Coherent-State Expansion

Formula

a|\alpha\rangle=\alpha|\alpha\rangle, \qquad |\alpha\rangle =e^{-|\alpha|^2/2} \sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle =D(\alpha)|0\rangle.

Useful identities are

P(n)=e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!}, \qquad \langle N\rangle=(\Delta N)^2=|\alpha|^2,

\langle\beta|\alpha\rangle =e^{-(|\beta|^2+|\alpha|^2)/2+\beta^*\alpha},

\int_{\mathbb C}\frac{d^2\alpha}{\pi} |\alpha\rangle\langle\alpha|=I,

and, for $H=\hbar\omega(N+1/2)$ ,

e^{-iHt/\hbar}|\alpha\rangle =e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle.

$[a,a^\dagger]=I$ and $\langle m|n\rangle=\delta_{mn}$ .
$D(\alpha)=\exp(\alpha a^\dagger-\alpha^*a)$ .
$\alpha$ is dimensionless; it labels both mean position and mean momentum.
Exact shape preservation assumes an ideal harmonic Hamiltonian. Linear driving displaces the packet, while changed quadratic terms can squeeze it.

Symbol	Meaning
$\alpha,\beta$	dimensionless complex phase-space labels
$a,a^\dagger$	annihilation and creation operators
$N=a^\dagger a$	number operator
$D(\alpha)$	displacement operator
$d^2\alpha$	area measure on the complex label plane

Distinct coherent states are normalized but not orthogonal.
A coherent state is not an energy eigenstate unless $\alpha=0$ .
The $1/\pi$ factor is part of the identity-resolution measure.
The replacement $a\to\alpha$ is valid for normally ordered expressions; for example $\langle aa^\dagger\rangle=|\alpha|^2+1$ .
Optical coherence and detection claims require the quantum-optics measurement framework, not oscillator algebra alone.

Derivations, coordinate wavefunctions, displacement algebra, energy moments, examples, exercises, and references are at Coherent States.