Harmonic oscillator coherent state wavefunction

[quantum539]

Hi, I am trying to find the wavefunction of a coherent state of the harmonic oscillator ( potential mw²x²/2 ) with eigenvalue of the lowering operator: b.

I know you can do this is many ways, but I cannot figure out why this particular method does not work.

It can be shown (and you can find this easily on the internet) that the eigenvalue b evolves as:

b(t)=b₀e^-iwtWhat I did was find the coherent state wavefunction u(x,t) by using the eigenfunction equation with the lowering operator a:

a[u(x,t)]=b*u(x,t)

with a=(mwx+(h-bar)*d/dx)/sqrt(2mw(h-bar))

that gives

u(x,t)=const*e^{-(1/l2)*(x-l*b)2}

where l=sqrt(2(h-bar)/mw)
now, when I put in the time evolution of b I get:

u(x,t)=const*e^{-(1/l2)*(x-lb₀e-iwt)2}

(I plugged in b(t) from before in for b)
This state does not satisfy the Schrödinger equation, for one thing it cannot be normalized because the integral over all x of the norm squared of the wavefunction varies in time.

This confuses me because b(t) comes from treating the coherent state as a superposition of harmonic oscillator energy eigenstates, which come from the Schrödinger equation. Since the Schrödinger equation conserves the integral over all x of probability density, why do I get a state which does not do so from harmonic oscilator states (and thus, by extension, the Schrödinger equation)?

Thanks so much in advance, I have done this over several times over the last day and cannot find out anything I did wrong nor a solution online.

 

[vanhees71]

Have a look at this manuscript, which I found with Google (unfortunately, the author is not given), but at least on the first glance it looks pretty nice and comprehensive:

http://www.fysik.su.se/~hansson/KFT2/extra notes/cstates copy.pdf

 

[quantum539]

Thanks,

But I was wondering if someone knows why the method I described does not work.

 

[stevendaryl]

Thanks,

But I was wondering if someone knows why the method I described does not work.

You said in your original post:

It can be shown (and you can find this easily on the internet) that the eigenvalue b evolves as:
$b(t)=b_0e^{-iwt}$

Could you give a URL to where this is shown? It doesn't make sense to me.

 

[vanhees71]

This relation is true in the Heisenberg picture of time evolution, and one must not mix these pictures as in the original posting. I have a treatment of the problem in my QM lecture notes, but only in German, but there are many formulae; so perhaps it's possible to understand the calculations:

http://theory.gsi.de/~vanhees/faq/quant/node51.html