Probability current

For a nonrelativistic particle of mass $m$, the probability current is

Here $\alpha >0$ and $q,p,\gamma$ are real functions of time. Differentiation in position gives ${\psi }_x=[-\alpha (x-q)+ip/\hslash ]\psi$. The standard probability current is thereforeFor a coherent state moving in a quadratic Hamiltonian, the centre follows the classical equation $\dot{q}=p/m$, so the current is the normalized Gaussian probability density transported at its centre velocity. The factor $\sqrt{\alpha /\pi }$ is essential: omitting it changes the current by a dimensional constant and violates the continuity equation for the normalized density. A spatially constant phase does not affect the current.

Where a nonzero wavefunction can be written as $\psi =\sqrt{\rho }{e}^{i\theta }$, and there is no vector potential, substituting into the Schrodinger probability current gives $\mathbf{j}=\rho \hslash \nabla \theta /m$. The direction of probability flow is determined by the phase gradient rather than the slope of the real part. A plane wave and its complex conjugate have the same real part and density but opposite currents. In a vector potential the gauge-invariant kinetic momentum replaces $\hslash \nabla \theta$.

For a Hermitian local mass coefficient and $H=c\mathbf{\sigma }\cdot \mathbf{p}+V{\sigma }_z$, the density and current obey ${\partial }_t({\Psi }^{†}\Psi )+\nabla \cdot (c{\Psi }^{†}\mathbf{\sigma }\Psi )=0$. A normalized ${\sigma }_x=-1$ interface spinor carries $j_x=-c\rho$ and $j_y=0$.

For a wavefunction satisfying the Schrödinger equation with a real potential,This continuity equation expresses local conservation of probability.

If the probability current of a normalizable wavefunction vanishes at spatial infinity, integrating the probability continuity equation over all space shows that the total probability $\int |\psi {|}^2$ is constant in time.