When Do the Energy Eigenstates Align Again?

[E&M]

Homework Statement

$$\Psi$$(x,0) = A[$$\psi$$₁ + $$\psi$$₂ + $$\psi$$₃]

How long will $$\Psi$$(x,t) take to go back to $$\Psi$$(x,0) ?

E₁, E₂, E₃ are the energies associated with each $$\psi$$₁, $$\psi$$₂, $$\psi$$₃.

Homework Equations

TISE

The Attempt at a Solution

I multiplied $$\Psi$$(x,0) by e^-iE_nt/h. The problem is that, there are three different values of energy associated with each $$\psi$$

 

[lanedance]

show your work...

its been a while since I've done this stuff, but i think that E_n should be thought of as an operator, so it will operate on each energy eigenstate to gives its energy in the phase term

the question will then amount to, based on the different energies, at what time, does psi(x,t) differ from psi(x,0) in only an overall phase.

IF you need more, think of the phase as being an arrow on a clock. The energy determines how fast each arrow rotates. So if the 3 arrows start of together all rotating at different speeds, at what time do they all line up again?