Free particle


In physics a free particle is a particle that is never under the influence of an external force

Classical Free Particle

The classical free particle is characterized simply by a fixed velocity. The momentum is given by :\\mathbf{p}=m\\mathbf{v} and the 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

The Schroedinger equation for a free particle is: : - \\frac{\\hbar^2}{2m} \ abla^2 \\ \\psi(\\mathbf{r}, t) = i\\hbar\\frac{\\partial}{\\partial t} \\psi (\\mathbf{r}, t) The solution for a particular momentum is given by a plane wave: : \\psi(\\mathbf{r}, t) = e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} with the constraint : \\frac{\\hbar^2 k^2}{2m}=\\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
  • over all space must be unity, there will be a problem normalizing this momentum eigenfunction. This will not be a problem for a general free particle which is somewhat localized in momentum and position. (See particle in a box for a further discussion.)
The expectation value of the momentum p is : \\langle\\mathbf{p}\angle=\\langle \\psi
i\\hbar\ abla|\\psi\angle = \\hbar\\mathbf{k}
The expectation value of the energy E is : \\langle E\angle=\\langle \\psi |i\\hbar\\frac{\\partial}{\\partial t}|\\psi\angle = \\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 \angle =\\frac{\\langle p \angle^2}{2m} where p=|'''p'''|. The group velocity of the wave is defined as :\\left.\ight. v_g= d\\omega/dk = dE/dp = v where v is the classical velocity of the particle. The phase velocity of the wave is defined as :\\left.\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: :\\left.\ight. \\psi(\\mathbf{r}, t) = \\int A(\\mathbf{k})e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} d\\mathbf{k} where the integral is over all k-space. Category:Physics

Relativistic free particle (Klein-Gordon equation)

If the particle is charge-neutral and spinless, and relativistic effects cannot be ignored, we may use the Klein-Gordon equation to describe the wave function. The Klein-Gordon equation for a free particle is written : \ abla^2\\psi-\\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}\\psi = \\frac{m^2c^2}{\\hbar^2}\\psi with the same solution as in the non-relativistic case: : \\psi(\\mathbf{r}, t) = e^{i(\\mathbf{k}\\cdot\\mathbf{r}-\\omega t)} except with the constraint : -k^2+\\frac{\\omega^2}{c^2}=\\frac{m^2c^2}{\\hbar^2} Just as with the non-relativistic particle, we have for energy and momentum: : \\langle\\mathbf{p}\angle=\\langle \\psi |- i\\hbar\ abla|\\psi\angle = \\hbar\\mathbf{k} : \\langle E\angle=\\langle \\psi |i\\hbar\\frac{\\partial}{\\partial t}|\\psi\angle = \\hbar\\omega Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles: :\\left.\ight. \\langle E \angle^2=m^2c^4+\\langle p \angle^2c^2 For massless particles, we may set m=0 in the above equations. We then recover the relationship between energy and momentum for massless particles: :\\left.\ight. \\langle E \angle=\\langle p \angle c Category:Physics

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