Free particle

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In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of constant potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point (or surface in three dimensions) in space.

Classical Free Particle[edit]

The classical free particle is characterized simply by a fixed velocity v. The momentum is given by

\mathbf{p}=m\mathbf{v}

and the kinetic energy (equal to total energy) by

E=\frac{1}{2}mv^2

where m is the mass of the particle and v is the vector velocity of the particle.

Non-Relativistic Quantum Free Particle[edit]

Propagation of de Broglie waves in 1d - real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the colour opacity) of finding the particle at a given point x is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the curvature decreases, so the decreases again, and vice versa - the result is an alternating amplitude: a wave. Top: Plane wave. Bottom: Wave packet.

Mathematical description[edit]

A free quantum particle is described by the Schrödinger equation:

 - \frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\partial}{\partial t} \psi (\mathbf{r}, t)

where ψ is the wavefunction of the particle at position r and time t. The solution for a particle with momentum p or wave vector k, at angular frequency ω or energy E, is given by the complex plane wave:

 \psi(\mathbf{r}, t) = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-E t)/\hbar}

with amplitude A. As for all quantum particles free or bound, the Heisenberg uncertainty principles

 \Delta p_x \Delta x \geq \frac{\hbar}{2}, \quad \Delta E \Delta t \geq \frac{\hbar}{2}

(similarly for the y and z directions), and the De Broglie relations:

 \mathbf{p} = \hbar \mathbf{k}, \quad E = \hbar \omega

apply. Since the potential energy is (set to) zero, the total energy E is equal to the kinetic energy, which has the same form as in classical physics:

 E = T \,\rightarrow \,\frac{\hbar^2 k^2}{2m} =\hbar \omega

Measurement and calculations[edit]

The integral of the probability density function

 \rho(\mathbf{r},t) = \psi^*(\mathbf{r},t)\psi(\mathbf{r},t) = |\psi(\mathbf{r},t)|^2

where * denotes complex conjugate, over all space is the probability of finding the particle in all space, which must be unity if the particle exists:

 \int_\mathrm{all\,space} |\psi(\mathbf{r},t)|^2 d^3 \mathbf{r}=1

This is the normalization condition for the wave function. The wavefunction is not normalizable for a plane wave, but is for a wavepacket.


Increasing amounts of wavepacket localization, meaning the particle becomes more localized.
In the limit ħ → 0, the particle's position and momentum become known exactly.
Interpretation of wave function for one spin-0 particle in one dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (which can measure in %) of finding the particle at the points on the x-axis.


In this case, the free particle wavefunction may be represented by a superposition of free particle momentum eigenfunctions ϕ(k), the Fourier transform of the momentum space wavefunction:

 \psi(\mathbf{r}, t) = \frac{1}{(\sqrt{2\pi\hbar})^3}\int_\mathrm{all \, \textbf{p} \, space}  A (\mathbf{p})e^{i(\mathbf{p}\cdot\mathbf{r}-E t)/\hbar} d^3 \mathbf{p} = \frac{1}{\sqrt{2\pi}}\int_\mathrm{all \, \textbf{k} \, space}  A (\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} d^3 \mathbf{k}

where the integral is over all k-space.

The expectation value of the momentum p is

 \langle\mathbf{p}\rangle=\left\langle \psi \left|-i\hbar\nabla\right|\psi\right\rangle = \int_\mathrm{all\,space} \psi^*(\mathbf{r},t)(-i\hbar\nabla)\psi(\mathbf{r},t) d^3 \mathbf{r} = \hbar\mathbf{k}

The expectation value of the energy E is

 \langle E\rangle=\left\langle \psi \left|i\hbar\frac{\partial}{\partial t}\right|\psi\right\rangle = \int_\mathrm{all\,space} \psi^*(\mathbf{r},t)\left(i\hbar\frac{\partial}{\partial t}\right)\psi(\mathbf{r},t) d^3 \mathbf{r} = \hbar\omega

Solving for k and ω and substituting into the constraint equation yields the familiar relationship between energy and momentum for non-relativistic massive particles

 \langle E \rangle = \frac{\langle p^2 \rangle}{2m}

where p = |p| is the magnitude of the momentum vector. The group velocity of the wave is defined as

 v_g= \frac{d\omega}{dk}

which turns out to be the classical velocity of the particle. The phase velocity of the wave is defined as

 v_p=\frac{\omega}{k} = \frac{E}{p} = \frac{p}{2m} = \frac{v}{2}

Relativistic free particle[edit]

Main article: Quantum field theory

There are a number of equations describing relativistic particles: see relativistic wave equations.

See also[edit]

Sources[edit]

  • Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
  • Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition), R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
  • Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
  • Elementary Quantum Mechanics, N.F. Mott, Wykeham Science, Wykeham Press (Taylor & Francis Group), 1972, ISBN 0-85109-270-5
  • Stationary States, A. Holden, College Physics Monographs (USA), Oxford University Press, 1971, ISBN 0-19-851121-3
  • Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Oulines, Mc Graw Hill (USA), 1998, ISBN (10-) 007-0540187

Further reading[edit]

  • The New Quantum Universe, T.Hey, P.Walters, Cambridge University Press, 2009, ISBN 978-0-521-56457-1.
  • Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
  • Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6