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Expectation values for a harmonic oscillator

  1. Mar 31, 2010 #1
    1. The problem statement, all variables and given/known data
    I need to find <x>, <x2>, <p>, and <p2> for a particle in the first state of a harmonic oscillator.

    2. Relevant equations
    The harmonic oscillator in the first state is described by [tex]\psi[/tex](x)=A[tex]\alpha[/tex]1/2*x*e-[tex]\alpha[/tex]*x2/2. I'm using the definition <Q>=([tex]\int[/tex][tex]\psi[/tex]1*Q*[tex]\psi[/tex])dx where [tex]\psi[/tex]1 is the complex conjugate of [tex]\psi[/tex], and Q is the specific operator.

    3. The attempt at a solution
    I solved for <x>, and found it was zero. <p> I'll solve for in a similar fashion. However, for <x2> and <p2>, I am unsure of what operators I use. For the <x> operator, it is simply x, so for <x2>, would I use x2 as an operator?
    (note that I the only superscripts here are the ones above e, I don't know why latex is putting all of my symbols so high
  2. jcsd
  3. Mar 31, 2010 #2


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    Gold Member

    Yes, that's what you should use.
  4. Mar 31, 2010 #3


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    Gold Member

    (Put the whole equation in the tex barackets instead of individual symbols)

    The operator for <x²> is simply multiplication by x², so <x²> = ∫ψ*(x)x²ψ(x)dx, and <p²> is

    [tex] -\int \psi^{*}(x) \hbar^2\frac{\partial^2}{\partial x^2}\psi(x) dx [/tex]
  5. Mar 31, 2010 #4
    ok, thank you so much guys. It's a good thing I have physics forum to at least make my new ventures into the realm of quantum mechanics a bit easier :)
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