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QM harmonic oscillator - integrating over a gaussian?

  1. Dec 4, 2017 #1
    1. The problem statement, all variables and given/known data

    For the first excited state of a Q.H.O., what is the probability of finding the particle in -0.2 < x < 0.2

    2. Relevant equations

    Wavefunction for first excited state: Ψ= (√2) y e-y2/2

    where: [​IMG]
    3. The attempt at a solution

    To find the probability, I tried the integral of : |Ψ|2

    but this gives the integral of gaussian. From what I've read, the integral of a gaussian can only be solved from -infinity to infinity. So how can I find it from -0.2 to 0.2?
  2. jcsd
  3. Dec 4, 2017 #2


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    Look up error function, then use a canned algorithm such as on a spreadsheet to find its value.
  4. Dec 4, 2017 #3


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  5. Dec 4, 2017 #4


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    Use that ##e^x=\sum\limits_{n=0}^{\infty}{\frac{x^n}{n!}}## so that ##e^{-\frac{x^2}{2}}=\sum\limits_{n=0}^{\infty}{(-1)^n\frac{x^{2n}}{2^nn!}}##.

    So you can integrate like it is a polynomial with infinite terms. You can choose up to which term of n to keep but I think for your value of x between 0.2 and -0.2 , the first three or four terms of integration are enough. You ll probably have to use a computer program or at least a calculator if you choose a very high value for n like the first 10 terms or more.
    Last edited: Dec 4, 2017
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