Eigenvalues and eigenfunctions of the lowering operator

[Ed Quanta]

Homework Statement

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?

Homework Equations

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

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

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

 

[ziad1985]

a-Ψn(x)= sqrt(n)Ψn-1(x) in this example Ψ is not an eigenfunction of a-, this have no use here as far I can think of..
I'm thinking of applying a+ to the eigenfunction of a-, and see what it should give you..

 

[dextercioby]

You're looking for the wavefunctions of the coherent states for the harm. osc. See the treatment in Galindo and Pascual, vol. 1. It turns out that the spectrum of the lowering ladder operator is the entire complex plane.