Solving 2-D Schrodinger Equation?

[sciguy2010]

Homework Statement

A particle of mass m is confined in a two-dimensions by the potential energy V = 1/2k(x²+4y²). Write down the Schrodinger equation for the system. Write down the ground state wave function and find the lowest four energy levels in terms of the quantities ħ, k, m etc. Make clear which, if any, of the levels is degenerate. Use any of the results you need from the one-dimensional harmonic oscillator without proof.

Homework Equations

Schrodinger's equation, two-dimensional:

[(-ħ/2m)(d²/d²x + d²/d²y) + V(x)] [itex]\psi[/itex]_E(x,y)]=E [itex]\psi[/itex]_E(x,y)

One-dimensional harmonic oscillator equations:

[itex]\psi[/itex]₀ = (m[itex]\omega[/itex]₀/ħ[itex]\pi[/itex])e^-ax2

[itex]\omega[/itex]_o=[itex]\sqrt{k/m}[/itex]

a=[itex]\sqrt{km}[/itex]/2ħ

E_n = (n+1/2)ħ[itex]\omega[/itex]₀

The Attempt at a Solution

Schrodinger's equation:

[(-ħ²/2m)(d²/d²x + d²/d²y) + 1/2k(x²+4y²)] [itex]\psi[/itex]_E(x,y)=E[itex]\psi[/itex]_E(x,y)

After this I'm not really sure what to do. As for what to plug in where, and how to solve for the energy levels. Do I use the equation to solve for the energy levels? How?

 

[jfy4]

I think you can treat x and y separately and simply write down the 1D stuff you already know, you will have to find out what the scaling factors are in each case if they differ from the canonical HO, but, say, for the energy eigenvalues, I think it will just be a sum of the energies in each direction with independent integer labels, n, m.

 

[vela]

Assume a solution of the form [itex]\psi(x,y) = X(x)Y(y)[/itex] and use separation of variables.