Expectatoon value particle in superposition of momentum states

[Fixxxer125]

Homework Statement

Demonstrate the relation between the expectation value and the measurement outcomes of an observable of a particle by conisdering as an observable the kinetic energy operator
E=p^/2m when the particle is in a superposition of 2 momentum eigenstates

Homework Equations

<O> = Int (from -inf -> inf) [(Psi*)O(Psi)] dx

The Attempt at a Solution

I am taking the superposition of 2 momentum eigenstates as

Psi= square root (1/L) [ A*exp(ikx)exp(-iEt/Hbar) +B*exp(ikx)exp(-iEt/Hbar) ]

And then putting this into the integral

<O> = Int (from 0->L) [(Psi*)(-hbar/2m*d²/dx²(Psi)] dx

However I end up with a very long equation for the expectation value whereas I thought the expectation value would be something along the lines of
A²hbar²k₁²/2m + B²hbar²k₂²/2m as this looks like an eigenvalue

 

[Modulated]

I'd take out the $\sqrt{1/L}$ since there's no reason to consider any sort of box here (and besides you can absorb it into the $A$ and $B$ terms). Get rid of the time-dependent bit (since you're going to ignore it anyway - you're integrating over $x$) and make sure you label your two $k$s differently, like you have in your final suggestion: $k_1$ and $k_2$. And then the approach you're using should work! You seem to have some idea what you expect to find, which is good - if you can't get there post where you get up to.