Expected Energy Value of a Time Independent Wave

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Homework Statement

For a region where the potential V=0, the wave function is given by ψ(x)=2αsin(3πx/a). Calculate the energy expectation value of this system. Note that α and a are two different constants.

Homework Equations

ψ(x)=2αsin(3πx/a)
E=K+V=K
K=p^2/2m
∫ψ*Pψdx=Expected momentum, where P=p operator=-iħ∂/∂x and the integral is from -∞ to +∞

The Attempt at a Solution

The energy operator is time dependent and the given equation is time independent so I have to use the momentum operator. This should be okay because V=0 ∴ E=K. Once I have the expected momentum I can use it to find the kinetic energy.

I don't see any way to solve the integral with -iħ∂/∂x inside the integral. No matter what I do the -i will be left at the end result leaving me with no real answer.

I'm thinking that I could use the Energy operator if I set the operator in terms of momentum (E=K=P^2/2m) since V=0. This would square the i making it -1. But I'm pretty sure that it would still be time dependent and therefor be meaningless.

 

[BvU]

Hi,

The energy operator is time dependent

Is it ? Can you demonstrate that >

 

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I think I got it.

If I Put the energy operator in terms of momentum then E=P^2/2m then the operator is time independent. Since the integral from -∞ to +∞ = 1 I can take the second derivative of ψ(x) and move the constants outside the integral. This leaves only the constants from the derivative multiplied by 1.

Final answer is E=(9πħ^2)/(2ma^2)

Since m and a are unknown this is good enough for an answer. Note that the constant a is not acceleration (the ma in the answer is not force).