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chris_avfc

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## Homework Statement

Particle of mass m undergoes simple harmonic motion along the x axis

Normalised eigenfunctions of the particle correspond to the energy levels

[tex] E_n = (n+ 1/2)\hbar\omega\ \ \ \ (n=0,1,2,3...) [/tex]

For the two lowest energy levels the eigenfunctions expressed in natural units are:

[tex] u_0 = C_0 \exp^{-q^2 /2} [/tex]

[tex] u_1 = C_1 q \exp^{-q^2 /2} [/tex]

At time [itex] t = 0 [/itex] the wave function of the particle is given by an equal superposition of the two eigenstates represented by the two eigenfunctions [itex] u_0 [/itex] and [itex] u_1 [/itex].

Assume [itex] \psi [/itex] is normalised.

Calculate the expectation values of the momentum operator [itex] \hat{P} [/itex] and position operator [itex] \hat{X} [/itex] at time [itex] t [/itex].

## Homework Equations

Given two standard integrals:

[tex] \int e^{-ax^2}\,dx = 1/2 \sqrt{\pi/a} [/tex][tex] \int x^2 e^{-ax^2}\,dx = 1/4 \sqrt{\pi/a^2}[/tex]

## The Attempt at a Solution

I've calculated the two normalisation constants, but then I am seriously stuck, I don't have a clue what to do, could somebody point me in the right direction?

Thanks,

Chris

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