String Waves and Matter Waves — Question 8

Question 8

Quadratic mode frequencies produce an exact full revival

A particle in a hard-wall interval has eigenstates $u_n$ with energies $E_n=n^2E_1$. An arbitrary finite superposition is $\Psi(x,t)=\sum_n c_nu_n(x)e^{-in^2E_1t/\hbar}$.

1. Find a positive time at which every component has its initial phase.

2. Compare that time with the classical round-trip period of a particle having momentum $p_n=n\pi\hbar/L$, using $E_1=\pi^2\hbar^2/(2mL^2)$.

Question 8 - Solution

Quadratic mode frequencies produce an exact full revival

(a) Method and calculation

$$n^2E_1T_{\mathrm{rev}}/\hbar=2\pi n^2\Rightarrow\boxed{T_{\mathrm{rev}}=2\pi\hbar/E_1}.$$$$\boxed{\Psi(x,T_{\mathrm{rev}})=\Psi(x,0)},\qquad T_{\mathrm{rev}}=4mL^2/(\pi\hbar).$$

(b) Interpretation and further calculation

$$T_{{\mathrm{cl}},n}=2L/(p_n/m)=2mL^2/(n\pi\hbar).$$$$\boxed{T_{\mathrm{rev}}/T_{{\mathrm{cl}},n}=2n}.$$

Physical check

This is a common revival time for any finite superposition. A special choice of occupied modes can revive earlier, so it need not be the shortest recurrence for every state.