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Question Regarding Harmonic Oscillator Eigenkets

  1. Oct 26, 2011 #1
    Hi everyone!

    Given that a harmonic oscillator has eigenkstates |n> where n = 1,2,3,..., how can we calculate <X>, <P>, <X^2>, etc. Is there a need to define a wavefunction in the |n> basis?

  2. jcsd
  3. Oct 26, 2011 #2


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    Essentially no, because |n>'s are eigenkets of the number operator/Hamiltonian and X and P, though unbounded & with purely continuous spectrum, can be expressed as linear combinations of the raising & lowering ladder operators whose action on the eigenket's space becomes known once you establish that |n>'s are eigenkets of H and N.
  4. Oct 26, 2011 #3
    So that means just express the |n> kets as linear combinations of the ladder operators, and then use them as ψ in the formula <X> = <ψ|X|ψ>? But how would you deal with the infinite dimensionality? Will the answer be finite in that case?

    Thank you by the way for the idea!
  5. Oct 26, 2011 #4
    Consider that
    You can use these to write [itex]\hat{x}[/itex] and [itex]\hat{p}[/itex] in terms of [itex]\hat{a}[/itex] and [itex]\hat{a}^{\dagger}[/itex]. Then you know
    \hat{a}|n\rangle = \sqrt{n}|n-1\rangle
    You have the tools to take the expectation value.
  6. Oct 27, 2011 #5


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    You don't express the kets as ladder operators acting on the vacuum, you express x and p as ladder operators.
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