Phase Space

Phase space is the classical state space whose coordinates are generalized positions and their conjugate momenta. For $n$ degrees of freedom, a point of phase space is labeled locally by

$$(q^1,\ldots,q^n,p_1,\ldots,p_n).$$

A classical state at one instant is a phase-space point. Classical time evolution is a trajectory through phase space. Quantum mechanics uses the same symbols $x$ and $p$, but its state space is a Hilbert space, not phase space.

Why Quantum Mechanics Needs This

Phase-space language appears throughout quantum mechanics:

• canonical variables $q^i$ and $p_i$ become position and momentum operators;
• wave packets are compared with localized regions of phase space;
• semiclassical approximations follow families of classical trajectories;
• path integrals and propagators use classical actions along phase-space-related paths;
• the Heisenberg group encodes noncommuting phase-space translations;
• quantum-classical comparison often asks how many effective quantum states fit into a phase-space region.

The goal is not to make quantum states classical. The goal is to know which classical geometric picture is being invoked when a quantum page talks about trajectories, actions, momenta, or classical limits.

Configuration Space Versus Phase Space

Configuration space records positions or generalized coordinates. If a particle moves in one dimension, its configuration space coordinate is $x$. If a rigid rotor is described by angles, those angles are configuration coordinates.

Phase space adds momenta. For one particle in one dimension, phase space has coordinates

$$(x,p).$$

For one particle in three dimensions, it has coordinates

$$(x,y,z,p_x,p_y,p_z).$$

For $n$ generalized coordinates $q^i$, phase space is locally $2n$-dimensional:

$$(q^1,\ldots,q^n,p_1,\ldots,p_n).$$

Geometrically, the natural phase space of a configuration space $Q$ is its cotangent bundle $T^*Q$. The covectors are momenta. This geometric statement becomes important in symplectic geometry, but most elementary calculations use local canonical coordinates.

Canonical Coordinates and Momenta

In a regular Lagrangian system, the conjugate momenta are

$$p_i = \frac{\partial L}{\partial\dot q^i}.$$

The pair $(q^i,p_i)$ is called canonical when it has the standard Hamiltonian form

$$\dot q^i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = - \frac{\partial H}{\partial q^i}.$$

The word “canonical” carries structure. It is not just a convenient list of variables. Canonical coordinates are adapted to the symplectic form and to Poisson brackets. Changes of variables that preserve this structure are canonical transformations. Here the point is that phase-space coordinates come in conjugate pairs.

In curvilinear coordinates, canonical momenta need not equal mass times coordinate velocities. For planar polar coordinates,

$$p_\theta = mr^2\dot\theta,$$

which is angular momentum, not $m\dot\theta$.

Phase-Space Trajectories

A Hamiltonian $H(q,p,t)$ defines a velocity at each phase-space point through Hamilton’s equations. Starting from initial data

$$(q(t_0),p(t_0)),$$

the solution traces a curve

$$t\mapsto(q(t),p(t)).$$

For a time-independent Hamiltonian, the trajectory lies on an energy surface

$$H(q,p)=E.$$

In one degree of freedom, the energy surface is usually a curve in the two-dimensional $(q,p)$ plane. In more degrees of freedom, it is a higher-dimensional hypersurface inside phase space.

Free Particle Example

For a free particle in one dimension,

$$H(x,p) = \frac{p^2}{2m}.$$

Hamilton’s equations are

$$\dot x = \frac{p}{m}, \qquad \dot p=0.$$

Thus phase-space trajectories are horizontal lines in the $(x,p)$ plane:

$$p(t)=p_0, \qquad x(t)=x_0+\frac{p_0}{m}t.$$

The momentum stays fixed, while the position moves at constant velocity. This is the phase-space version of inertial motion.

Harmonic Oscillator Example

For the harmonic oscillator,

$$H(x,p) = \frac{p^2}{2m} + \frac12m\omega^2x^2.$$

The energy contours are ellipses:

$$\frac{p^2}{2m} + \frac12m\omega^2x^2 = E.$$

Hamilton’s equations give

$$\dot x=\frac{p}{m}, \qquad \dot p=-m\omega^2x.$$

The phase-space point moves around an ellipse. The center $(0,0)$ is the stable equilibrium. This classical picture is useful when comparing the quantum oscillator to coherent states and minimum-uncertainty packets, but the quantum ground state is not a point at the origin.

Phase-Space Volume

Hamiltonian flow preserves phase-space volume. In canonical coordinates, this is Liouville’s theorem. Intuitively, a cloud of nearby classical initial conditions may stretch and fold, but its phase-space volume is unchanged.

This volume preservation is one reason phase-space cells are useful in statistical mechanics and semiclassical counting. Quantum mechanics introduces a natural action scale. A rough one-dimensional cell has area on the order of

$$2\pi\hbar.$$

This is not a statement that phase space is literally tiled by little boxes. It is a semiclassical counting rule tied to Fourier analysis, boundary conditions, and the noncommutativity of $X$ and $P$.

Contrast with Hilbert Space

Classical phase space and quantum Hilbert space are different state spaces.

A classical pure state is a point $(q,p)$.

A quantum pure state is a ray in Hilbert space:

$$\lvert\psi\rangle \sim e^{i\alpha}\lvert\psi\rangle.$$

A position-space wavefunction $\psi(x)$ is not a distribution over phase space. It is a coordinate representation of a Hilbert-space vector. A momentum-space wavefunction $\phi(p)$ is another representation of the same state, not a second half of a phase-space pair.

The uncertainty principle blocks the idea that an arbitrary quantum state has simultaneously sharp $x$ and $p$ values:

$$\Delta X\,\Delta P \ge \frac{\hbar}{2}.$$

There are phase-space-like quantum tools, such as Wigner functions and coherent-state representations, but they have their own rules. Wigner functions, for example, can be negative and are not ordinary probability densities.

Semiclassical Use

Semiclassical methods use phase-space trajectories without turning quantum mechanics into classical mechanics.

For wave packets, one often tracks the center

$$(\langle X\rangle,\langle P\rangle)$$

and compares it with a classical trajectory. This comparison works best when the packet is narrow relative to the scales of the problem and remains narrow for the time interval of interest.

For WKB and propagator methods, phase-space data enter through classical momenta, actions, and stability of trajectories. The stationary path contributes a phase

$$\exp \left( \frac{i}{\hbar}S_{\mathrm{cl}} \right),$$

but the quantum amplitude still includes interference, prefactors, and caustic phases.

Common Mistakes

• Treating a quantum wavefunction as an ordinary probability density on phase space.
• Thinking that position representation plus momentum representation together specify a classical phase-space point.
• Forgetting that canonical momenta depend on coordinates and on the Lagrangian.
• Assuming every pair of variables called $(q,p)$ is canonical.
• Confusing an energy contour in phase space with a quantum energy eigenstate.
• Treating the semiclassical cell size $2\pi\hbar$ as a literal microscopic grid.
• Assuming a wave packet center following a classical path means the whole quantum state is classical.

Cross-Links

• Lagrangian Mechanics Review
• Hamiltonian Mechanics Review
• Poisson Brackets
• Canonical Transformations
• Symplectic Vector Spaces
• Symplectic Manifolds, First Look
• Position and Momentum Representations
• Heisenberg Group
• Gaussian Wave Packets
• Classical Limit
• Semiclassical Limit Overview
• Semiclassical Propagator
• WKB Approximation

References

• V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd ed., Springer, 1989.
• H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, 3rd ed., Addison-Wesley, 2002.
• R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008.
• M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1990.
• M. Brack and R. K. Bhaduri, Semiclassical Physics, Westview Press, 2003.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. What is the dimension of phase space for a particle moving in three-dimensional space?

Solution

The configuration space has three coordinates, for example $(x,y,z)$. Phase space also includes three conjugate momenta $(p_x,p_y,p_z)$. Therefore the phase-space dimension is $6$.

2. Show that free-particle phase-space trajectories are horizontal lines in the $(x,p)$ plane.

Solution

For

$$H=\frac{p^2}{2m},$$

Hamilton’s equations give

$$\dot x=\frac{p}{m}, \qquad \dot p=0.$$

Thus $p(t)=p_0$ is constant, and

$$x(t)=x_0+\frac{p_0}{m}t.$$

In the $(x,p)$ plane, this is motion along the horizontal line $p=p_0$.

3. For the harmonic oscillator, show that constant-energy curves are ellipses.

Solution

The energy equation is

$$\frac{p^2}{2m} + \frac12m\omega^2x^2 = E.$$

Rewrite it as

$$\frac{p^2}{2mE} + \frac{x^2}{2E/(m\omega^2)} = 1.$$

This is the equation of an ellipse in the $(x,p)$ plane for $E>0$.

4. Why is a momentum-space wavefunction not the momentum half of a classical phase-space point?

Solution

A momentum-space wavefunction $\phi(p)$ is a representation of an entire Hilbert-space state in the generalized momentum basis. It gives probability amplitudes for different momentum outcomes. A classical phase-space point instead assigns one momentum value and one position value simultaneously. Quantum states generally cannot be reduced to such simultaneous sharp values because $X$ and $P$ do not commute.