Free particle explained

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 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 v. The momentum is given by$$\backslash mathbf=m\backslash mathbf$$and the kinetic energy (equal to total energy) by$$E=\backslash fracmv^2=\backslash frac$$where m is the mass of the particle and v is the vector velocity of the particle.

Quantum free particle

Mathematical description

See main article: Schrödinger equation and Matter wave. A free particle with mass

m in non-relativistic quantum mechanics is described by the free Schrödinger equation:$$-\; \backslash frac\; \backslash nabla^2\; \backslash \; \backslash psi(\backslash mathbf,\; t)\; =\; i\backslash hbar\backslash frac\; \backslash psi\; (\backslash mathbf,\; t)$$

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 a complex plane wave:

$$\backslash psi(\backslash mathbf,\; t)\; =\; Ae^\; =\; Ae^$$

with amplitude A and has two different rules according to its mass:

1. if the particle has mass
   m : $\backslash omega\; =\; \backslash frac$ (or equivalent $E\; =\; \backslash frac$).
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

p .

The De Broglie relations:

p = \hbar k , 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:

$$E\; =\; T\; \backslash ,\backslash rightarrow\; \backslash ,\backslash frac\; =\backslash hbar\; \backslash omega$$

As for all quantum particles free or bound, the Heisenberg uncertainty principles $\backslash Delta\; p\_x\; \backslash Delta\; x\; \backslash geq\; \backslash frac$ 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.^[1]

Measurement and calculations

The integral of the probability density function

$$\backslash rho(\backslash mathbf,t)\; =\; \backslash psi^*(\backslash mathbf,t)\backslash psi(\backslash mathbf,t)\; =\; |\backslash psi(\backslash mathbf,t)|^2$$

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:$$\backslash int\_\backslash mathrm\; |\backslash psi(\backslash mathbf,t)|^2\; d^3\; \backslash mathbf=1$$

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

Fourier decomposition

The free particle wave function may be represented by a superposition of momentum eigenfunctions, with coefficients given by the Fourier transform of the initial wavefunction:^[2]

$$\backslash psi(\backslash mathbf,\; t)\; =\backslash frac\; \backslash int\_\backslash mathrm\; \backslash hat\; \backslash psi\_0\; (\backslash mathbf)e^\; d^3\; \backslash mathbf$$

where the integral is over all k-space and $\backslash omega\; =\; \backslash omega(\backslash mathbf)\; =\; \backslash frac$ (to ensure that the wave packet is a solution of the free particle Schrödinger equation). Here

\psi _{0} is the value of the wave function at time 0 and \hat\psi _{0} is the Fourier transform of \psi _{0} . (The Fourier transform \hat\psi _{0(k)} is essentially the momentum wave function of the position wave function \psi _{0(r)} , but written as a function of k rather than p=\hbar k .)

The expectation value of the momentum p for the complex plane wave is

$$\backslash langle\backslash mathbf\backslash rangle=\backslash left\backslash langle\; \backslash psi\; \backslash left|-i\backslash hbar\backslash nabla\backslash right|\backslash psi\backslash right\backslash rangle\; =\; \backslash hbar\backslash mathbf,$$

and for the general wave packet it is

$$\backslash langle\backslash mathbf\backslash rangle\; =\; \backslash int\_\backslash mathrm\; \backslash psi^*(\backslash mathbf,t)(-i\backslash hbar\backslash nabla)\backslash psi(\backslash mathbf,t)\; d^3\; \backslash mathbf\; =\; \backslash int\_\backslash mathrm\; \backslash hbar\; \backslash mathbf\; |\backslash hat\backslash psi\_0(\backslash mathbf)|^2\; d^3\; \backslash mathbf.$$

The expectation value of the energy E is

$$\backslash langle\; E\backslash rangle=\backslash left\backslash langle\; \backslash psi\; \backslash left|-\; \backslash frac\; \backslash nabla^2\; \backslash right|\backslash psi\backslash right\backslash rangle\; =\; \backslash int\_\backslash text\; \backslash psi^*(\backslash mathbf,t)\backslash left(-\; \backslash frac\; \backslash nabla^2\; \backslash right)\backslash psi(\backslash mathbf,t)\; d^3\; \backslash mathbf\; .$$

Group velocity and phase velocity

The phase velocity is defined to be the speed at which a plane wave solution propagates, namely

$$v\_p=\backslash frac=\backslash frac\; =\; \backslash frac.$$

Note that

(p)/(2 m) 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 whose Fourier transform \hat\psi _{0} is concentrated near a particular wave vector k . Then the group velocity of the plane wave is defined as$$v\_g=\; \backslash nabla\backslash omega(\backslash mathbf\; k)=\backslash frac=\backslash frac,$$

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.^[3] 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 .^[4] 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 _{\psi(t)} X as a function of time, where X is the position operator. Working in one spatial dimension for simplicity, we have:^[5] $$(\backslash Delta\_X)^2\; =\; \backslash frac(\backslash Delta\_P)^2+\backslash frac\backslash left(\backslash left\backslash langle\; \backslash tfrac\backslash right\backslash rangle\_\; -\; \backslash left\backslash langle\; X\backslash right\backslash rangle\_\; \backslash left\backslash langle\; P\backslash right\backslash rangle\_\; \backslash right)+(\backslash Delta\_X)^2,$$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 _{\psi _{0}} 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

See main article: Quantum field theory.

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

See also

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

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,

Specific

1. Web site: Lecture 9.
2. Section 4.1
3. Sections 4.3 and 4.4
4. Equation 4.24
5. Proposition 4.10

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,