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I Constraints on potential for normalizable wavefunction

  1. Apr 10, 2017 #1
    We know that in one dimension if ##E>V(\infty)## or ##E>V(-\infty)## then the resulting wave function will not be normalizable. The basic argument is that if ##E>V(\infty)##, then a stationary solution to the Schrodinger equation will necessarily have a concavity with the same sign as the solution itself for all ##x## greater than some value ##a##. So if the resulting wavefunction was positive after ##a##, then the wavefunction would also be concave up and curve away from the x-axis. Then the integral of the squared wavefunction would tend towards infinity since the wavefunction would never again go to ##0##. A similar argument can be made if the wavefunction was negative after ##a## and again for ##E>V(-\infty)##. We don't have this problem with the infinite square well or simple harmonic oscillator because the potentials go to infinity at ##\pm \infty##.

    In short, my question is this: What are necessary and sufficient conditions for a potential to generate normalizable solutions?

    I would prefer to keep things simple with a one-dimensional treatment that doesn't need to be as rigorous as say a functional analysis proof (so maybe constrain ourselves to potentials that are at least C2). But after that's accomplished, I definitely wouldn't mind a more advanced discussion.
     
  2. jcsd
  3. Apr 16, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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