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Quantum Mechanics Question

  1. Sep 8, 2008 #1
    1. The problem statement, all variables and given/known data
    I am currently doing undergraduate research and was assigned this as sort of an introduction. I am sure this is a very rudimentary problem and appreciate any help.

    Basically, its your regular old H*psi = E*psi.

    Well, knowing that H is psi(-D^2/Dx^2 + cosx + sin^2x = E*psi

    and I want to prove that Ne^(lambda(1-cos)) is where E is equal to zero where N is the normalization constant and lambda is an arbitrary constant.

    2. The attempt at a solution

    The thing that I tried to do was divide by psi, and set E=0 to completely get rid of psi. I am not sure if I am allowed to do this but I did. This left me with

    -D^2/Dx^2 + cosx + sin^2x = 0.

    From there, I moved them to seperate sides and doubly integrated. I got

    ln|x| = cos^2(x)/4 - cos(x) +x^2/4

    and exponentating I got

    x = e^(Cos^2(x)/4 - cos(x) +x^2/4)

    The problem is, without psi, I don't think that derivitave means anything and so I think I need to somehow keep the psi in there but I don't know what to do other than to divide out psi.

    I would also rather a few hints instead of an explicit solution; I am sure that its just something that I am over looking and with a hint or two, I could do this.

    I am very gracious of your help,

  2. jcsd
  3. Sep 8, 2008 #2


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    H operates on psi. So the equation is -psi(x)''+(cos(x)+sin^2(x))*psi(x)=0. And you don't have to solve it. Just substitute psi(x)=N*exp(lambda*(1-cos(x)). I think you'll find it works for only one choice of lambda.
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