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?
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