Quantum Oscillator with different frequencies

[mgal95]

Homework Statement

Solve the Schrödinger Equation for an harmonic potential of the form [itex](1/2)m\omega_+^2x^2[/itex] for x>0 and [itex](1/2)m\omega_-^2x^2[/itex] for x<0. Find the equation that determines the energy spectrum. You can use [itex]m=1/2[/itex] and [itex]\hbar=1[/itex]

Homework Equations

[/B]
I wrote down Schrödinger Equation for this potential
[tex]-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}\psi_E+\left(\frac{1}{2}m\omega_+^2x^2\theta(x)+\frac{1}{2}m\omega_-^2x^2\theta(-x)\right)\psi_E=E\psi_E[/tex]

where [itex]\theta(x)[/itex] is the step function.

The Attempt at a Solution

I tried several tricks. First I tried to solve the equation for x>0 and x<0 separately and then impose the wavefunction and its first derivative to be continuous to x=0. However, solving the Schrödinger for the Harmonic Oscillator demands the use of the Frobenius method (solution in the form of a series) and that quantizes the energy as [itex]E_{+,-}=\hbar\omega_{+,-}(n+1/2)[/itex]. However, if I solve the equation right and left I get different values for the constant E which, for arbitrary frequencies, do not have to be equal. I guess the problem is that in the Frobenius method one uses the fact that the symmetric potential admits solutions that are eigenstates of parity also, which is not the case here since the potential is not symmetric.

Another way I tried was to expand in the basis of let's say the eigenfunctions for the harmonic oscillator with [itex]\omega_+[/itex] frequency, since this is a complete set for the linear operator in the Schrödinger equation with the given boundary conditions (same as in the standard quantum harmonic oscillator of course). I derived an equation for the coefficients of the expansion, but I actually have an infinite system of linear equations to solve, because the coefficients seem to mix, due to terms of the form

[tex]\int_0^\infty\psi_n\psi_mx^2dx[/tex]

which do not vanish for [itex]n\neq m[/itex]. [itex]\psi_k[/itex] is the k-th eigenfunction of the harmonic oscillator with frequency [itex]\omega_+[/itex]. On top of that I could not determine an equation that could give me the energy spectrum. I should impose (maybe) some boundary condition, since this is why quantization arises in such systems, but I cannot find one.

Do you have any ideas?

Thanks!

 

[blue_leaf77]

Your first attempt seems more doable. But we cannot comment on anything unless you show us what you got in your first method.

 

[Dr Transport]

There is a trick you haven't considered for the solution of each side independently...