Overview
The probabilistic interpretation of quantum mechanics assigns a meaning to the wave function specified by the Schrodinger Equation. It basically asserts that the probability of finding a particle with a wave function {% \psi(x) %} in the interval {% dx %} is {% |\psi(\vec{x})|^2 dx %}.
Equation
The probability of finding a particle between {% a %} and {% b %}
{% \int_a ^b | \psi(x,t) |^2 dx %}
{% \int_{- \infty} ^{\infty} | \psi(x,t) |^2 dx = 1 %}
{% i \hbar (\psi^* \frac{\partial \psi}{\partial t} + \psi \frac{\partial \psi^*}{\partial t} ) =
-\frac{\hbar^2}{2m} (\psi^* \nabla ^2 \psi - \psi \nabla ^2 \psi^*)
%}
Define the probability current to be
{% J(x) = \frac{i \hbar}{2m} (\psi^* \nabla \psi - \psi \nabla \psi^*) %}
Then the conservation law can be expressed as
{% \frac{\partial \rho}{\partial t} = - \nabla \cdot J %}