Introduction: Conservation of Probability
Probability is conserved—it can flow from place to place but can't be created or destroyed. The probability current \(\vec{j}\) describes this flow, satisfying a continuity equation.
The Probability Current
Define probability density \(\rho = |\psi|^2\) and probability current:
\[j = \frac{\hbar}{2mi}\left(\psi^*\frac{\partial\psi}{\partial x} - \psi\frac{\partial\psi^*}{\partial x}\right) = \frac{\hbar}{m}\text{Im}\left(\psi^*\frac{\partial\psi}{\partial x}\right)\]The Continuity Equation
\[\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} = 0\]This says: rate of change of probability = net inflow of probability current.
Worked Examples
Example 1: Plane Wave Current
For \(\psi = Ae^{i(kx - \omega t)}\):
\[j = \frac{\hbar k}{m}|A|^2 = \frac{p}{m}|A|^2 = v|A|^2\]Probability flows at the classical velocity.
Example 2: Stationary State
For real \(\phi(x)\), \(j = 0\). No probability flows—consistent with "stationary."
Example 3: Current at a Barrier
Incident current minus reflected current equals transmitted current. This gives the transmission coefficient \(T = j_T/j_I\).
The Quantum Connection
The continuity equation ensures the Schrödinger equation conserves total probability. In scattering, current conservation gives transmission + reflection = 1. In higher dimensions, \(\vec{j} = \frac{\hbar}{m}\text{Im}(\psi^*\nabla\psi)\) and \(\partial_t\rho + \nabla \cdot \vec{j} = 0\).