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In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, a free particle is a particle that, in some sense, is not bound by an external
force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
, or equivalently not in a region where its
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
varies. In
classical physics Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, this means the particle is present in a "field-free" space. In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, it means the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.


Classical free particle

The classical free particle is characterized by a fixed
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
''v''. The
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
of a particle with mass ''m'' is given by p=mv and the
kinetic energy In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
(equal to total energy) by E=\fracmv^2=\frac.


Quantum free particle


Mathematical description

A free particle with mass m in non-relativistic quantum mechanics is described by the free
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
: - \frac \nabla^2 \ \psi(\mathbf, t) = i\hbar\frac \psi (\mathbf, t) where ''ψ'' is the
wavefunction In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
of the particle at position r and time ''t''. The solution for a particle with momentum p or
wave vector In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength) ...
k, at
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
''ω'' or energy ''E'', is given by a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
: \psi(\mathbf, t) = Ae^ = Ae^ with
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
''A'' and has two different rules according to its mass:
  1. if the particle has mass m: \omega = \frac (or equivalent E = \frac ).
  2. if the particle is a massless particle: \omega=kc.
The eigenvalue spectrum is infinitely degenerate since for each eigenvalue ''E''>0, there corresponds an infinite number of eigenfunctions corresponding to different directions of \mathbf. The De Broglie relations: \mathbf = \hbar \mathbf, E = \hbar \omega apply. Since the potential energy is (stated to be) 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 \omega As for ''all'' quantum particles free ''or'' bound, the
Heisenberg uncertainty principle The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
s \Delta p_x \Delta x \geq \frac apply. It is clear that since the plane wave has definite momentum (definite energy), the probability of finding the particle's location is uniform and negligible all over the space. In other words, the wave function is not normalizable in a Euclidean space, ''these stationary states can not correspond to physical realizable states''.


Measurement and calculations

The normalization condition for the wave function states that if a wavefunction belongs to the
quantum state space In physics, a quantum state space is an abstract space in which different "positions" represent not literal locations, but rather quantum states of some physical system. It is the quantum analog of the phase space of classical mechanics. Relativ ...
\psi \in L^2(\mathbb^3), then the integral of the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
\rho(\mathbf,t) = \psi^*(\mathbf,t)\psi(\mathbf,t) = , \psi(\mathbf,t), ^2, where * denotes
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
, over all space is the probability of finding the particle in all space, which must be unity if the particle exists: \int_ , \psi(\mathbf,t), ^2 d^3 \mathbf=1. The state of a free particle given by plane wave solutions is ''not'' normalizable as Ae^ \notin L^(\mathbb^3), for any fixed time t. Using
wave packet In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an Envelope (waves), envelope. A wave packet can be analyzed into, or can be synthesized from, a ...
s, however, the states can be expressed as functions that ''are'' normalizable.


Wave packet

Using the Fourier inversion theorem, the free particle wave function may be represented by a superposition of ''momentum'' eigenfunctions, or, ''wave packet'': \psi(\mathbf, t) =\frac \int_\mathrm \hat \psi_0 (\mathbf)e^ d^3 \mathbf, where \omega(\mathbf) = \frac, and \hat \psi_0 (\mathbf) is the
Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of a " sufficiently nice" initial wavefunction \psi(\mathbf,0). The expectation value of the momentum p for the complex plane wave is \langle\mathbf\rangle=\left\langle \psi \left, -i\hbar\nabla\\psi\right\rangle = \hbar\mathbf , and for the general wave packet it is \langle\mathbf\rangle = \int_\mathrm \psi^*(\mathbf,t)(-i\hbar\nabla)\psi(\mathbf,t) d^3 \mathbf = \int_\mathrm \hbar \mathbf , \hat\psi_0(\mathbf), ^2 d^3 \mathbf. The expectation value of the energy E is \langle E\rangle=\left\langle \psi \left, - \frac \nabla^2 \\psi\right\rangle = \int_\text \psi^*(\mathbf,t)\left(- \frac \nabla^2 \right)\psi(\mathbf,t) d^3 \mathbf .


Group velocity and phase velocity

The
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, t ...
is defined to be the speed at which a plane wave solution propagates, namely v_p=\frac=\frac = \frac. Note that \frac is ''not'' the speed of a classical particle with momentum p; rather, it is half of the classical velocity. Meanwhile, suppose that the initial wave function \psi_0 is a
wave packet In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an Envelope (waves), envelope. A wave packet can be analyzed into, or can be synthesized from, a ...
whose Fourier transform \hat\psi_0 is concentrated near a particular wave vector \mathbf k. Then the
group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
of the plane wave is defined as v_g= \nabla\omega(\mathbf k)=\frac=\frac, which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move. The figure illustrates this phenomenon, with the individual peaks within the wave packet propagating at half the speed of the overall packet.


Spread of the wave packet

The notion of group velocity is based on a linear approximation to the dispersion relation \omega(k) near a particular value of k. In this approximation, the amplitude of the wave packet moves at a velocity equal to the group velocity ''without changing shape''. This result is an approximation that fails to capture certain interesting aspects of the evolution a free quantum particle. Notably, the width of the wave packet, as measured by the uncertainty in the position, grows linearly in time for large times. This phenomenon is called the spread of the wave packet for a free particle. Specifically, it is not difficult to compute an exact formula for the uncertainty \Delta_X as a function of time, where X is the position operator. Working in one spatial dimension for simplicity, we have: Proposition 4.10 (\Delta_X)^2 = \frac(\Delta_P)^2+\frac\left(\left\langle \tfrac()\right\rangle_ - \left\langle X\right\rangle_ \left\langle P\right\rangle_ \right)+(\Delta_X)^2, where \psi_0 is the time-zero wave function. The expression in parentheses in the second term on the right-hand side is the quantum covariance of X and P. Thus, for large positive times, the uncertainty in X grows linearly, with the coefficient of t equal to (\Delta_P)/m. If the momentum of the initial wave function \psi_0 is highly localized, the wave packet will spread slowly and the group-velocity approximation will remain good for a long time. Intuitively, this result says that if the initial wave function has a very sharply defined momentum, then the particle has a sharply defined velocity and will (to good approximation) propagate at this velocity for a long time.


Relativistic quantum free particle

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


See also

*
Wave packet In physics, a wave packet (also known as a wave train or wave group) is a short burst of localized wave action that travels as a unit, outlined by an Envelope (waves), envelope. A wave packet can be analyzed into, or can be synthesized from, a ...
*
Group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
*
Particle in a box In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes the movement of a free particle in a small space surrounded by impenetrable barriers. The model is mainly used a ...
* Finite square well * Delta potential


Notes


References

* * * ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, * ''Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition)'', R. Eisberg, R. Resnick, John Wiley & Sons, 1985, * ''Stationary States'', A. Holden, College Physics Monographs (USA), Oxford University Press, 1971, * * ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, * ''Elementary Quantum Mechanics'', N.F. Mott, Wykeham Science, Wykeham Press (Taylor & Francis Group), 1972, * ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum's Outlines, Mc Graw Hill (USA), 1998,


Further reading

* ''The New Quantum Universe'', T.Hey, P.Walters, Cambridge University Press, 2009, . * ''Quantum Field Theory'', D. McMahon, Mc Graw Hill (USA), 2008, * ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum's Easy Outlines Crash Course, Mc Graw Hill (USA), 2006, {{DEFAULTSORT:Free Particle Concepts in physics Classical mechanics Quantum models