An Electron in a Finite Well — Question 8

Question 8

Deep-well leakage can be approximated as an increase of effective width

A deep finite well has physical width L and a nearly constant exterior decay constant $\kappa$. Use the approximation $L_{\mathrm{eff}}=L+2/\kappa$ and $E_n\simeq n^2\pi^2\hbar^2/(2mL_{\mathrm{eff}}^2)$. The corresponding infinite-well energy is $E_n^{(\infty)}=n^2\pi^2\hbar^2/(2mL^2)$.

1. Derive the energy ratio and its first correction for $\kappa L\gg1$.

2. Evaluate the effective-width ratio and its linear approximation at $\kappa L=20$.

Question 8 - Solution

Deep-well leakage can be approximated as an increase of effective width

(a) Method and calculation

$$\boxed{\frac{E_n}{E_n^{(\infty)}}\simeq\left(1+\frac2{\kappa L}\right)^{-2}}.$$$$\boxed{E_n/E_n^{(\infty)}\simeq1-\frac4{\kappa L}+O((\kappa L)^{-2})}.$$

(b) Interpretation and further calculation

$$\kappa L=20:\quad\boxed{(1+.1)^{-2}=.826446}.$$$$\boxed{1-4/20=.800000}.$$

Physical check

The numerical ratio is exact only within the effective-width approximation. A true finite-well calculation must account for the energy dependence of the decay constant.