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Eigenvalues and eigenfunctions of the lowering operator

  1. Mar 10, 2007 #1
    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. jcsd
  3. Mar 10, 2007 #2
    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..
  4. Mar 12, 2007 #3


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    Homework Helper

    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.
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