1. The problem statement, all variables and given/known data
A particle is in a 1D harmonic oscillator potential. Under what conditions will the
expectation value of an operator Q (no explicit time dependence) depend on time if
(i) the particle is initially in a momentum eigenstate?
(ii) the particle is initially in an energy eigenstate?
2. Relevant equations
The first two parts of this question required me to show that
[itex]\frac{d}{dt}[/itex]<Q> = [itex]\frac{i}{hbar}[/itex] <[H,Q]> + <[itex]\frac{d}{dt}[/itex]Q>
Q is any hermitian operator. I did this fine and then derived the virial theorem from this, which is where the rate of change of the expectation for Q is zero. I'm assuming I'm supposed to use this equation to find the conditions, but to be perfectly honest I have no idea how to approach this at all.
I know that if the operator commutes with the Hamiltonian H then it will have no dependence on time, but how can I use this to answer the question?
A particle is in a 1D harmonic oscillator potential. Under what conditions will the
expectation value of an operator Q (no explicit time dependence) depend on time if
(i) the particle is initially in a momentum eigenstate?
(ii) the particle is initially in an energy eigenstate?
2. Relevant equations
The first two parts of this question required me to show that
[itex]\frac{d}{dt}[/itex]<Q> = [itex]\frac{i}{hbar}[/itex] <[H,Q]> + <[itex]\frac{d}{dt}[/itex]Q>
Q is any hermitian operator. I did this fine and then derived the virial theorem from this, which is where the rate of change of the expectation for Q is zero. I'm assuming I'm supposed to use this equation to find the conditions, but to be perfectly honest I have no idea how to approach this at all.
I know that if the operator commutes with the Hamiltonian H then it will have no dependence on time, but how can I use this to answer the question?
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