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## Homework Statement

(See attachment)

## Homework Equations

[itex] x = \sqrt{\frac{\hbar}{2m \omega}} ( a + a^{\dagger} )[/itex]

[itex] x = i \sqrt{\frac{\hbar m \omega}{2}} ( a^{\dagger} - a )[/itex]

## The Attempt at a Solution

In part a) I was able to construct a separable Hamiltonian for the harmonic oscillators in the x and y direction.

The x Hamiltonian includes the term [itex] - \frac{λ x'^{2}}{2}[/itex]

and the y Hamiltonian includes the term [itex] \frac{λ y'^{2}}{2}[/itex]

Before moving on to part b), at my professor's advice, I collected the squared terms like so:

(for x): [itex]\frac{m}{2} ( \omega^{2} - \frac{λ}{m} ) x'^{2}[/itex]

Calling the term within the brackets α (for y I called it β since there is a + instead of a -)

moving on to part b) I attempted to solve for the energy states by expressing all of the position and momentum operators in terms of the raising and lowering operators.

(for x) after expanding:

[itex]H_{x'} = -\frac{\hbar \omega}{4}(a_{x'}^{\dagger 2} - a_{x'}a_{x'}^{\dagger}- a_{x'}^{\dagger}a_{x'} + a_{x'}^{2}) + \frac{\hbar \omega}{4α}(a_{x'}^{2} + a_{x'}a_{x'}^{\dagger} + a_{x'}^{\dagger}a_{x'} + a_{x'}^{\dagger 2})[/itex]

After simplifying using the commutator between a and a dagger and a few steps of algebra:

[itex]H_{x'} = \frac{\hbar \omega}{4α} [ (2n_{x} + 1)(α + 1) + (1 - α)(a_{x'}^{\dagger 2} + a_{x'}^{2})][/itex]

I'm pretty sure I can't have those raising and lowering operators in my energy eigenvalues but I can't see any way to eliminate them, I know that in the unperturbed oscillator, the squared terms from the position and momentum operators will cancel, but the alpha and beta are causing problems.

Thanks in advance.