Phy5645/HO problem2: Difference between revisions
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(New page: Problem 2: The expectation value of p in eigenstate. <math>\langle\Psi_n|p|\Psi_n\rangle=-i\sqrt{\frac{m\hbar\omega}{2}}\langle\Psi_n|\hat{a}-\hat{a}^{\dagger}|\Psi_n\rangle</math> <ma...) |
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In terms of the raising and lowering operators, the momentum operator is | |||
<math>\hat{p}=-i\sqrt{\frac{m\hbar\omega}{2}}(\hat{a}-\hat{a}^{\dagger}).</math> | |||
We now take its expectation value with respect to an arbitrary eigenstate of the harmonic oscillator: | |||
<math>==-i\sqrt{\frac{m\hbar\omega}{2}}(\langle | <math>\langle n|\hat{p}|n\rangle=-i\sqrt{\frac{m\hbar\omega}{2}}\langle n|(\hat{a}-\hat{a}^{\dagger})|n\rangle</math> | ||
<math>=-i\sqrt{\frac{m\hbar\omega}{2}}(\langle n|\hat{a}|n\rangle-\langle n|\hat{a}^{\dagger}|n\rangle)</math> | |||
<math> | <math>=-i\sqrt{\frac{m\hbar\omega}{2}}(\sqrt{n}\langle n|n-1\rangle-\sqrt{n+1}\langle n|n+1\rangle)</math> | ||
<math>=0\!</math> | |||
A similar intuitive argument as before would lead us to expect this result; one may think of the problem in momentum space as an effective [[Schrödinger Equation|Schrödinger equation]] in the "coordinate" <math>p\!</math> with a harmonic potential. | |||
Back to [[Harmonic Oscillator Spectrum and Eigenstates#Problems|Harmonic Oscillator Spectrum and Eigenstates]] | |||
Latest revision as of 13:33, 18 January 2014
In terms of the raising and lowering operators, the momentum operator is
We now take its expectation value with respect to an arbitrary eigenstate of the harmonic oscillator:
A similar intuitive argument as before would lead us to expect this result; one may think of the problem in momentum space as an effective Schrödinger equation in the "coordinate" with a harmonic potential.
