An Electron in a Finite Well — Question 9

Question 9

Sensitivity to wall height directly measures exterior probability

A symmetric finite well has $V=0$ for $|x|<a$ and $V=V_0>0$ outside. Bound energies satisfy $0<E<V_0$. Define $k=\sqrt{2mE}/\hbar$, $\kappa=\sqrt{2m(V_0-E)}/\hbar$, $z=ka$, $w=\kappa a$ and $z_0=a\sqrt{2mV_0}/\hbar$. Use the Hellmann–Feynman relation $dE/dV_0=\langle\partial H/\partial V_0\rangle$ for a normalized nondegenerate bound state at fixed a. The potential is zero inside and $V_0$ outside.

1. Relate the energy slope to the probability outside.

2. If $dE/dV_0=.180$, find the slope of the binding energy $B=V_0-E$ and both spatial probabilities.

Question 9 - Solution

Sensitivity to wall height directly measures exterior probability

(a) Method and calculation

$$\partial H/\partial V_0=0\ (|x|<a),\quad1\ (|x|>a).$$$$\boxed{dE/dV_0=\int_{|x|>a}|\psi|^2dx=P_{\mathrm{out}}}.$$

(b) Interpretation and further calculation

$$\boxed{P_{\mathrm{out}}=.180,\qquad P_{\mathrm{in}}=.820}.$$$$\boxed{dB/dV_0=1-dE/dV_0=P_{\mathrm{in}}=.820}.$$

Physical check

Both slopes lie between zero and one. This derivative holds at fixed width and mass, with the energy zero fixed at the well bottom.