Harmonic oscillator coherent state wavefunction

  • #1

Main Question or Discussion Point

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-iwt


What 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 schodinger 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 schrodinger equation. Since the schrodinger 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 schrodinger 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.
 
Last edited:

Answers and Replies

  • #3
Thanks,

But I was wondering if someone knows why the method I described does not work.
 
  • #4
stevendaryl
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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:
[itex]b(t)=b_0e^{-iwt}[/itex]
Could you give a URL to where this is shown? It doesn't make sense to me.
 
  • #5
vanhees71
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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
 

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