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Calculating a Quantum Nomalization Constant

  1. Oct 16, 2008 #1
    1. The problem statement, all variables and given/known data

    If I have a wave function that = Ce((-mwx2)/(2*hbar))

    (Its the wave function of the ground state of a simple harmonic oscilator)


    How do I calculate C?



    2. Relevant equations

    Quantum Normalization condition I think is all i need.



    3. The attempt at a solution


    C2 is pulled out of the integral because its a constant

    Leaving me with the integral from - infinity to infinity of e-mwx2/hbar

    How do I integrate that? Is there an easier way to solve for C?

    Everythings 1D
     
  2. jcsd
  3. Oct 16, 2008 #2
  4. Oct 16, 2008 #3
    Ok, now I know that

    C= sqrt[1/(sqrt((pi*hbar)/(m*w)))]


    However, I dont know w.

    Odd....
     
  5. Oct 16, 2008 #4
    Ok, now I need to solve for <x2>

    Which means I obviously end up with:

    C2 times the integral from - infinity to inifinty of x2*e-ax2

    where a = mw/hbar

    I can seem to find a solution to this integral in my handbook. How do you intgrate that?
     
  6. Oct 17, 2008 #5
    Last edited: Oct 17, 2008
  7. Oct 17, 2008 #6
  8. Oct 17, 2008 #7
    [tex]

    \begin{align*}
    \int_{ - \infty }^\infty {x^2 \exp \left( { - \lambda x^2 } \right)} dx = - \int_{ - \infty }^\infty {\frac{\partial }{{\partial \lambda }}\exp \left( { - \lambda x^2 } \right)} dx = - \frac{\partial}{{\partial\lambda }}\int_{ - \infty }^\infty {\exp \left( { - \lambda x^2 } \right)} dx = - \frac{d}{{d\lambda }}\sqrt {\frac{\pi }{\lambda }}
    \end{align*}

    [/tex]

    - this is what the link contains. It is called differentiation under integration.

    Of course partial integration can be used too, but the above method simplifies things greatly.
     
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