in this state on a wall perpendicular to the x-axis is given by Fx=-partial derivative of E with respect to partial derivative of Lx

2. Relevant equations

Xa,r=-partial derivative of Er wit respect to partial derivative of xa

where Xa,r is the generalized force (conjugate to the external parameter xa) in the state r

3. The attempt at a solution

I derived an expression for the quantized energy which is (h2(nx2+ny2+nz2)/(8mLx2)

Do I have to use pressure to some extent? Any advice would be greatly appreciated

2. Jan 24, 2010

### Maxim Zh

The absolute value of the energy shift which takes place when the length of the box changes is equal to the absolute value of the work done by the system. This work is

$$dA = F_x dL_x$$

It's easier to read the following:

$$F_x = - \frac{\partial E}{\partial L_x}.$$

See