Multiple Electrons in Rectangular Traps — Question 6

Question 6

Fermi filling produces a large zero-temperature force on a movable wall

For $N=2K$ independent electrons in a one-dimensional well, $E_g=\pi^2\hbar^2K(K+1)(2K+1)/(6mL^2)$. Define the force on the right wall as $F=-\partial E_g/\partial L$ at fixed N.

1. Derive F exactly.

2. For large N, express F in terms of the line density $n_1=N/L$ and discuss its length dependence at fixed density.

Question 6 - Solution

Fermi filling produces a large zero-temperature force on a movable wall

(a) Method and calculation

$$E_g=C(N)L^{-2}\Rightarrow\boxed{F=2E_g/L}.$$$$\boxed{F=\frac{\pi^2\hbar^2K(K+1)(2K+1)}{3mL^3}}.$$

(b) Interpretation and further calculation

$$K\gg1:\quad K(K+1)(2K+1)\simeq2K^3=N^3/4.$$$$\boxed{F\simeq\frac{\pi^2\hbar^2}{12m}(N/L)^3=\frac{\pi^2\hbar^2n_1^3}{12m}}.$$

Physical check

At fixed particle number the force decreases as the box expands. In the large-system limit at fixed line density it remains finite and independent of overall length.