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The Virial Theorem problem

  1. Apr 12, 2010 #1
    1. The problem statement, all variables and given/known data
    A particle is moving along the x-axis in the potential:

    [tex]\[V\left( x \right)=k{{x}^{n}},\][/tex]
    where [itex]k[/itex] is a constant, and [itex]n[/itex] is a positive even integer. [itex]\left| \psi \right\rangle [/itex] is described as a normed eigenfunction for the Hamiltonoperator with eigenvalue E.

    Show through the "Virial Theorem" that:

    [tex]\[\begin{align}
    & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle =\frac{2}{n+2}E \\
    & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle =\frac{2}{n+2}E,
    \end{align}\]
    [/tex]
    where [itex]\hat{V}\[/itex] and [itex]\hat{T}\[/itex] denotes the operators respectively for potential and kinetic energy.


    2. Relevant equations
    The Virial Theorem:

    [tex]\[2\left\langle T \right\rangle =\left\langle x\frac{dV}{dx} \right\rangle \][/tex]

    3. The attempt at a solution
    Well, I'm kinda lost.
    I'm not sure how to calculate anything tbh...

    The thing that confuses me, which is what I think I should do, is calculating:

    [tex]\[\begin{align}
    & \left\langle \psi \right|\hat{V}\left| \psi \right\rangle \\
    & \left\langle \psi \right|\hat{T}\left| \psi \right\rangle \\
    \end{align}\]
    [/tex]

    But can't find anything in my book that shows how to calculate anything that looks like that.

    So a hint would be very helpful :)


    Regards
     
  2. jcsd
  3. Apr 12, 2010 #2

    George Jones

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    Start with

    [tex]
    \left\langle \psi \right|\hat{H}\left| \psi \right\rangle = \left\langle \psi \right|\hat{T} + \hat{V} \left| \psi \right\rangle.
    [/tex]

    What is the left side? What is the right side?
     
  4. Apr 14, 2010 #3
    Sorry for the late reply...

    But that is my problem. I'm not sure how to calculate that ?
    Is it an integral, a commutator trick, or...? As I said, I can't seem to find anything in my book that shows how to calculate that.
     
  5. Apr 15, 2010 #4

    George Jones

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    Start at the beginning. What is

    [tex]\left\langle \psi \right|\hat{H}\left| \psi \right\rangle?[/tex]
     
  6. Apr 15, 2010 #5

    Cyosis

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    Homework Helper

    Hint to George Jones' question:

    [itex] \left| \psi \right\rangle [/itex] is described as a normed eigenfunction for the Hamiltonoperator with eigenvalue E.
     
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