Physical interpretation of this coherent state

[ergospherical]

Given the usual raising & lowering operators $A^{\dagger}$ & $A$ for a quantum harmonic oscillator, consider a coherent state $|\alpha\rangle \equiv e^{\alpha A^{\dagger} - \bar{\alpha} A} |0\rangle$. I first check that $|\alpha\rangle$ is an eigenvector of $A$. I already proved that if $X$, $Y$ commute with $[X,Y]$ then $e^{X+Y} = e^{X} e^{Y} e^{-\frac{1}{2}[X,Y]}$, which is applicable here because both $A^{\dagger}$ & $A$ clearly commute with $[A^{\dagger}, A] = 1$, therefore\begin{align*}
|\alpha \rangle &= e^{\alpha A^{\dagger}} e^{-\bar{\alpha} A} e^{-\frac{1}{2}[\alpha A^{\dagger},-\bar{\alpha} A]} |0\rangle = e^{\frac{1}{2}|\alpha|^2} e^{\alpha A^{\dagger}} |0\rangle
\end{align*}where I used that $e^{-\bar{\alpha} A}|0 \rangle = (1 - \bar{\alpha} A + \dots)|0\rangle = |0 \rangle$. Upon application of $A$,\begin{align*}
A|\alpha\rangle = e^{\frac{1}{2}|\alpha|^2} (A e^{\alpha A^{\dagger}}) |0 \rangle = e^{\frac{1}{2}|\alpha|^2} ([A , e^{\alpha A^{\dagger}}] + e^{\alpha A^{\dagger}} A) |0 \rangle &= e^{\frac{1}{2}|\alpha|^2} (\alpha e^{\alpha A^{\dagger}} + e^{\alpha A^{\dagger}} A) |0 \rangle \\
&= \alpha e^{\frac{1}{2}|\alpha|^2} e^{\alpha A^{\dagger}} |0\rangle \\
&= \alpha |\alpha \rangle
\end{align*}which means that $|\alpha \rangle$ is of eigenvalue $\alpha$. I make use of a similar operator identity $e^X e^Y = e^Y e^X e^{[X,Y]}$ to calculate the inner product of two general states:\begin{align*}
\langle \alpha | \beta \rangle = e^{\frac{1}{2} (|\alpha|^2 + |\beta|^2)}\langle 0 | e^{\bar{\beta} A} e^{\alpha A^{\dagger}}| 0 \rangle &= e^{\frac{1}{2} (|\alpha|^2 + |\beta|^2)}\langle 0 | e^{\alpha A^{\dagger}} e^{\bar{\beta} A} e^{[\bar{\beta} A, \alpha A^{\dagger}]} | 0 \rangle \\
&= e^{\frac{1}{2}(|\alpha|^2 + |\beta|^2)}e^{-2\alpha \bar{\beta}} \langle 0 | e^{\alpha A^{\dagger}} e^{\bar{\beta} A} | 0 \rangle \\
&= e^{\frac{1}{2}(|\alpha|^2 -2\alpha \bar{\beta} + |\beta|^2))}
\end{align*}where I used the fact that $\langle 0 | e^{\alpha A^{\dagger}} \leftrightarrow e^{\bar{\alpha} A} |0\rangle$. It is therefore also the case that $|\langle \alpha | \beta \rangle|^2 = e^{|\alpha - \beta|^2}$, that the set $\{ |\alpha \rangle \}_{\alpha \in \mathbf{C}}$ spans the space and that one can select a basis from a suitable subset of $\{ |\alpha \rangle \}_{\alpha \in \mathbf{C}}$.

To consider the physical interpretation of $|\alpha(t) \rangle$ for a general complex $\alpha \in \mathbf{C}$, it is advised to calculate $\langle \alpha | P | \alpha \rangle$,\begin{align*}
\langle \alpha | P | \alpha \rangle = \frac{i}{\sqrt{2}} \langle \alpha | (A^{\dagger} - A) | \alpha \rangle &= \frac{i}{\sqrt{2}} \langle \alpha | (\bar{\alpha} - \alpha) |\alpha \rangle \\
&= \frac{i}{\sqrt{2}} (\bar{\alpha} - \alpha) \\
&= \sqrt{2} \mathrm{Im}(\alpha)
\end{align*}How am I supposed to interpret that the mean value of the momentum is proportional to the imaginary part of $\alpha$? Also, how would I use this result to describe, qualitatively, how the position and momentum space wavefunctions evolve (I already worked out that $|\alpha(t) \rangle = e^{-i\omega t/2} | e^{-i\omega t} \alpha \rangle$?

 

[Demystifier]

You need the position and momentum representations of the coherent state. Many textbooks discuss that, and so does the wikipedia.
https://en.wikipedia.org/wiki/Coherent_state#Quantum_mechanical_definition

 

[ergospherical]

Thanks for the link. Is there a way to see that the position and momentum basis wavefunctions oscillate like in a classical system without further calculation apart from the OP? I only ask because my problem sheet suggests that it is not necessary to work anything else out explicitly.

 

[vanhees71]

You can refer to Ehrenfest's theorem. Since the harmonic oscillator has a linear set of equations of motion for $x$ and $p$ the expectation values fulfill precisely the same equations of motion as the classical harmonic oscillator. A coherent state is one of minimal uncertainty product, $\Delta x \Delta p=\hbar/2$, and thus are closest to a classical description of the system, especially for large $|\alpha|$.