Energies of a Trapped Electron — Question 8

Question 8

The relative level spacing approaches a continuum only at large mode index

An infinite well has $E_n=n^2E_1$. Define relative adjacent spacing by $r_n=(E_{n+1}-E_n)/E_n$.

1. Derive $r_n$ and its leading large-n behavior.

2. Find the smallest integer n for which $r_n<.0100$.

Question 8 - Solution

The relative level spacing approaches a continuum only at large mode index

(a) Method and calculation

$$\boxed{r_n=\frac{2n+1}{n^2}=\frac2n+\frac1{n^2}}.$$$$n\gg1:\quad\boxed{r_n\sim2/n\to0}.$$

(b) Interpretation and further calculation

$$.01n^2-2n-1>0\Rightarrow n>\frac{2+\sqrt{4.04}}{.02}=200.499\ldots.$$$$\boxed{n_{\min}=201},\quad r_{200}=.010025,\quad r_{201}=.00997500.$$

Physical check

The absolute gap grows with n, while its ratio to the total energy shrinks. A quasi-continuum claim therefore needs a specified resolution or relative scale.