Is state an energy eigenstate of the infinite square well

[acdurbin953]

Homework Statement

Is state ψ(x) an energy eigenstate of the infinite square well?

ψ(x) = aφ₁(x) + bφ₂(x) + cφ₃(x)

a,b, and c are constants

Homework Equations

Not sure... See attempt at solution.

The Attempt at a Solution

I have no idea how to solve, and my book does not address this type of problem.
My one guess was to let the potential V(x) of the infinite square well be analogous to the Hamiltonian operator, and to then find the eigenstates of V(x). But I don't know how to do that, nor do I know if that is even right.
It would be helpful if someone could point me in the right direction on this one. Thank you.

 

[Simon Bridge]

The potential V is not "analogous" to the hamiltonian - it is part of the hamiltonian.
The hamiltonian operator is: $$\hat H = \frac{\hat p}{2m} + V$$ ... but what are $\varphi$ ?

 

[acdurbin953]

φ is the eigenstate of H, right? How do you calculate the eigenstates of H? Are they solutions of the differential equation that represents H?

Side question: In the case of the infinite well is it correct that H = V because the momentum is always going to be 0?