1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

QM - Eigenfunction / Eigenvalue Problem

  1. Aug 31, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the eigenfunctions and eigenvalues for the operator:

    a = x + [tex]\frac{d}{dx}[/tex]

    2. The attempt at a solution

    a = x + [tex]\frac{d}{dx}[/tex]

    a[tex]\Psi[/tex] = [tex]\lambda[/tex][tex]\Psi[/tex]

    x[tex]\Psi[/tex] + [tex]\frac{d\Psi}{dx}[/tex] = [tex]\lambda[/tex][tex]\Psi[/tex]

    x + [tex]\frac{1}{\Psi} [/tex][tex]\frac{d\Psi}{dx}[/tex] = [tex]\lambda[/tex]

    x + [tex]\frac{d}{dx}[/tex] ln([tex]\Psi[/tex])= [tex]\lambda[/tex]

    [tex]\frac{1}{2}[/tex]x[tex]^{2}[/tex] + ln([tex]\Psi[/tex]) = [tex]\lambda[/tex]x +c

    [tex]\Psi[/tex] = e^(-[tex]\frac{1}{2}[/tex]x[tex]^{2}[/tex]+[tex]\lambda[/tex]x+c)

    [tex]\Psi[/tex] = e^(-[tex]\frac{1}{2}[/tex]x[tex]^{2}[/tex]+[tex]\lambda[/tex])[tex]\Psi[/tex](0)

    Not sure from here... I think I plug into initial equation?

    a[tex]\Psi[/tex] = [tex]\lambda[/tex][tex]\Psi[/tex]

    substituting and using chain rule I obtain...

    x[tex]\Psi[/tex] + ([tex]\lambda[/tex]-x)[tex]\Psi[/tex] = [tex]\lambda[/tex][tex]\Psi[/tex]

    so x + ([tex]\lambda[/tex] - x) = [tex]\lambda[/tex]

    so [tex]\lambda[/tex] = [tex]\lambda[/tex] nope I guess I don't do that... good check though, so I guess this is the correct way to solve for eigenfunction... Can someone help me with finding [tex]\lambda[/tex]?
     
  2. jcsd
  3. Aug 31, 2007 #2
    Normally, the spectrum of allowed eigenvalues [itex] \lambda [/itex] is obtained from the normalization condition. I.e., there should be a non-zero normalization constant (which you denoted by [itex] \Psi(0) [/itex]) such that the integral of the square of the modulus of the wavefunction over entire space (the total probability of finding the particle) is equal to 1. In your case, this doesn't matter, because the wavefunction is normalizable for all real [itex] \lambda [/itex]. So, all values of [itex] \lambda [/itex] are allowed. However, in more complex cases such as a particle in a potential well - the normalization condition would allow you to find the discrete spectrum of energy eigenvalues.

    Eugene.

    P.S. Perhaps, it is not a good idea to ask the same question in three different threads.

    https://www.physicsforums.com/showthread.php?t=182374
    https://www.physicsforums.com/showthread.php?t=182369
     
  4. Aug 31, 2007 #3
    Thanks Eugene...

    Sorry about that... Kind of new to this... I havn't seen a way to delete your own posts... but there probably is one...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: QM - Eigenfunction / Eigenvalue Problem
  1. Eigenvalues in QM (Replies: 8)

  2. Eigenvalue problem (Replies: 5)

  3. Qm problem (Replies: 0)

Loading...