I What's the significance of the phase in a coherent state?

1. May 17, 2017

vancouver_water

In a coherent state defined by $$|\alpha\rangle = \exp{\left(-\frac{|\alpha|^2}{2}\right)}\exp{\left(\alpha \hat{a}^\dagger\right)} |0\rangle$$ there is a definite phase associated with the state by $$\alpha = |\alpha| \exp{\left(i\theta\right)}$$ where the number operator and phase operator are conjugates, $$-i\partial_{\theta} = \hat{n}.$$ The meaning of the number operator is obvious but what is the significance of the phase in this state? What would be a consequence of picking a new phase for this state?

2. May 17, 2017

DrDu

Take the expectation value of x and p and you will see.

3. May 23, 2017

vancouver_water

So i get $\langle x\rangle \propto \Re{(\alpha)}$ and $\langle p \rangle \propto \Im{(\alpha)}$. Which gives the relation $\langle p \rangle = m\omega\langle x \rangle \tan\theta$. So it gives the relation between expectations of x and p.

4. May 23, 2017

DrDu

Yes. Are you familiar with the phase space from classical mechanics, or action-angle variables?

5. May 23, 2017

vancouver_water

With phase space yes, but not with action angle variables. I'll read about them though, Thanks!

6. May 23, 2017

DrDu

Also take in mind that alpha is time dependent as coherent states aren't energy eigenstates.