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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= rac{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 Schrödinger Equation for a free particle is:

:
- rac{\hbar^2}{2m}
abla^2 \ \psi(\mathbf{r}, t) =
i\hbar rac{\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

:
rac{\hbar^2 k^2}{2m}=\hbar \omega


  • 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

:
  Abla\psi Angle \hbar\mathbf{k}
  \langle E Angle \langle \psi i\hbar rac{\partial}{\partial t}\psi angle = \hbar\omega
  Where P '''p''' The group velocity of the wave is defined as