Proving Completeness of SHO's Coherent States

[WisheDeom]

Homework Statement

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.

Homework 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.

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?

 

[dextercioby]

Can you show the series involved in the ket-bra converges ? What criteria do you know for an infinite series to converge ?

 

[WisheDeom]

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

[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.

 

[dextercioby]

But you know that

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

 

[paranormal]

I understand the importance of proving the completeness of a set in order to establish its basis for a given system. In this case, we are looking at the set of coherent states for the quantum simple harmonic oscillator (SHO).

First, we need to understand the definition of completeness in this context. Completeness means that any vector in the Hilbert space of the SHO can be expressed as a linear combination of the coherent states, or in other words, the coherent states span the entire Hilbert space.

To prove this, we need to show that the sum in the given equation converges and is non-zero. The sum is over all possible coherent states, which are defined as eigenkets of the creation operator with eigenvalue \lambda . This means that for any given value of \lambda , we have a corresponding coherent state.

To compute the sum, we can use the orthonormality property of the energy eigenkets, which states that \left\langle m | n \right\rangle = \delta_{mn} . This means that the only term that will contribute to the sum is when the coherent state matches the energy eigenket, i.e. when \lambda = n .

Using this property, we can rewrite the sum as:

\sum_{\left\{ \left| \lambda \right\rangle \right\}} \left| \lambda \right\rangle \left\langle \lambda \right| = \sum_n \left| n \right\rangle \left\langle n \right|

This sum is now over all the energy eigenkets, which we know to be complete. Therefore, the sum converges to the identity operator, which is non-zero.

This proves that the set of coherent states is complete and forms a basis for the Hilbert space of the SHO. As for the value of \pi , it is not related to the completeness of the set, but rather to the normalization constant in the definition of the coherent states. This can be computed using the integral of the Gaussian function in the definition of the coherent states.

In conclusion, we have shown that the sum in the given equation converges and is non-zero, proving the completeness of SHO's coherent states.