Statistical Mechanics and Thermodynamics

[vladittude0583]

Homework Statement

Consider a particle confined within a box in the shape of a cube of edges L_x=L_y=L_z.

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

Homework Equations

X_a,r=-partial derivative of E_r wit respect to partial derivative of x_a

where X_a,r is the generalized force (conjugate to the external parameter x_a) in the state r

The Attempt at a Solution

I derived an expression for the quantized energy which is (h²(n_x²+n_y²+n_z²)/(8mL_x²)

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

 

[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

$$dA = F_x dL_x$$

F_x=-partial derivative of E with respect to partial derivative of L_x

It's easier to read the following:

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

See
https://www.physicsforums.com/showthread.php?t=8997&highlight=LaTeX+TeX"