- #1

BOAS

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## Homework Statement

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Particle in one dimensional box, with potential ##V(x) = 0 , 0 \leq x \leq L## and infinity outside.

##\psi (x,t) = \frac{1}{\sqrt{8}} (\sqrt{5} \psi_1 (x,t) + i \sqrt{3} \psi_3 (x,t))##

Calculate the expectation value of the Hamilton operator ##\hat{H}## . Compare it with the energy eigenvalues ##E_1##, ##E_2##, and ##E_3##.

## Homework Equations

## The Attempt at a Solution

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The subscript refers to the different eigenvalue solutions.

##\psi_1 = e^{-i \frac{E_1 t}{\hbar} } \sqrt{\frac{2}{L}} \sin \frac{\pi}{L} x##

and

##\psi_3 = e^{-i \frac{E_3 t}{\hbar} } \sqrt{\frac{2}{L}} \sin \frac{3 \pi}{L} x##

Using the fact that ##\hat{H} \phi = E \phi## I find that

##\hat{H} \phi = -\frac{\hbar^2}{2m} \frac{\partial^2 \phi}{\partial x^2}## and cancelling out the stationary wave equation, I get that

##E = \frac{\hbar^2 \pi^2}{2m L^2 \sqrt{8}} (\sqrt{5} + 9 \sqrt{3})##

and since the expectation value of the hamiltonian is the total energy, that should be what I'm looking for.

I am confused about how I compare this value to ##E_n## when I have a superposition of stationary states. Does the expression ##E_n = \frac{\hbar^2}{2m} (\frac{n \pi}{L})## apply to all stationary states?

i.e is it literally a case of plugging in ##n = 1,2,3## ?

I'm not sure if I'm explaining myself clearly. How does the case where ##n = 1## effect my ##\psi_3##?