Free Particle

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Free Particle

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: : $- rac{\hbar^2}{2m} abla^2 \ \psi(\mathbf{r}, t) = i\hbarrac{\partial}{\partial t} \psi (\mathbf{r}, t)$ 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\hbarrac{\partial}{\partial t}\psi angle = \hbar\omega
	Where P	'''p''' The group velocity of the wave is defined as