Energy Levels of Half Harmonic Oscillator

[samgrace]

Homework Statement

A harmonic oscillator of mass m and angular frequency ω experiences the potential:

V(x) = 1/2mω^{2}x^{2} between -infinity < x < +infinity


and solving the Schrödinger equation for this potential yields the energy levels

E_n = (n + 1/2) h_bar ω


Determine the energy levels for the half oscillator for which

V(x) = 1/2mω^{2}x^{2} between -infinity < x < 0

= infinity otherwise

The Attempt at a Solution

-h_bar^2/2m *d^2ψ(x)/dx^2 + 1/2mω^2x^2 = Eψ(x)


so d^2ψ(x)/dx^2 = -(E - 1/2mω^2x^2)*2m/h_bar^2 ψ(x) ==> d^2ψ(x)/dx^2 = k^2ψ(x)



So the general solution is ψ(x) = Ae^kx + Be^-kx

 

[Orodruin]

Your k is not constant and generally depends on x, which means your differential equation is more difficult than that to solve.

The fact that you have been given the energy levels for the full oscillator should be a hint. Can you think of a way to relate the problem of the half-oscillator to the full oscillator?