Question 39.56: Let ΔEadj be the energy difference between two adjacent ener......
Let ΔE_{adj} be the energy difference between two adjacent energy levels for an electron trapped in a one-dimensional infinite potential well. Let E be the energy of either of the two levels. (a) Show that the ratio ΔE_{adj}/E approaches the value 2/n at large values of the quantum number n. As n → ∞ , does (b) ΔE_{adj}, (c) E, or (d) ΔE_{adj}/E approach zero? (e) What do these results mean in terms of the correspondence principle?
(a) The allowed energy values are given by E_n=n^2 h^2 / 8 m L^2 . The difference in energy between the state n and the state n + 1 is
\Delta E_{ adj }=E_{n+1}-E_n=( n+1 )^2-n^2 \frac{h^2}{8 m L^2}=\frac{ ( 2 n+1 ) h^2}{8 m L^2}
and
\frac{\Delta E_{\text {adj }}}{E}=\left(\frac{( 2 n+1 ) h^2}{8 m L^2} \right) \left( \frac{8 m L^2}{n^2 h^2}\right)=\frac{2 n+1}{n^2} .
As n becomes large, 2 n+1 \rightarrow 2 n \text { and } ( 2 n+1) / n^2 \rightarrow 2 n / n^2=2 / n \text {. }
(b) No. As n \rightarrow \infty, \Delta E_{\text {adj }} and E do not approach 0, but \Delta E_{\text {adj }} / E does.
(c) No. See part (b).
(d) Yes. See part (b).
(e) ΔE_{adj}/E is a better measure than either ΔE_{adj} or E alone of the extent to which the quantum result is approximated by the classical result.