Proving Completeness of SHO's Coherent States

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Homework Help Overview

The original poster attempts to prove the completeness of the coherent states of the quantum simple harmonic oscillator (SHO), specifically that they form a basis for the Hilbert space of the SHO. The coherent states are defined as eigenkets of the creation operator, and the poster is exploring the convergence of an infinite series related to these states.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to show the convergence of the series involved in the ket-bra representation. The original poster questions how to approach the integration over the complex plane and considers different parametrizations for the variable λ. Others inquire about the criteria for convergence of infinite series and express uncertainty about applying these concepts to operators.

Discussion Status

The discussion is ongoing, with participants exploring various aspects of the problem. Some have suggested that the original poster assume convergence to facilitate further exploration, while others are questioning the foundational concepts and seeking clarification on the mathematical tools available for this type of analysis.

Contextual Notes

There is mention of the original poster's class not covering rigorous operator calculus, which may limit their ability to apply certain mathematical techniques. Additionally, the complexity of the integration over the complex plane is noted as a challenge in the discussion.

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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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