Continuity Equation

This is a compact lookup projection. Continuity Equation owns the full derivation, boundary analysis, generalizations, and exercises. Probability Current owns the choice and interpretation of the Hamiltonian-dependent flux.

Helpful background. Continuity Equation derives the balance law. Probability Current supplies the Hamiltonian-specific flux when the model differs from the constant-mass scalar case below.

Source-free local and integral balance

For one spinless particle with

$$\hat H=-\frac{\hbar^2}{2m}\nabla^2+V(\mathbf r,t), \qquad V=V^*,$$

define

$$\rho=|\psi|^2, \qquad \mathbf j =\frac{\hbar}{2mi} \left(\psi^*\nabla\psi-\psi\nabla\psi^*\right).$$

Then

$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j=0.$$

For a fixed region $\Omega$ with outward-oriented $d\mathbf S$,

$$\frac{d}{dt}\int_\Omega\rho\,d^3r =-\int_{\partial\Omega}\mathbf j\cdot d\mathbf S.$$

Positive outward flux therefore decreases the probability stored in $\Omega$.

Electromagnetic current

For charge $q$ and minimal coupling to $(\Phi,\mathbf A)$,

$$\mathbf j =\frac1m\operatorname{Re} \left[ \psi^*(-i\hbar\nabla-q\mathbf A)\psi \right].$$

Under

$$\mathbf A'=\mathbf A+\nabla\chi, \qquad \Phi'=\Phi-\partial_t\chi, \qquad \psi'=e^{iq\chi/\hbar}\psi,$$

the kinetic derivative of $\psi$ is gauge covariant, whereas $\rho$, the bilinear current, and the continuity equation are gauge invariant.

Source and sink terms

For the same kinetic operator but a complex local potential $V$,

$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j =\frac{2}{\hbar}\operatorname{Im}V\,\rho.$$

Thus the effective absorber

$$V_{\mathrm{eff}}=V_R-\frac{i}{2}\Gamma, \qquad \Gamma\ge 0,$$

gives

$$\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf j =-\frac{\Gamma}{\hbar}\rho.$$

Here $\Gamma$ has units of energy. A positive imaginary part instead acts as a local source. These effective terms intentionally describe nonunitary probability balance within the modeled sector.

Local identity versus global unitarity

• Real, local $V$ cancels from the pointwise derivation wherever $\psi$ is sufficiently regular. That cancellation is a local differential statement.
• Integrating the local equation conserves the total norm only when the net boundary flux vanishes, including any flux at infinity.
• For a time-independent closed Hamiltonian, self-adjointness on its specified domain generates unitary evolution. A merely formal Hermiticity or symmetry calculation that ignores the boundary form is not enough.
• Position-dependent mass, nonlocal kernels, spin terms, lattice Hamiltonians, and reduced many-particle descriptions require the density and current derived from their actual dynamics.
• Stationary density need not imply zero current, and continuity fixes only the divergence of a current when divergence-free additions are possible.

Symbols and units

Symbol	Meaning
$\rho$	probability density
$\mathbf j$	probability current density
$m$	particle mass in the scalar formula
$q$	particle charge
$d\mathbf S$	outward-oriented surface element
$\Gamma$	local loss coefficient in energy units

In $d$ spatial dimensions,

$$[\rho]=L^{-d}, \qquad [\mathbf j]=L^{1-d}T^{-1}.$$

Calculation checks

• The source-free equation has units $L^{-d}T^{-1}$ in every term.
• Integrating $\nabla\cdot\mathbf j$ reproduces the outward surface flux with the minus sign shown above.
• A plane wave has constant density and constant current, so both local derivatives vanish even when the current is nonzero.
• Setting $\Gamma=0$ recovers the source-free equation.
• The electromagnetic current is unchanged by the displayed gauge transformation.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. I, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018, doi:10.1017/9781316995433.
• J. G. Muga, J. P. Palao, B. Navarro, and I. L. Egusquiza, “Complex absorbing potentials,” Physics Reports 395, 357–426 (2004), doi:10.1016/j.physrep.2004.03.002.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, doi:10.1017/9781108587280.