3D Quantum harmonic Oscillator

[c299792458]

Homework Statement

What are the stationary states of an isotropic 3D quantum harmonic oscillator in a potential U(x,y,z) = {1\over2}m\omega^2 (x^2+y^2+z^2) in the form \psi(x,y,z)=f(x)g(y)h(z) and how many linearly independent states have energy E=({3\over 2}+n)\hbar\omega?

Homework Equations

See above.

The Attempt at a Solution

The solution of a HO in 1D, say the x-direction, is c H_n e^{\sqrt{m\omega\over\hbar}x} where H_n is the nth Hermite polynomial. So I am guessing \psi = c^3 H_n^3 e^{{m\omega\over\hbar}(x^2+y^2+z^2)}. I don't quit understand how to count L.I. states, though. I am guessing the number of combinations of n_x,n_y,n_z such that n_x+n_y+n_z=n? Please help! Thanks

 

[vela]

Homework Statement

What are the stationary states of an isotropic 3D quantum harmonic oscillator in a potential U(x,y,z) = {1\over2}m\omega^2 (x^2+y^2+z^2) in the form \psi(x,y,z)=f(x)g(y)h(z) and how many linearly independent states have energy E=({3\over 2}+n)\hbar\omega?

Homework Equations

See above.

The Attempt at a Solution

The solution of a HO in 1D, say the x-direction, is c H_n e^{\sqrt{m\omega\over\hbar}x} where H_n is the nth Hermite polynomial.

That's not quite right.

So I am guessing \psi = c^3 H_n^3 e^{{m\omega\over\hbar}(x^2+y^2+z^2)}.

Decent guess. Try plugging \psi(x,y,z) into the Schrödinger equation. You should find the equation separates, so you can solve it.

I don't quit understand how to count L.I. states, though. I am guessing the number of combinations of n_x,n_y,n_z such that n_x+n_y+n_z=n? Please help! Thanks

Yes, that's correct.

 

[c299792458]

Thanks, vela. So is \psi=cH_{n_x}H_{n_y}H_{n_z}e^{{m\omega\over\hbar}{(x^2+y^2+z^2)}}, where c is some normalization constant?

 

[vela]

Close. You're missing a factor of -1/2 in the exponent. What's the argument of the Hermite polynomials?

 

[c299792458]

Ah, thanks. Is the argument \sqrt{m\omega\over\hbar}x_i, where x_i\in\{x,y,z\}?

 

[vela]

Yup!

 

[c299792458]

Thanks! :-)