Free Particle

One-Sentence Description

A free particle is a nonrelativistic particle on an unconfined space with no potential energy, so momentum diagonalizes the Hamiltonian and energy is continuous.

Physical Setup

The ideal model is a particle of mass $m$ moving on $\mathbb R^d$ with no external potential. It describes particles far from interactions, asymptotic scattering states, Fourier-mode benchmarks, and the local starting point for wave-packet motion.

Changing the spatial domain changes the model. A periodic box is a finite-volume regulator. A hard-wall box is the infinite square well. A ring is a compact angular system. The free particle card refers to the unconfined model unless a regulator is explicitly stated.

Hilbert Space

For the unconfined model,

$$\mathcal H=L^2(\mathbb R^d),$$

with generalized momentum eigenstates used as a spectral basis. Physical localized states are normalizable wave packets, not single plane waves.

Hamiltonian

The Hamiltonian is

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

In coordinate representation,

$$\hat H = -\frac{\hbar^2}{2m}\nabla^2,$$

with domain chosen so that the kinetic-energy operator is self-adjoint. On the full line or full space, Fourier transformation diagonalizes the operator.

Parameters

Symbol	Meaning
$m$	particle mass
$d$	spatial dimension
$\mathbf p$	momentum spectral label
$\mathbf k$	wave-vector label, with $\mathbf p=\hbar\mathbf k$

Solvability

The model is exactly solvable by Fourier transform. Momentum components are constants of motion, and each momentum component evolves by a phase.

For a momentum component,

$$E(\mathbf p) = \frac{\mathbf p^2}{2m} = \frac{\hbar^2\mathbf k^2}{2m}.$$

Spectrum and Eigenstates

The momentum spectrum is continuous on $\mathbb R^d$. The energy spectrum is continuous for $E\ge 0$:

$$E\in[0,\infty).$$

Plane waves are generalized eigenfunctions. A common convention in one dimension is

$$\psi_k(x)=\frac{1}{\sqrt{2\pi}}e^{ikx}, \qquad \hat H\psi_k = \frac{\hbar^2k^2}{2m}\psi_k.$$

Because $k$ and $-k$ have the same energy in one dimension, energy eigenstates are degenerate except at the threshold. In higher dimensions, all momenta on a sphere of fixed $\lvert\mathbf p\rvert$ have the same energy.

Key Observables

• Momentum is conserved.
• Velocity satisfies $\mathbf v=\mathbf p/m$ for a narrow packet.
• Probability current is the diagnostic for flux-normalized scattering states.
• Wave-packet spreading follows from the nonlinear dispersion relation $\omega(\mathbf k)=\hbar\mathbf k^2/(2m)$.

What It Teaches

The free particle teaches continuous spectra, generalized eigenvectors, Fourier-space dynamics, group velocity, phase velocity, and the need to distinguish ideal basis states from physical wave packets.

It is also the background model behind scattering theory: interactions are measured relative to asymptotic free motion.

Canonical Links

• Free Particle
• Free Particle Hamiltonian
• Momentum Eigenstates
• Plane Waves and Delta Normalization
• Gaussian Wave Packets
• Free-Particle Propagator: First Encounter

Variants

• periodic-box free particle;
• free particle on a ring;
• relativistic free particle;
• charged particle in a uniform magnetic field;
• asymptotic free states in scattering theory.

Common Mistakes

• Treating a plane wave as a normalizable localized particle.
• Forgetting that $E=p^2/(2m)$ is not one-to-one in momentum.
• Reusing the continuum spectrum after imposing hard-wall boundary conditions.
• Calling a particle “free” while leaving an implicit external potential or boundary condition in the problem.

Quick Checks

1. Why does a one-dimensional free particle have two momentum labels for most positive energies?

Solution

For $E=p^2/(2m)>0$, the two momenta $p=\sqrt{2mE}$ and $p=-\sqrt{2mE}$ have the same energy. They correspond to right-moving and left-moving generalized states.

2. What changes if the particle is placed in a periodic box of length $L$?

Solution

The momentum labels become discrete, $k=2\pi n/L$, while the local Hamiltonian remains $-\hbar^2d^2/(2m\,dx^2)$. The box is a regulator with periodic boundary conditions, not the same model as a hard-wall well.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.