Correction for attenuation

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

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

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

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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:

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

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

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.

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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:

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

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

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

See also

• Particle in a box
• Finite square well
• Delta potential
• Wave packet

Sources

• 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

• 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