SHO in 2D - ground state energy

[nhanle]

Homework Statement

A two-dimensional harmonic oscillator is described by a potential of the form
V(x,y) = 1/2 m $$\omega^{2}$$(x$$^{2}$$+y$$^{2}$$ + $$\alpha (x-y)^{2}$$
where $$\alpha$$ is a positive constant.

Homework Equations

Find the ground-state energy of the oscillator

The Attempt at a Solution

I have tried to plug in the energy of SHO for each dimension x,y; yielding E = h_bar $$\omega$$(n_x+1/2) + h_bar $$\omega$$(n_y+1/2)
which method should I use to solve the third term i.e. $$\alpha (x-y)^{2}$$?

 

[vela]

You can only use $E=\hbar\omega(n+1/2)$ if the Hamiltonian is of the form

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m\omega^2\hat{x}^2$$

In this problem, the xy cross term messes things up, so you can't just look at it as two harmonic oscillators with the same frequency. You want to find a change of coordinates that will get rid of the cross term.

 

[nhanle]

Thank you, after your suggestion

here is my attempt

x' = x + y
y' = x - y

So the Hamiltonian becomes

$$\hat{H}= \frac{\hat{p'^2}}{2m} + m\omega^2( \frac{1}{2} (\hat{x'^2}) + \frac{(1+ 2\alpha)}{2}(\hat{y'^2}))$$

where $$\hat{p'}^2 = (\hat{p_{x'}})^2 + (\hat{p_{y'}})^2$$

Question:
- Is it similar to the 2D SHO with the perturbed term $$\hat{H^{1}}} = \alpha\:\hat{y'}^2$$?

 

[vela]

Shouldn't there be a factor of sqrt(2) in your coordinate transformation?

You don't need to treat it as a perturbation. Your Hamiltonian is now that of two harmonic oscillators of different frequencies:

$$\hat{H}= \left[\frac{\hat{p}'_x^2}{2m} + \frac{1}{2} m\omega^2 \hat{x}'^2\right] + \left[\frac{\hat{p}'_y^2}{2m} + \frac{1}{2} m\omega'^2 \hat{y}'^2\right]$$

where $\omega'^2 = \omega^2(1+2\alpha)$.

 

[nhanle]

wow, thank you so much.

Just to extend this arguments further. If the question was about the eigenstates of the modified Hamiltonian
$$\hat{H} = \frac{\hat{p}^2}{2m} + m\omega^2 ( x^2 + y^2 + \alpha (y-x)^2 + z^2)$$

, can I assume:

1) the ground state |n,l,m> has its symmetry skewed in the direction of (x-y)

2) the n =2 state has mixing between 211 and 21-1 that split the energy

Is it correct assumption? Just want to know so that I won't get lost when trying to do the calculation

 

[vela]

wow, thank you so much.

Just to extend this arguments further. If the question was about the eigenstates of the modified Hamiltonian
$$\hat{H} = \frac{\hat{p}^2}{2m} + m\omega^2 ( x^2 + y^2 + \alpha (y-x)^2 + z^2)$$

, can I assume:

1) the ground state |n,l,m> has its symmetry skewed in the direction of (x-y)

Yes, the spherical symmetry is broken by the (x-y) term.

Did you mean |n_x,n_y,n_z>? Or did you solve the 3D harmonic oscillator, with the non-spherically symmetric potential, in spherical coordinates?

2) the n =2 state has mixing between 211 and 21-1 that split the energy

Is it correct assumption? Just want to know so that I won't get lost when trying to do the calculation

I don't know.