# Free particle
A free particle is a quantum-mechanical particle subject to no force, so its potential energy is zero everywhere and its Hamiltonian is purely kinetic, \( \hat{H} = \hat{p}^2/2m \). It is the first system treated in most quantum mechanics textbooks, and even this minimal case displays the characteristic quantum phenomena: delocalized plane-wave states, wave packets built from them, and the spreading of those packets in time.[1](https://qonticlab.rice.edu/apps/comparative/free-particle/free-particle.html)
| Key fact | Detail |
|---|---|
| Hamiltonian | \( \hat{H} = \hat{p}^2/2m \) with \( V = 0 \); the time-dependent Schrödinger equation is \( i\hbar\,\partial\psi/\partial t = -(\hbar^2/2m)\,\partial^2\psi/\partial x^2 \) in one dimension[2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) |
| Energy spectrum | \( E = \hbar^2 k^2/2m \ge 0 \); any non-negative energy is allowed, so the spectrum is continuous and not quantized[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle) |
| Stationary states | Plane waves \( e^{\pm ikx} \), which are momentum eigenstates with eigenvalues \( p = \pm\hbar k \) but are completely delocalized in space[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle) • [2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) |
| Physical states | Localized wave packets, superpositions of plane waves with a spread of momenta; a Gaussian packet is the minimum-uncertainty example[2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) • [1](https://qonticlab.rice.edu/apps/comparative/free-particle/free-particle.html) |
| Spreading | A Gaussian packet of initial width \( \sigma \) broadens as \( \sigma^2(t) = \sigma^2\left(1 + t^2/4m^2\sigma^4\right) \)[4](https://academic.oup.com/book/57152/chapter/457410668) |
| Propagator | \( K_{\mathrm{free}} = \sqrt{m/(2\pi i\hbar(t-t_0))}\,\exp\left[im(x'-x)^2/(2\hbar(t-t_0))\right]\,\Theta(t-t_0) \)[5](https://etneil.github.io/grad_qm_lec_notes/propagators_paths.html) |
| Relativistic case | The free relativistic particle is described by the Klein-Gordon equation (spin 0) or the Dirac equation (spin 1/2)[6](https://www.math.uni-tuebingen.de/de/forschung/maphy/lehre/ws-2018-19/we_rqm/home?set_language=de) |
## Plane-wave solutions
Because the free-particle [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) contains only the kinetic energy operator, it separates directly, and the stationary solutions are plane waves of fixed momentum \( p \):
\[ \psi_p(x,t) = \exp\left(\frac{i}{\hbar}px\right)\exp\left(-\frac{i}{\hbar}\frac{p^2}{2m}t\right). \]
Substituting this form into the Schrödinger equation reduces it to the classical free-particle relation \( E = p^2/2m \), and the solution is valid for any real value of \( p \): one right-traveling wave for \( p > 0 \) and one left-traveling wave for \( p < 0 \).[2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) The plane waves \( e^{ikx} \) and \( e^{-ikx} \) are eigenfunctions of the momentum operator with eigenvalues \( p = \pm\hbar k \); the real functions \( \sin kx \) and \( \cos kx \) are standing-wave superpositions of the two and are eigenfunctions of the Hamiltonian but not of momentum.[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle)
Since no boundary condition restricts \( k \), the energy \( E = \hbar^2k^2/2m \) takes any non-negative value, matching the continuum of kinetic energies of a classical free particle.[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle) This is the opposite of the particle in a box, where confinement forces discrete energy levels.
## Why plane waves are not physical states
A plane wave has uniform probability density over all space, so it cannot describe a particle localized in some region, such as an electron traveling between the cathode and the screen of a tube. It also cannot be normalized in the usual sense: its squared magnitude is a constant, so the total probability integrated over an infinite line diverges.[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle)
**Two ways around the problem** are used. The first is to build a wave packet, a superposition of plane waves with a spread of momenta around an average value, which is localized in space and can represent a particle or a beam of particles.[2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) The second is to place the particle in a large finite volume and impose periodic boundary conditions; the plane waves can then be normalized, but the price is that only a discrete, volume-dependent set of momenta is allowed.[2](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html) Formally, the plane waves and the position delta functions serve as generalized eigenstates of momentum and position, and rigorous treatments handle them with continuum normalization or rigged [Hilbert space](https://www.edgepedia.org/hilbert-space) methods.[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle)
## Wave packets and spreading
Because the Schrödinger equation is linear, any superposition of plane-wave solutions is again a solution. A localized packet is written as an integral over momenta, and each component evolves with its own phase \( e^{-ip^2t/2m\hbar} \).[3](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle) The packet's center moves at the group velocity fixed by the dispersion relation \( \omega(k) \), which the Schrödinger equation was constructed to match to the classical energy-momentum relation.[7](https://phisx.org/lib/download/qm.pdf)
The packet also spreads, and while a classical ensemble of free particles with a spread of initial velocities also spreads, the specifically quantum feature is that localization imposes a spread in velocity.[11](https://quantumlectures.org/en/chapters/postulates/lesson-5) Different momentum components travel at different speeds, so the superposition goes out of phase and the packet pulls apart.[1](https://qonticlab.rice.edu/apps/comparative/free-particle/free-particle.html) For the standard Gaussian initial state
\[ \psi(q,0) = \left(\frac{1}{2\pi\sigma^2}\right)^{1/4}\exp\left[-\frac{(q-a)^2}{4\sigma^2}\right], \]
the probability density remains Gaussian at all later times,
\[ |\psi(q,t)|^2 = \left[\frac{1}{2\pi\sigma^2(t)}\right]^{1/2}\exp\left[-\frac{(q-a)^2}{2\sigma^2(t)}\right], \]
with the width growing as
\[ \sigma^2(t) = \sigma^2\left(1 + \frac{t^2}{4m^2\sigma^4}\right). \]
Equivalently, \( \sigma(t) = \sigma_0\sqrt{1 + (\hbar t/m\sigma_0^2)^2} \): the narrower the initial packet, the faster it spreads, because a small \( \sigma_0 \) implies a large momentum spread by the Heisenberg relation \( \Delta x\,\Delta p \ge \hbar/2 \).[4](https://academic.oup.com/book/57152/chapter/457410668) • [1](https://qonticlab.rice.edu/apps/comparative/free-particle/free-particle.html)
## The free-particle propagator
Time evolution of any free-particle state can be written with the propagator, the transition amplitude for the particle to go from \( x \) at time \( t_0 \) to \( x' \) at time \( t \). It is defined as \( K(x',t;x,t_0) = \langle x'|e^{-i\hat{H}(t-t_0)/\hbar}|x\rangle \) and satisfies the Schrödinger equation with a delta-function source, vanishing for \( t < t_0 \).[5](https://etneil.github.io/grad_qm_lec_notes/propagators_paths.html) Evaluating the momentum integral by completing the square gives
\[ K_{\mathrm{free}}(x',t;x,t_0) = \sqrt{\frac{m}{2\pi i\hbar(t-t_0)}}\,\exp\left[\frac{im(x'-x)^2}{2\hbar(t-t_0)}\right]\,\Theta(t-t_0). \]
The same result follows from the Feynman path integral, where the free-particle action is quadratic and the path integral can be evaluated exactly; in three dimensions the prefactor becomes the power \( 3/2 \).[5](https://etneil.github.io/grad_qm_lec_notes/propagators_paths.html) • [8](https://www.ks.uiuc.edu/Services/Class/PHYS480/first.pdf) The propagator depends only on the differences \( x'-x \) and \( t-t_0 \), reflecting the translation invariance of free-particle propagation in space and time.[9](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/02%3A_1D_Wave_Mechanics/2.02%3A_Free_Particle-_Wave_Packets) Convolving an arbitrary initial wavefunction with this kernel yields its later form, and the propagator itself is just the wavefunction evolved from a delta-function initial condition.[9](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/02%3A_1D_Wave_Mechanics/2.02%3A_Free_Particle-_Wave_Packets)
## The relativistic free particle
When the particle's speed is a significant fraction of the speed of light, the non-relativistic Hamiltonian \( p^2/2m \) no longer applies. A quantum-mechanical description of a relativistic free particle instead starts from the relativistic energy-momentum relation, producing the Klein-Gordon equation for spin-0 particles and the [Dirac equation](https://www.edgepedia.org/dirac-equation) for spin-1/2 particles; the Dirac equation is the one used for all known fermions such as electrons and quarks.[10](https://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp10.pdf) • [6](https://www.math.uni-tuebingen.de/de/forschung/maphy/lehre/ws-2018-19/we_rqm/home?set_language=de) These equations admit free-particle plane-wave solutions as well, and their rigorous analysis establishes a conserved probability current and propagation speeds below the speed of light.[6](https://www.math.uni-tuebingen.de/de/forschung/maphy/lehre/ws-2018-19/we_rqm/home?set_language=de)
## References
1. [Free Particle, Q-Ontic Lab, Rice University](https://qonticlab.rice.edu/apps/comparative/free-particle/free-particle.html)
2. [Wavepackets in Quantum Mechanics, York University](https://www.yorku.ca/marko/ComPhys/Wavepacket/Wavepacket.html)
3. [3.4: The Quantum Mechanical Free Particle, Chemistry LibreTexts](https://chem.libretexts.org/Courses/BethuneCookman_University/B-CU%3ACH-331_Physical_Chemistry_I/CH-331_Text/CH-331_Text/03._The_Schrodinger_Equation_and_a_Particle_In_a_Box/3.04%3A_The_Quantum_Mechanical_Free_Particle)
4. [Section 1.7: Spreading of a Wave Packet, Gauge Theory of Elementary Particle Physics, Oxford Academic](https://academic.oup.com/book/57152/chapter/457410668)
5. [Propagators and Path Integration, Graduate Quantum Mechanics Lecture Notes](https://etneil.github.io/grad_qm_lec_notes/propagators_paths.html)
6. [Wave Equations of Relativistic Quantum Mechanics, University of Tübingen](https://www.math.uni-tuebingen.de/de/forschung/maphy/lehre/ws-2018-19/we_rqm/home?set_language=de)
7. [PhiSX: Quantum Physics (section 4.4, Free Particles and Spreading Wave Packets)](https://phisx.org/lib/download/qm.pdf)
8. [Notes on Quantum Mechanics, section 2.4: Propagator for a Free Particle, University of Illinois at Urbana-Champaign](https://www.ks.uiuc.edu/Services/Class/PHYS480/first.pdf)
9. [2.2: Free Particle – Wave Packets, Physics LibreTexts (Likharev, Essential Graduate Physics)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Essential_Graduate_Physics_-_Quantum_Mechanics_%28Likharev%29/02%3A_1D_Wave_Mechanics/2.02%3A_Free_Particle-_Wave_Packets)
10. [Relativistic Quantum Mechanics, chapter 10: Klein-Gordon Equation, University of Illinois at Urbana-Champaign](https://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp10.pdf)
11. [Theme 3, Lesson 5: Dynamics of a particle](https://quantumlectures.org/en/chapters/postulates/lesson-5)
---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Exactly solvable quantum systems*
*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*
License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license