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## Homework Statement

We're given an unnormalized state function ψ(x) of an electron in a 1 dimensional box of length pi. The state function is a polynomial. We're asked to find the probability that a measurement of its energy would find it in the lowest possible energy state.

## Homework Equations

H = -h

^{2}/ 2m d

^{2}/dx

^{2}

E = h

^{2}n

^{2}π

^{2}/ 2mL

^{2}

## The Attempt at a Solution

I took several approaches to finding the lowest possible energy state probability (n=1). I attempted to do it in the same way that you would do a position probability, except I also included the Hamiltonian. So I tried something along the lines of <E> = ∫(0 to pi) ψ(x)*⋅ H⋅ ψ(x) dx which I thought might give me a number that I could then divide into the second equation above (using n=1 to find energy). This returned a number that is clearly not a probability. I'm not going to explain every approach I took because I don't want to take away from my main questions. But another approach I took is finding the expectation value and dividing the first energy level value by this expectation value, which returned a very small value for n=1 (I assume n=1 should be somewhat probable, with decreasing values for n=2 and so forth).

I do not understand the energy dependence on position. How does energy only depend on position? Also, is it necessary to normalize my energy equations (since it has to do with probability). I'm not sure I'm even on the right track with expectation values or my integration of

**ψ(x)*⋅ H⋅ ψ(x)**, there's some kind of connection between the quantum mechanical ideas that I'm just not grasping. Any help is appreciated.