Wave Functions of a Trapped Electron — Question 8

Question 8

Sudden expansion preserves the initial wave function but changes its energy decomposition

An electron starts in the ground state of $0<x<L$. The right wall suddenly moves to $2L$. Immediately afterward the old wave function is unchanged on $[0,L]$ and zero on $(L,2L)$. New modes are $v_j=L^{-1/2}\sin(j\pi x/2L)$.

1. Calculate the overlap with new mode $j=2$.

2. Calculate the overlap with new mode $j=1$ and the probability outside those two modes.

Question 8 - Solution

Sudden expansion preserves the initial wave function but changes its energy decomposition

(a) Method and calculation

$$c_2=\frac{\sqrt2}{L}\int_0^L\sin^2(\pi x/L)dx=\boxed{1/\sqrt2},\quad\boxed{P_2=1/2}.$$$$c_1=\frac{\sqrt2}{L}\int_0^L\sin(\pi x/L)\sin(\pi x/2L)dx.$$

(b) Interpretation and further calculation

$$\boxed{c_1=\frac{4\sqrt2}{3\pi}},\quad\boxed{P_1=\frac{32}{9\pi^2}=.360253}.$$$$\boxed{P_{\mathrm{other}}=1-P_1-P_2=.139747}.$$

Physical check

The electron does not instantly become the new ground state. Sudden changes preserve the wave function at the instant of change, while the new Hamiltonian has different eigenstates.