# Homework Help: Return time to its original state of a particle in the double infinite well

1. Dec 29, 2017

### Javier141241

1. The problem statement, all variables and given/known data
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) .

2. Relevant equations
Ψ(x,t)=∑cnψn(x)·exp(-i·Ent/(ħ))

En=n2π2ħ2/(2ma2)

3. 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 doesnt 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
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 Im 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 cant take common factor the same way.)

2. Dec 29, 2017

### kuruman

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

3. Dec 30, 2017

### Javier141241

I just checked and it seems that doesnt 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 couldnt all be 2π at the same time, but wasnt thinking of multiples of 2π).