1. The problem statement, all variables and given/known data
The expectation value of the time derivative of an arbitrary quantum operator [itex]\hat{O}[/itex] is given by the expression:
Obtain an expression for [itex]\langle[/itex]d[itex]\hat{L}[/itex]x/dt+d[itex]\hat{L}[/itex]y/dt[itex]\rangle[/itex] where [itex]\hat{H}[/itex]=[itex]\hat{H}[/itex]0+μ0Bz[itex]\hat{L}[/itex]z/hbar
2. Relevant equations
[itex][[/itex][itex]\hat{L}[/itex]x,[itex]\hat{L}[/itex]y[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]z
[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]x
[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]y
[itex][[/itex]A,B[itex]][/itex]=AB-BA
3. The attempt at a solution
[itex]\langle[/itex]d[itex]\hat{L}[/itex]x/dt+d[itex]\hat{L}[/itex]y/dt[itex]\rangle[/itex]=d[itex]\langle[/itex][itex]\hat{L}[/itex]x+[itex]\hat{L}[/itex]y[itex]\rangle[/itex]
=[itex]\frac{1}{ih}[/itex] d[itex]\langle[/itex][itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex][itex]\rangle[/itex]/dt
=[itex]\frac{1}{ih}[/itex][itex]\langle[/itex] ∂[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex]/∂t[itex]\rangle[/itex]+[itex]\frac{i}{hbar}[/itex][itex]\langle[/itex][itex][[/itex][itex]\hat{H}[/itex],[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex][itex]][/itex][itex]\rangle[/itex]
I'm not sure how to continue on from this
The expectation value of the time derivative of an arbitrary quantum operator [itex]\hat{O}[/itex] is given by the expression:
d[itex]\langle[/itex][itex]\hat{O}[/itex][itex]\rangle[/itex]/dt[itex]\equiv[/itex][itex]\langle[/itex]d[itex]\hat{O}[/itex]/dt[itex]\rangle[/itex]=[itex]\langle[/itex]∂[itex]\hat{O}[/itex]/∂t[itex]\rangle[/itex]+i/hbar[itex]\langle[/itex][itex][[/itex][itex]\hat{H}[/itex],[itex]\hat{O}[/itex][itex]][/itex][itex]\rangle[/itex]
Obtain an expression for [itex]\langle[/itex]d[itex]\hat{L}[/itex]x/dt+d[itex]\hat{L}[/itex]y/dt[itex]\rangle[/itex] where [itex]\hat{H}[/itex]=[itex]\hat{H}[/itex]0+μ0Bz[itex]\hat{L}[/itex]z/hbar
2. Relevant equations
[itex][[/itex][itex]\hat{L}[/itex]x,[itex]\hat{L}[/itex]y[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]z
[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]x
[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex]=i*hbar[itex]\hat{L}[/itex]y
[itex][[/itex]A,B[itex]][/itex]=AB-BA
3. The attempt at a solution
[itex]\langle[/itex]d[itex]\hat{L}[/itex]x/dt+d[itex]\hat{L}[/itex]y/dt[itex]\rangle[/itex]=d[itex]\langle[/itex][itex]\hat{L}[/itex]x+[itex]\hat{L}[/itex]y[itex]\rangle[/itex]
=[itex]\frac{1}{ih}[/itex] d[itex]\langle[/itex][itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex][itex]\rangle[/itex]/dt
=[itex]\frac{1}{ih}[/itex][itex]\langle[/itex] ∂[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex]/∂t[itex]\rangle[/itex]+[itex]\frac{i}{hbar}[/itex][itex]\langle[/itex][itex][[/itex][itex]\hat{H}[/itex],[itex][[/itex][itex]\hat{L}[/itex]y,[itex]\hat{L}[/itex]z[itex]][/itex]+[itex][[/itex][itex]\hat{L}[/itex]z,[itex]\hat{L}[/itex]x[itex]][/itex][itex]][/itex][itex]\rangle[/itex]
I'm not sure how to continue on from this
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