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The probability current is the flux associated with a quantum state's probability density: its outward surface integral gives the rate at which probability leaves a region. Subtracting the Schrödinger equation times the conjugate wavefunction from its complex conjugate gives the probability continuity equation, identifying the current below. The displayed expression assumes a real scalar potential and no electromagnetic vector potential. In an electromagnetic field, the gauge-invariant current instead uses the kinetic momentum operator, including its vector-potential contribution. A changing spatial phase can therefore produce flow even where the probability density is constant.
