# SHO in 2D - ground state energy

## 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 $$\omega$$(nx+1/2) + h_bar $$\omega$$(ny+1/2)
which method should I use to solve the third term i.e. $$\alpha (x-y)^{2}$$?

vela
Staff Emeritus
Homework Helper
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.

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 $$\hat{H^{1}}} = \alpha\:\hat{y'}^2$$?

Last edited:
vela
Staff Emeritus
Homework Helper
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 $\omega'^2 = \omega^2(1+2\alpha)$.

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
Staff Emeritus
Homework Helper
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 |nx,ny,nz>? 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.