Free particle

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

and the kinetic energy (equal to total energy) by

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

Main articles: Schrödinger equation and Matter wave

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)$$-{\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 }$$\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 \hbar$$\Delta p_{x}\Delta x\geq {\frac {\hbar }{2}},\quad \Delta E\Delta t\geq \hbar$

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

$\mathbf {p} =\hbar \mathbf {k} ,\quad E=\hbar \omega$${\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$$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}$$\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$$\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} -Et)/\hbar }d^{3}\mathbf {p} ={\frac {1}{({\sqrt {2\pi }})^{3}}}\int _{\mathrm {all\,{\textbf {k}}\,space} }A(\mathbf {k} )e^{i(\mathbf {k} \cdot \mathbf {r} -\omega t)}d^{3}\mathbf {k}$$\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 }})^{3}}}\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, and $E=E(\mathbf {p} )={\frac {\mathbf {p} ^{2}}{2m}}$$E=E({\mathbf {p}})={\frac {{\mathbf {p}}^{2}}{2m}}$ and $\omega =\omega (\mathbf {k} )={\frac {\hbar \mathbf {k} ^{2}}{2m}}$$\omega =\omega ({\mathbf {k}})={\frac {\hbar {\mathbf {k}}^{2}}{2m}}$ (to ensure that the wavepacket is a solution of the free particle Schrödinger equation). Note that here we abuse notation and denote $A(\mathbf {p} )=A(\hbar \mathbf {k} )$$A({\mathbf {p}})=A(\hbar {\mathbf {k}})$ and $A(\mathbf {k} )$$A({\mathbf {k}})$ with the same symbol, when we should denote ${\hat {A}}(\mathbf {k} )=A(\hbar \mathbf {k} )$${\hat {A}}({\mathbf {k}})=A(\hbar {\mathbf {k}})$, where $A$$A$ is the p-space and ${\hat {A}}$${\hat {A}}$ the k-space function.

The expectation value of the momentum p for the complex plane wave 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}$$\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}}$

and for the general wavepacket it is

$\langle \mathbf {p} \rangle =\int _{\mathrm {all\,space} }\psi ^{*}(\mathbf {r} ,t)(-i\hbar \nabla )\psi (\mathbf {r} ,t)d^{3}\mathbf {r} =\int _{\mathrm {all\,{\textbf {k}}\,space} }\hbar \mathbf {k} |A(\mathbf {k} )|^{2}d^{3}\mathbf {k}$$\langle {\mathbf {p}}\rangle =\int _{{\mathrm {all\,space}}}\psi ^{*}({\mathbf {r}},t)(-i\hbar \nabla )\psi ({\mathbf {r}},t)d^{3}{\mathbf {r}}=\int _{{\mathrm {all\,{\textbf {k}}\,space}}}\hbar {\mathbf {k}}|A({\mathbf {k}})|^{2}d^{3}{\mathbf {k}}$

The expectation value of the energy E is (for both plane wave and general wave packet; here one can observe the special status of time and hence energy in quantum mechanics as opposed to space and momentum)

$\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$$\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$

For the plane wave, 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 \mathbf {p} \rangle ^{2}}{2m}}.$$\langle E\rangle ={\frac {\langle {\mathbf {p}}\rangle ^{2}}{2m}}.$

In general, the identity holds in the form

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

where p = |p| is the magnitude of the momentum vector.

The group velocity of the plane wave is defined as

$v_{g}={\frac {d\omega }{dk}}$$v_{g}={\frac {d\omega }{dk}}$

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

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

Relativistic quantum free particle

Main article: Quantum field theory

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 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 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 978-007-145533-6