1. Limited time only! Sign up for a free 30min personal 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 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
    [​IMG]
    [​IMG]

    where: [​IMG]
    [​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

    kuruman

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

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

    TSny

    User Avatar
    Homework Helper
    Gold Member

  5. Dec 4, 2017 #4

    Delta²

    User Avatar
    Gold Member

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: QM harmonic oscillator - integrating over a gaussian?
  1. QM: Harmonic oscillator (Replies: 13)

Loading...