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In physics, a free particle is a particle that, in some sense, is not bound. In the classical case, this is represented with the particle not being influenced by any external force.
Classical Free Particle The classical free particle is characterized simply by a fixed velocity. The momentum is given by and the energy by where m is the mass of the particle and v is the vector velocity of the particle. Non-Relativistic Quantum Free Particle The Schrödinger equation for a free particle is: - rac abla^2 psi(mathbf, t) = ihbar rac psi (mathbf, t) The solution for a particular momentum is given by a plane wave: psi(mathbf, t) = e^ with the constraint rac=hbar omega where r is the position vector, t is time, k is the wave vector, and ω is the angular frequency. Since the integral of ψψ The expectation value of the momentum p is langlemathbf angle=langle psi |-ihbar abla|psi angle = hbarmathbf The expectation value of the energy E is langle E angle=langle psi |ihbar rac|psi angle = hbaromega 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 angle = rac where p=|p|. The group velocity of the wave is defined as ight. v_g= domega/dk = dE/dp = v where v is the classical velocity of the particle. The phase velocity of the wave is defined as ight. v_p=omega/k = E/p = p/2m = v/2 A general free particle need not have a specific momentum or energy. In this case, the free particle wavefunction may be represented by a superposition of free particle momentum eigenfunctions: ight. psi(mathbf, t) = int A(mathbf)e^ dmathbf where the integral is over all k-space. Relativistic free particle There are a number of equations describing relativistic particles. For a description of the free particle solutions, see the individual articles. | ||||||||
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