Expectatoon value particle in superposition of momentum states

  • Thread starter Fixxxer125
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  • #1
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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
 

Answers and Replies

  • #2
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I'd take out the [itex]\sqrt{1/L}[/itex] since there's no reason to consider any sort of box here (and besides you can absorb it into the [itex]A[/itex] and [itex]B[/itex] terms). Get rid of the time-dependent bit (since you're going to ignore it anyway - you're integrating over [itex]x[/itex]) and make sure you label your two [itex]k[/itex]s differently, like you have in your final suggestion: [itex]k_1[/itex] and [itex]k_2[/itex]. 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.
 

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