Proving Completeness of SHO's Coherent States

  • Thread starter Thread starter WisheDeom
  • Start date Start date
  • Tags Tags
    Coherent States
WisheDeom
Messages
11
Reaction score
0

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?
 
Physics news on Phys.org
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}
 
##|\Psi|^2=\frac{1}{\sqrt{\pi b^2}}\exp(\frac{-(x-x_0)^2}{b^2}).## ##\braket{x}=\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dx\,x\exp(-\frac{(x-x_0)^2}{b^2}).## ##y=x-x_0 \quad x=y+x_0 \quad dy=dx.## The boundaries remain infinite, I believe. ##\frac{1}{\sqrt{\pi b^2}}\int_{-\infty}^{\infty}dy(y+x_0)\exp(\frac{-y^2}{b^2}).## ##\frac{2}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,y\exp(\frac{-y^2}{b^2})+\frac{2x_0}{\sqrt{\pi b^2}}\int_0^{\infty}dy\,\exp(-\frac{y^2}{b^2}).## I then resolved the two...
Hello everyone, I’m considering a point charge q that oscillates harmonically about the origin along the z-axis, e.g. $$z_{q}(t)= A\sin(wt)$$ In a strongly simplified / quasi-instantaneous approximation I ignore retardation and take the electric field at the position ##r=(x,y,z)## simply to be the “Coulomb field at the charge’s instantaneous position”: $$E(r,t)=\frac{q}{4\pi\varepsilon_{0}}\frac{r-r_{q}(t)}{||r-r_{q}(t)||^{3}}$$ with $$r_{q}(t)=(0,0,z_{q}(t))$$ (I’m aware this isn’t...
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
Back
Top