Jonginn Yun

Classical Motivation

In classical mechanics, the state of a one-dimensional oscillator is a point in phase space:

$$(x,p).$$

If one wants a quantum state that behaves as classically as possible, the natural idea is to associate a quantum state with a point in phase space and require that translations of phase space move the state covariantly.

Quantum mechanics prevents a state from being an actual point in phase space. Wave behavior and the uncertainty principle force any localized state to have finite spread. The closest replacement is a Gaussian packet centered at a phase-space point.

This is also the equality case of the uncertainty relation: the Cauchy-Schwarz argument implies that minimum-uncertainty wave packets satisfy a linear relation between the fluctuation operators, and the nontrivial normalizable solutions are Gaussian.

Displacement In Phase Space

The position-translation operator is generated by momentum, while the momentum-translation operator is generated by position. A simultaneous phase-space displacement by $(x_0,p_0)$ is therefore implemented by

$$D(x_0,p_0) = \exp\left[ \frac{i}{\hbar}(p_0 x-x_0 p) \right].$$

For the harmonic oscillator,

$$x=x_{\mathrm{zpf}}(a+a^\dagger), \qquad p=i p_{\mathrm{zpf}}(a^\dagger-a).$$

Substituting these into the phase-space displacement gives an operator of the form

$$D(\alpha) = \exp(\alpha a^\dagger-\alpha^*a),$$

where

$$\alpha = \frac{i p_0 x_{\mathrm{zpf}}}{\hbar} + \frac{x_0 p_{\mathrm{zpf}}}{\hbar}.$$

Thus the displacement operator is the quantum operator that moves the vacuum Gaussian packet to a packet centered around the phase-space point represented by $\alpha$.

Coherent States

The coherent state is defined by displacing the oscillator vacuum:

$$|\alpha\rangle = D(\alpha)|0\rangle.$$

It is the harmonic oscillator state most closely aligned with a classical phase-space point.

Using the displacement identity

$$D^\dagger(\alpha)aD(\alpha)=a+\alpha,$$

one finds

$$a|\alpha\rangle=\alpha|\alpha\rangle.$$

So coherent states are eigenstates of the annihilation operator.

Why This Is The Classical State

A coherent state is classical-looking in three connected senses:

1. It is centered at a phase-space point.
2. It is generated by translating the vacuum Gaussian packet.
3. Its uncertainty is minimal and remains Gaussian under harmonic evolution.

The state is not a literal phase-space point, but it is the most localized phase-space packet compatible with quantum mechanics.

Related Topics

The same oscillator language leads naturally to:

• beam-splitter transformations,
• squeezing,
• two-mode squeezing,
• input-output theory,
• Poincare recurrence in closed oscillator systems.

These deserve separate notes because each changes the phase-space picture in a distinct way.