probability current

From Physics wiki

The probability density $\rho(\mathbf{r}) = \left| \psi(\mathbf{r}) \right|^2\,$ corresponding to the wavefunction of a particle may in general evolve dynamically along with the wavefunction. Since the probability of finding the particle somewhere must be $1\,$ , there has to be a flow, or flux of probability, with an associated conservation equation, the continuity equation,

$\frac{\partial \rho(\mathbf{r},t)}{\partial t} + \boldsymbol{\nabla} \cdot \mathbf{j}(\mathbf{r},t)\,$ ,

where $\mathbf{j}(\mathbf{r},t)\,$ is called the probability current. In one dimension, this is simply

$\frac{\partial \rho(x,t)}{\partial t} + \frac{\partial j(x,t)}{\partial x}\,$ .

The probability current roughly tells us how the particle is moving, since the rate of change of the probability of finding the particle in a region $V\,$ is given by

$\frac{\partial}{\partial t} \int_V\!dV\, \left| \psi \right|^2 = -\int_V\!dV\,\boldsymbol{\nabla}\cdot \mathbf{j}\,$ ,

which, by the divergence theorem, is equal to the rate at which probability is flowing into that region, i.e.,

$\frac{\partial}{\partial t} \int_V\!dV\, \left| \psi \right|^2 = -\int_{\partial V}\!dS\, \mathbf{j}\cdot \hat\mathbf{n}\,$ .

Derivation

The above conseration law is guaranteed only when the wavefunction obeys the Schrödinger equation:

$\frac{\partial \rho}{\partial t}\,$	$=\frac{\partial}{\partial t} (\psi \psi^*)\,$ ,
	$=\left(\frac{\partial}{\partial t} \psi\right) \psi^* + \psi \left( \frac{\partial}{\partial t} \psi^*\right)\,$ ,
	$=\frac{1}{i\hbar}\left( -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi\right) \psi^* + \frac{1}{i\hbar}\left( -\frac{\hbar^2}{2m} \nabla^2 \psi^* + V \psi^*\right) \psi\,$ ,
	$=-\frac{\hbar}{2mi} \left( \psi^* \nabla^2 \psi - \psi \nabla^2 \psi^* \right)\,$ ,
	$=-\frac{\hbar}{2mi} \boldsymbol{\nabla}\cdot \left( \psi^* \boldsymbol\nabla \psi - \psi \boldsymbol\nabla \psi^* \right)\,$ ,