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

[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?

 

[DrDu]

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

 

[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.

 

[DrDu]

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.

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

 

[vancouver_water]

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

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

 

[DrDu]

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