Return time to its original state of a particle in the double infinite well

[Javier141241]

Homework Statement

First sorry for the traduction mistakes.

Prove that any wave function of a particle in a 1 dimensional infinite double well of width a, returns to its original state in time T=4ma²/(πħ) .

Homework Equations

Ψ(x,t)=∑c_nψ_n(x)·exp(-i·E_nt/(ħ))

E_n=n²π²ħ²/(2ma²)

The Attempt at a Solution

I will explain my reasoning for a simpler case (combination of first two stationary states)
Ψ(x,t)=c₁ψ₁exp(-i·E₁t/(ħ))+c₂ψ₂exp(-i·E₂t/(ħ))
Since the global phase of Ψ doesn't matter ( |Ψ|² ) you obtain the ω of oscillation taking common factor exp(-i·E₁t/(ħ)) and obtaining ω=(E₁-E₂)/(ħ). Therefore for the wave function to return to the same state it was at t=0, t must be
t·(E₁-E₂)/(ħ)=2π
this leads to
t=4a²m/(²ħπ)·1/Q where Q its a term dependent of n

Since de statement says for any wave function and I get a similar result but depending of the Ψ in question, what I am missing? I wrote it for a combination of the two first stationary states, how would it be for a combination of n states? (since you can't take common factor the same way.)

 

[kuruman]

The answer is staring at you and you almost had it. You have

Ψ(x,t)=∑c_nψ_n(x)·exp(-i·E_nt/(ħ))

Suppose you factor out exp(-i·E₁t/ħ). What happens to the common factor exp(-i·E₁t/ħ) at t = T? What happens to each term in the summation at t = T?

 

[Javier141241]

I just checked and it seems that doesn't matter which n's you take in (∑Ei)T/ħ, it just turns into multiples of 2π, diferent for each term. Should have checked it before more rigourous ( I was thinking that they couldn't all be 2π at the same time, but wasnt thinking of multiples of 2π).
Thanks for your time!

 

[kuruman]

The wavefunctions of a particle in a box are mathematically the same as the standing waves in a string tied at both ends. The ground state (n=1) of the particle in a box corresponds to the fundamental frequency of the string. Any harmonic frequency is a multiple of the fundamental which means that in the time required for the string to complete one oscillation at the fundamental frequency, it has completed an integer number of oscillations at all harmonic frequencies.