1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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?
  2. jcsd
  3. Oct 9, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    But you know that

    [tex] |m\rangle\langle n| = \delta_{mn} [/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook