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Quantum Theory: Completeness of Coherent States of the Simple Harmonic Oscillator

  1. Oct 9, 2012 #1
    1. The problem statement, all variables and given/known data
    I must prove that the set of coherent states [itex]\left\{ \left| \lambda \right\rangle \right\}[/itex] of the quantum simple harmonic oscillator (SHO) is a complete set, i.e. it forms a basis for the Hilbert space of the SHO.

    2. Relevant equations
    The coherent states are defined as eigenkets of the creation operator with eigenvalue [itex] \lambda [/itex]; in terms of the energy eigenkets they can be written

    [tex] \left| \lambda \right\rangle = \exp \left( -\frac{|\lambda|^2}{2} \right) \sum_n \frac{\lambda^n}{\sqrt{n!}} \left| n \right\rangle [/tex]

    Completeness means the sum (infinite series in this case)

    [tex]\sum_{\left\{ \left| \lambda \right\rangle \right\}} \left| \lambda \right\rangle \left\langle \lambda \right|[/tex]

    converges and is non-zero. Sites have told me the sum should converge to [itex] \pi [/itex], but I don't know how to compute that.

    3. The attempt at a solution

    I'm not even quite sure how to start. The eigenvalues are complex numbers, so I know the sum (integration) must be over the complex plane, but how should I do this? I tried parametrizing [itex] \lambda = x + iy [/itex], and then separately by [itex] \lambda = r e^{i \theta} [/itex], but both got very messy quickly, and I'm not sure what to do. Am I on the right track at all?
     
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  3. Oct 9, 2012 #2

    dextercioby

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    Can you show the series involved in the ket-bra converges ? What criteria do you know for an infinite series to converge ?
     
  4. Oct 10, 2012 #3
    I know there are a number of tests for series of numbers, but I'm not sure how to translate this to operators. My class didn't do any rigorous operator calculus; we sort of played it by ear. But in this case I'm not even sure how to start really.

    If I assume it does converge, the sum should be

    [tex] \int_{-\infty}^{\infty} d^2 \lambda e^{|\lambda|^2} \sum_m \sum_n \frac{\lambda*^m \lambda^n}{\sqrt{m! n!}} \left| m \right\rangle \left\langle n \right| [/tex]

    but I don't know how to evaluate this.
     
  5. Oct 11, 2012 #4

    dextercioby

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    But you know that

    [tex] |m\rangle\langle n| = \delta_{mn} [/tex]
     
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