String Waves and Matter Waves — Question 10

Question 10

Changing the length shifts string and matter-wave frequencies at different rates

A fixed mode index n is followed while the confinement length changes slowly from L to $L+\delta L$. The string speed v and particle mass m remain constant. Assume ideal adiabatic tracking for the particle.

1. Derive the fractional first-order frequency changes for the string and matter wave.

2. Find the exact ratios when the length increases by 10 percent.

Question 10 - Solution

Changing the length shifts string and matter-wave frequencies at different rates

(a) Method and calculation

$$\omega_s\propto L^{-1}\Rightarrow\boxed{\delta\omega_s/\omega_s\simeq-\delta L/L}.$$$$\omega_m\propto L^{-2}\Rightarrow\boxed{\delta\omega_m/\omega_m\simeq-2\delta L/L}.$$

(b) Interpretation and further calculation

$$L\prime=1.10L:\quad\boxed{\omega_s\prime/\omega_s=1/1.10=.909091}.$$$$\boxed{\omega_m\prime/\omega_m=1/(1.10)^2=.826446}.$$

Physical check

The mode index stays fixed only under the specified tracking assumption. An abrupt boundary change creates a superposition and is a different problem.