# Eigenvalues and eigenfunctions of the lowering operator

1. Mar 10, 2007

### Ed Quanta

1. The problem statement, all variables and given/known data
Consider lowering and rising operators that we encountered in the harmonic oscillator problem.
1. Find the eigenvalues and eigenfunctions of the lowering operator.
2. Does the rising operator have normalizable eigenfunctions?

2. Relevant equations
a-= 1/sqrt(2hmw) (ip + mwx)
a+ = 1/sqrt(2hmw) (ip - mwx)

a-Ψ(x) = yΨ(x) where y is the eigenvalue

3. The attempt at a solution

So I applied a-, the lowering operator to Ψ(x) and eventually ended up with the differential equation

dΨ(x)/dx + (mwx/h - sqrt (2hmw)y/h)Ψ(x)=0

I believe I solved this differential equation correctly using separation of variables and ended up with

Ψ(x)= A exp (-(mwx^2)/h + sqrt(2hmw)(y)x/h)

What do I do now? How do I find eigenvalue y? I know that a-Ψn(x)= sqrt(n)Ψn-1(x)?

Am I supposed to be able to come up with this result? If so, how? Thanks

2. Mar 10, 2007