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I was looking over the simple case of a 1D particle restrained inside an infinite square well potential ("particle in a box") and was having some difficulty understanding the relationship between the energy states and the expectation value for the energy.

Using the time independent Schrödinger equation and normalizing the wave function I get:

ψ(x) = sqrt(2/L) * sin (n*pi*x / L)

Which implies:

k = n*pi / L = p / hbar = sqrt(2mE) / hbar

E = (n*pi*hbar)^2 / 2m*L^2

Then I try calculating the expectation value for the energy. (Here is where I have trouble.)

<E> = ∫ψ* i hbar ∂ψ/∂t dx = i hbar ∫ψ* 0 dx = 0

[Where the bounds of the integral are from -∞ to ∞]

How can both of these statements about the energy of the particle be true? I feel like I am missing something fundamental. Does the uncertainty principle play a role here? Or is the Energy operator simply not valid in the time independent case of Schrödinger's equation?

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# Energy of an Infinite Square Well

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