SHO in 2D - ground state energy

  • Thread starter nhanle
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Homework Statement


A two-dimensional harmonic oscillator is described by a potential of the form
V(x,y) = 1/2 m [tex] \omega^{2}[/tex](x[tex]^{2}[/tex]+y[tex]^{2}[/tex] + [tex]\alpha (x-y)^{2}[/tex]
where [tex]\alpha[/tex] 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 [tex]\omega[/tex](nx+1/2) + h_bar [tex]\omega[/tex](ny+1/2)
which method should I use to solve the third term i.e. [tex]\alpha (x-y)^{2}[/tex]?
 

Answers and Replies

  • #2
vela
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You can only use [itex]E=\hbar\omega(n+1/2)[/itex] if the Hamiltonian is of the form

[tex]\hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m\omega^2\hat{x}^2[/tex]

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.
 
  • #3
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Thank you, after your suggestion

here is my attempt

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

So the Hamiltonian becomes

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

where [tex]\hat{p'}^2 = (\hat{p_{x'}})^2 + (\hat{p_{y'}})^2 [/tex]

Question:
- Is it similar to the 2D SHO with the perturbed term [tex]\hat{H^{1}}} = \alpha\:\hat{y'}^2[/tex]?
 
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  • #4
vela
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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:

[tex]\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][/tex]

where [itex]\omega'^2 = \omega^2(1+2\alpha)[/itex].
 
  • #5
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wow, thank you so much.

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

, 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
 
  • #6
vela
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wow, thank you so much.

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

, 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.
 

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