String Waves and Matter Waves — Question 3

Question 3

Mixed boundary conditions shift the spatial spectrum by half an index

Consider a mode on $0<x<L$ satisfying $\psi(0)=0$ and $\psi\prime(L)=0$. Treat these as supplied mathematical boundary conditions, as for a string fixed at one end and free at the other.

1. Derive the positive allowed wave numbers.

2. Using $E=\hbar^2k^2/(2m)$, compare the first two mode energies for a particle with these boundary conditions.

Question 3 - Solution

Mixed boundary conditions shift the spatial spectrum by half an index

(a) Method and calculation

$$\psi=A\sin kx,\quad\psi\prime(L)=Ak\cos kL=0.$$$$\boxed{k_n=(n+\tfrac12)\pi/L},\quad n=0,1,2,\ldots.$$

(b) Interpretation and further calculation

$$E_n=\frac{\hbar^2\pi^2}{2mL^2}(n+\tfrac12)^2.$$$$\boxed{E_1/E_0=(3/2)^2/(1/2)^2=9}.$$

Physical check

These are not the two impenetrable-wall boundary conditions of the ordinary infinite well. The different boundary condition is what changes the spectrum.