Proving Completeness of SHO's Coherent States

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The discussion centers on proving the completeness of coherent states in the quantum simple harmonic oscillator (SHO), which are defined as eigenkets of the creation operator. Completeness requires demonstrating that the infinite series sum of these states converges to a non-zero value, with some sources suggesting it converges to π. The participant expresses uncertainty about how to approach the integration over the complex plane and the necessary criteria for convergence in this context. They attempt to parameterize the eigenvalues but find the calculations complex and challenging. The discussion highlights the need for a clearer understanding of operator calculus to evaluate the series involved in the completeness proof.
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Homework Statement


I must prove that the set of coherent states \left\{ \left| \lambda \right\rangle \right\} of the quantum simple harmonic oscillator (SHO) is a complete set, i.e. it forms a basis for the Hilbert space of the SHO.

Homework Equations


The coherent states are defined as eigenkets of the creation operator with eigenvalue \lambda; in terms of the energy eigenkets they can be written

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

Completeness means the sum (infinite series in this case)

\sum_{\left\{ \left| \lambda \right\rangle \right\}} \left| \lambda \right\rangle \left\langle \lambda \right|

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

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 \lambda = x + iy, and then separately by \lambda = r e^{i \theta}, 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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Can you show the series involved in the ket-bra converges ? What criteria do you know for an infinite series to converge ?
 
dextercioby said:
Can you show the series involved in the ket-bra converges ? What criteria do you know for an infinite series to converge ?

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

\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|

but I don't know how to evaluate this.
 
But you know that

|m\rangle\langle n| = \delta_{mn}
 
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