String Waves and Matter Waves — Question 1

Question 1

The same spatial boundary condition produces different frequency ladders

A string of wave speed v and a nonrelativistic particle of mass m occupy $0<x<L$ with zero displacement or wave function at both ends. Their dispersions are $\omega_s=vk$ and $\omega_m=\hbar k^2/(2m)$.

1. Derive the allowed wave numbers and wavelengths.

2. Compare the ratios of second-mode to first-mode angular frequencies.

Question 1 - Solution

The same spatial boundary condition produces different frequency ladders

(a) Method and calculation

$$\psi=A\sin kx+B\cos kx,\quad\psi(0)=0\Rightarrow B=0.$$$$\sin kL=0\Rightarrow\boxed{k_n=n\pi/L,\quad\lambda_n=2L/n},\ n\ge1.$$

(b) Interpretation and further calculation

$$\omega_{s,n}=vn\pi/L\Rightarrow\boxed{\omega_{s,2}/\omega_{s,1}=2}.$$$$\omega_{m,n}=\hbar n^2\pi^2/(2mL^2)\Rightarrow\boxed{\omega_{m,2}/\omega_{m,1}=4}.$$

Physical check

Boundary conditions select spatial modes. The dynamical dispersion relation determines how those modes evolve in time.