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! :-)