(adsbygoogle = window.adsbygoogle || []).push({}); 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=4ma^{2}/(πħ) .

2. Relevant equations

Ψ(x,t)=∑c_{n}ψ_{n}(x)·exp(-i·E_{n}t/(ħ))

E_{n}=n^{2}π^{2}ħ^{2}/(2ma^{2})

3. The attempt at a solution

I will explain my reasoning for a simpler case (combination of first two stationary states)

Ψ(x,t)=c_{1}ψ_{1}exp(-i·E_{1}t/(ħ))+c_{2}ψ_{2}exp(-i·E_{2}t/(ħ))

Since the global phase of Ψ doesnt matter ( |Ψ|^{2}) you obtain the ω of oscillation taking common factor exp(-i·E_{1}t/(ħ)) and obtaining ω=(E_{1}-E_{2})/(ħ). Therefore for the wave function to return to the same state it was at t=0, t must be

t·(E_{1}-E_{2})/(ħ)=2π

this leads to

t=4a^{2}m/(^{2}ħπ)·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.)

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

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