Two state oscillations and quantum oscillator

[Beer-monster]

Homework Statement

I was asked an interesting question once that I'd like to solve but have no idea where to start.

It's hard to remember the exact details but basically:

Two electrons are in a harmonic oscillator potential but in two separate states $\left | m \right \rangle$ and $\left | n \right \rangle$. I know these are stationary states but the combination of them are not.

I was asked something like what would the energy between the states need to be for the states to oscillate.

My understanding is that this means that the a certain condition electrons have equal probability of transition to the higher state m or dropping to the lower state n so the electrons basically hop back and forth between these states.

Does this make sense and how would I go about approaching this problem with the math?

 

[Almsco]

It sounds like you're trying to solve a two-state system problem. This type of problem is usually solved using the Rabi oscillation equation, which states that the probability of transition between two states is directly related to the energy difference between them. The equation is P(m,n) = (1/2)sin^2 (E(m,n)/h * t). Where E(m,n) is the energy difference between the two states and h is Planck's constant.

You can use this equation to calculate the energy difference between the two states that would cause the electrons to oscillate between them. Hope this helps!