Coherent states and minimum uncertainty

A coherent state is a normalized eigenstate of the annihilation operator, $a|\alpha\rangle=\alpha|\alpha\rangle$, with complex dimensionless parameter α. Expanding in number states and comparing adjacent coefficients gives a factorially weighted series. Normalization introduces the exponential prefactor. Squared coefficients form a Poisson distribution of occupation with mean $|\alpha|^2$. Despite the annihilation eigenvalue equation, it is not an eigenstate of the Hermitian number operator unless α is zero.

The complex parameter specifies two independent means: its real part fixes position and its imaginary part fixes momentum. Defining oscillator length $\ell_0=\sqrt{\hbar/(m\omega)}$, the means are $\sqrt2\ell_0\operatorname{Re}\alpha$ and $\sqrt2\hbar\operatorname{Im}\alpha/\ell_0$. A global phase of the whole ket does not change these means; changing the phase of α rotates the physical displacement in phase space. Confusing these two phases would erase the distinction between displaced position and displaced momentum.

A coherent state has the same position and momentum variances as the oscillator ground state. Displacement changes the means but not these intrinsic widths. Under the positive quadratic Hamiltonian, α rotates as $\alpha e^{-i\omega t}$, while an additional overall phase has no effect on density. The packet stays Gaussian, and its means follow the classical sinusoid. Its energy expectation includes both the classical displacement energy and the nonzero ground-state contribution.

Number-state phases are responsible for that shape-preserving motion. The oscillator's equally spaced spectrum makes all their relative phases advance in a compatible way. An anharmonic potential generally changes this property, allowing distortion, spreading, or more complicated revivals. The constant-width result therefore belongs to an exact harmonic Hamiltonian, rather than to any localized quantum packet. Coupling to an environment can also alter widths and purity even when a classical damped mean remains a useful description.

A displaced ground state saturates the position–momentum uncertainty product with the oscillator's natural equal scaled widths. A squeezed Gaussian can also saturate that product at a suitable phase while having a smaller position variance and larger momentum variance. General rotated squeezed states have position–momentum covariance and need the fuller uncertainty relation. Thus minimum product alone does not identify a coherent state unless its width and covariance properties are also specified.

A large $|\alpha|$ makes the mean excursion large compared with the fixed width and makes relative occupation fluctuations small. It does not remove quantum uncertainty or turn an ensemble of measurements into one disturbance-free path. A statistical mixture can reproduce some means without reproducing the pure state's off-diagonal coherence. To diagnose a coherent preparation one needs appropriate quadrature or interference information, rather than merely a sharply peaked classical-looking average signal.