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## Homework Statement

Part (a): Derive Ehrenfest's Theorem. What is a good quantum number?

Part (b): Write down the energy eigenvalues and sketch energy diagram showing first 6 levels.

Part (c): What's the symmetry of the new system and what happens to energy levels? Find a new good quantum number and corresponding operator. Use Ehrenfest's to show it is true.

Part (d): Write down the new wavefunctions.

## Homework Equations

## The Attempt at a Solution

__Part (a)__[tex]\frac{d}{dt}<\psi|Q|\psi> = -<\psi|HQ|\psi> + i\hbar<\psi|\frac{dQ}{dt}|\psi> + <\psi|QH|\psi>[/tex]

[tex] = <\psi|QH - HQ|\psi> + i\hbar<\psi|\frac{dQ}{dt}|\psi>[/tex]

Assuming observable doesn't change with time:

[tex] = <\psi|[Q,H]|\psi>[/tex]

If ##[Q,H] = 0##, then Q and H share a common ket ##|\psi>##such that ##Q|\psi> = q_o|\psi>##, where ##q_0## is the good quantum number.

Stationary state is when ##\frac{d<Q>}{dt} = 0##.

__Part (b)__[tex] E = (n_x + \frac{1}{2})\hbar \omega_x + (n_y + \frac{1}{2})\hbar \omega_y [/tex]

For ##\omega_x = \omega_y + \delta \omega##, ##E = \hbar \omega_y(n_x + n_y +1) + (n_x + \frac{1}{2})\hbar \delta \omega##.

Can I assume that energy contribution of ##\delta \omega## is small that when drawing the graph I can just ignore it?

In ascending order, the energy levels are: ##\hbar\omega_y##, ##2\hbar\omega_y##, ##3\hbar \omega_y## .... where (nx,ny): the first is nx = ny = 0, second is (0,1) or (1,0), the third is (1,1) or (2,0) or (0,2).

And how do I sketch the energy levels? Is it simply:

__Part (c)__There's hardly any change to the energy diagram, since we ignored the contribution due to ##\delta \omega##?

I'm guessing the symmetry is the good quantum number ##n = n_x + n_y## is conserved? What do they mean by the corresponding differential operator?

__Part (d)__I have no idea why the energy eigenstates will change. Are they the same as eigenfunctions?