Energies of a Trapped Electron — Question 4

Question 4

An energy cutoff permits only a finite number of box modes

An electron in an infinite well is prepared using only eigenstates with $E_n\le E_{\mathrm{cut}}=10E_1$. Spin is not included. The energy levels are $E_n=n^2E_1$.

1. List the permitted mode indices and count them.

2. Generalize the count to arbitrary positive $E_{\mathrm{cut}}$ and evaluate it just below $9E_1$.

Question 4 - Solution

An energy cutoff permits only a finite number of box modes

(a) Method and calculation

$$n^2\le10\Rightarrow n\le\sqrt{10}\Rightarrow\boxed{n=1,2,3}.$$$$\boxed{N=3}\quad\text{spatial eigenstates}.$$

(b) Interpretation and further calculation

$$\boxed{N(E_{\mathrm{cut}})=\left\lfloor\sqrt{E_{\mathrm{cut}}/E_1}\right\rfloor}.$$$$E_{\mathrm{cut}}=9E_1-\epsilon,\quad0<\epsilon\ll E_1\Rightarrow\boxed{N=2}.$$

Physical check

The count is a staircase, not a continuous density of states. The distinction between below and at a threshold matters for the highest permitted mode.