Two- and Three-Dimensional Electron Traps — Question 10

Question 10

Quantum wall forces satisfy a pressure-energy relation under isotropic scaling

In a three-dimensional rectangular infinite well, $E=C(n_x^2/L_x^2+n_y^2/L_y^2+n_z^2/L_z^2)$, with positive integer indices and $C=\pi^2\hbar^2/(2m)$. Ignore spin. A particle occupies the ground state of a cube of side L. Define the force on the x wall by $F_x=-\partial E/\partial L_x$ with the other sides fixed, and pressure by $P_x=F_x/(L_yL_z)$.

1. Find the force and pressure at the cubic geometry.

2. Express pressure using total energy and volume, and verify by differentiating energy during isotropic volume changes.

Question 10 - Solution

Quantum wall forces satisfy a pressure-energy relation under isotropic scaling

(a) Method and calculation

$$E=C(L_x^{-2}+L_y^{-2}+L_z^{-2})\Rightarrow\boxed{F_x=2C/L^3}.$$$$\boxed{P=2C/L^5},\quad E=3C/L^2,\quad\mathcal V=L^3.$$

(b) Interpretation and further calculation

$$\boxed{P=2E/(3\mathcal V)}.$$$$E(\mathcal V)=3C\mathcal V^{-2/3}\Rightarrow\boxed{-dE/d\mathcal V=2C\mathcal V^{-5/3}=P}.$$

Physical check

The partial derivative for one moving wall differs from differentiating all three lengths together. Both give the same pressure when their geometric factors are handled correctly.