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A particle that moves in 3 dimensions has that Hamiltonian

[tex] H=p^2/2m+\alpha*(x^2+y^2+z^2)+\gamma*z[/tex] where [tex] \alpha[/tex] and [tex]\gamma[/tex] are real nonzero constant numbers.

a) For each of the following observables , state whether or why the observable is conserved: parity , [tex]\Pi[/tex]; energy [tex]H[/tex] ; the z component of orbital angular momentum , [tex] L_z[/tex] ; the x component of orbital angular momentum , [tex] L_x[/tex] , the z componetm of the linear momentum [tex]p_z[/tex]

b) Reduce the expression for the time rate of change of the expectation value of the y component of orbital angular momentum , [tex]d<L_y>/dt[/tex] , to the simplest possible form. Find the classical analog to the result.

2. Relevant equations

3. The attempt at a solution

a) parity: [tex] \Pi \phi(r)=\phi(-r); [/tex] Have to show that H(r)=H(-r)

x -> -x

y-> -y

z -> -z

therefore , [tex] H=p^2/2m+\alpha*(x^2+y^2+z^2)+\gamma*z, H(-r)=] H=p^2/2m+\alpha*(-x)^2+(-y)^2+(-z)^2)+\gamma*(-z)=] H=p^2/2m+\alpha*(x^2+y^2+z^2)-\gamma*z[/tex]

observable for parity is not conserved since H(r) and H(-r) are not equal to each other.

For energy, I don't know how to show that the observable is observed, other than stating the Law of energy conservation, which I know thats not what you do; Same goes for the rest of the observables Should I take the commutator of : [H, L_z] , [H,L_x], [H,p_z]?

b) [tex] <L_y>=\varphi^2*L_ydy[/tex]. What do I set [tex] \varphi[/tex] equal to?

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# Hamiltonian problem: observables

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