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Free ParticleIn 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 -
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 Schroedinger equation for a free particle is: -
- \frac{\hbar^2}{2m} \nabla^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}\rangle=\langle \psi |-i\hbar\nabla|\psi\rangle = \hbar\mathbf{k} The expectation value of the energy E is -
\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \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 \rangle =\frac{\langle p \rangle^2}{2m} where p=|p|. The group velocity of the wave is defined as -
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 -
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: -
\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. 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 -
\nabla^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}\rangle=\langle \psi |-i\hbar\nabla|\psi\rangle = \hbar\mathbf{k} -
\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \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: -
\langle E \rangle^2=m^2c^4+\langle p \rangle^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: -
\langle E \rangle=\langle p \rangle c
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