Energies of a Trapped Electron — Question 6

Question 6

Adiabatic compression does work on a trapped electron

An electron remains in mode n while an infinite well contracts slowly from L to $L/2$. Its level is $E_n(L)=n^2\pi^2\hbar^2/(2mL^2)$. Define the force exerted by the electron on the movable right wall by $F=-dE_n/dL$.

1. Find the work done on the electron during compression.

2. Derive the wall force and verify the work by integration.

Question 6 - Solution

Adiabatic compression does work on a trapped electron

(a) Method and calculation

$$E_n(L/2)=4E_n(L)\Rightarrow\boxed{W_{\mathrm{on}}=3E_n(L)}.$$$$\boxed{F(L)=-\frac{dE_n}{dL}=\frac{n^2\pi^2\hbar^2}{mL^3}=\frac{2E_n(L)}L}.$$

(b) Interpretation and further calculation

$$W_{\mathrm{on}}=-\int_L^{L/2}F(\ell)d\ell=\left[\frac{n^2\pi^2\hbar^2}{2m\ell^2}\right]_L^{L/2}.$$$$\boxed{W_{\mathrm{on}}=3E_n(L)>0}.$$

Physical check

The outward wall force opposes compression. Adiabatic mode preservation is essential; a sudden change can populate several final modes.