Continuity Equation

The continuity equation is the local conservation law for quantum probability. For a nonrelativistic particle with wavefunction $\psi(\mathbf r,t)$, probability density

$$\rho(\mathbf r,t)=\lvert\psi(\mathbf r,t)\rvert^2$$

and probability current $\mathbf j(\mathbf r,t)$ obey

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

This equation says that probability cannot disappear from a small region unless it flows through the region’s boundary.

Required background. Probability Current supplies the Hamiltonian-dependent flux. Time-Dependent Schrödinger Equation in Coordinate Space supplies the evolution equation and the domain assumptions used below.

Local Conservation

For the ordinary Schrödinger Hamiltonian

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

with real scalar potential $V$, the current is

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

The continuity equation is local: it holds point by point wherever the wavefunction is sufficiently smooth and the Hamiltonian is the stated conservative one. It is stronger than merely saying that the total norm is constant.

In one dimension, the same statement is

$$\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} =0.$$

If the current increases with $x$ at a point, more probability is leaving the point to the right than entering from the left, so the local density decreases.

Compact Derivation

Start from the time-dependent Schrödinger equation and its complex conjugate:

$$\begin{aligned} i\hbar\frac{\partial\psi}{\partial t} &= -\frac{\hbar^2}{2m}\nabla^2\psi +V\psi,\\ -i\hbar\frac{\partial\psi^*}{\partial t} &= -\frac{\hbar^2}{2m}\nabla^2\psi^* +V\psi^*. \end{aligned}$$

Multiply the first equation by $\psi^*$, the second by $\psi$, and combine them to form $\partial_t(\psi^*\psi)$. The real potential terms cancel. The remaining kinetic terms can be written as a divergence:

$$\frac{\partial}{\partial t}\lvert\psi\rvert^2 = -\nabla\cdot \left[ \frac{\hbar}{2mi} \left( \psi^*\nabla\psi - \psi\nabla\psi^* \right) \right].$$

Identifying the bracketed quantity as $\mathbf j$ gives

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

The cancellation of $V$ is important. A real scalar potential can redirect probability flow but does not create or destroy total probability.

Integral Form

Let $\Omega$ be a fixed spatial region with boundary $\partial\Omega$. Integrating the local equation gives

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

By the divergence theorem,

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

The left side is the rate of change of probability inside $\Omega$. The surface integral is the outward probability flux. Positive outward flux means probability inside the region decreases.

In one dimension, for an interval $[a,b]$,

$$\frac{d}{dt} \int_a^b \rho(x,t)\,dx = j(a,t)-j(b,t).$$

Probability in the interval increases if more current enters through $a$ than leaves through $b$.

Conservation of Normalization

If $\Omega$ is all space and the wavefunction decays fast enough that the surface flux at infinity vanishes, then

$$\frac{d}{dt} \int_{\mathbb R^3} \rho\,d^3r =0.$$

Thus square normalization is conserved:

$$\int_{\mathbb R^3} \lvert\psi(\mathbf r,t)\rvert^2\,d^3r = \text{constant}.$$

For a normalized state, the constant is $1$. This is the coordinate-space form of unitary norm preservation.

On a finite domain, normalization is conserved only if the boundary conditions make the net outward flux vanish. Infinite-wall, periodic, and suitable self-adjoint boundary conditions all enforce this in different ways.

Examples

For a right-moving plane wave,

$$\psi=Ae^{i(kx-\omega t)},$$

the density and current are

$$\rho=\lvert A\rvert^2, \qquad j=\frac{\hbar k}{m}\lvert A\rvert^2.$$

Both are constant, so $\partial_t\rho=0$ and $\partial_xj=0$.

For a real bound-state eigenfunction in a time-independent real potential, the current vanishes:

$$j=0.$$

The density is time independent for a single stationary state, so the continuity equation reads $0+0=0$.

For a scattering state in a real one-dimensional potential, the stationary density may vary with $x$, but $\partial_t\rho=0$. The continuity equation then implies

$$\frac{dj}{dx}=0$$

in regions without sources or sinks. With signed currents this is the local reason

$$j_{\mathrm{inc}}+j_{\mathrm{ref}}=j_{\mathrm{trans}},$$

where $j_{\mathrm{ref}}<0$. In positive magnitudes the same statement is $j_{\mathrm{inc}}=|j_{\mathrm{ref}}|+j_{\mathrm{trans}}$.

For a wave packet crossing a detector region, $\rho$ inside the region can rise and fall even though total probability remains one. The change is accounted for by the boundary currents.

Boundary Conditions and Domains

Boundary conditions are not mathematical afterthoughts. They determine whether the Hamiltonian is a conservative quantum Hamiltonian on the chosen domain.

For a particle on an interval $[a,b]$, total probability is conserved when

$$j(a,t)=j(b,t).$$

Infinite-wall boundary conditions usually set $\psi=0$ at both endpoints, which makes the endpoint current vanish. Periodic boundary conditions identify the endpoints and make the current leaving one side enter the other.

Equality of the endpoint currents establishes norm balance. The full boundary form can establish symmetry on a proposed domain, but neither condition alone proves self-adjointness; one must also establish equality with the adjoint domain. Boundary Conditions treats that distinction as part of the physical problem.

Non-Hermitian Effective Hamiltonians

The local identity above assumes the stated real local differential expression and sufficient regularity. Global norm preservation additionally requires a self-adjoint Hamiltonian with its domain and boundary conditions. Effective models sometimes use a complex potential, for example

$$V_{\mathrm{eff}}(\mathbf r) = V_R(\mathbf r) - \frac{i}{2}\Gamma(\mathbf r), \qquad \Gamma(\mathbf r)\ge0.$$

Then probability in the explicit wavefunction is not conserved. The continuity equation becomes

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

Here $\Gamma$ has units of energy, so $\Gamma/\hbar$ is a local decay rate. More generally, a complex local potential contributes

$$\frac{2}{\hbar}\operatorname{Im}V\,\rho$$

to the right side. For $\operatorname{Im}V=-\Gamma/2$, this is the displayed sink. It may model absorption, loss into untracked channels, or an artificial absorbing boundary in a numerical calculation.

This does not mean fundamental quantum mechanics violates probability conservation. It means the effective description is not tracking all degrees of freedom. In a larger closed system, total probability is still conserved.

Current With Vector Potentials

When a charged particle is coupled to an electromagnetic vector potential, the continuity equation keeps the same form,

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

but the current changes to the gauge-invariant expression

$$\mathbf j = \frac{1}{m} \operatorname{Re} \left[ \psi^* \left( -i\hbar\nabla-q\mathbf A \right)\psi \right].$$

The kinetic derivative acting on $\psi$ transforms covariantly, while this bilinear physical current is gauge invariant. The local conservation law is therefore not just a formula for $\partial_t\rho$; it also identifies the current derived from the Hamiltonian being used.

Radial, Discrete, and Many-Particle Forms

For a spherically symmetric density and radial current,

$$\frac{\partial\rho}{\partial t} +\frac{1}{r^2}\frac{\partial}{\partial r} \left(r^2j_r\right)=0.$$

Equivalently, the shell probability $P(r,t)=4\pi r^2\rho(r,t)$ obeys

$$\frac{\partial P}{\partial t} +\frac{\partial}{\partial r} \left(4\pi r^2j_r\right)=0.$$

The factor $r^2$ is geometric; omitting it confuses local density with radial shell probability.

On a discrete orthonormal basis with amplitudes $c_n$ and Hermitian matrix $H_{nm}$, define the directed current into site $n$ from site $m$ by

$$J_{m\to n} =\frac{2}{\hbar} \operatorname{Im} \left(c_n^*H_{nm}c_m\right).$$

Then

$$\frac{d}{dt}\lvert c_n\rvert^2 =\sum_mJ_{m\to n}, \qquad J_{m\to n}=-J_{n\to m}.$$

This is the lattice analogue of a divergence law; its sign depends on the declared direction convention.

For $N$ nonrelativistic particles with configuration $X=(\mathbf r_1,\ldots,\mathbf r_N)$,

$$\frac{\partial}{\partial t}\lvert\Psi(X,t)\rvert^2 +\sum_{k=1}^N\nabla_k\cdot\mathbf j_k(X,t)=0,$$

where, in the absence of vector potentials,

$$\mathbf j_k =\frac{\hbar}{m_k} \operatorname{Im}\left(\Psi^*\nabla_k\Psi\right).$$

This conservation law lives on configuration space, not ordinary three-dimensional space. The one-particle density current used in many-body physics is obtained by integrating over the other coordinates and summing the appropriate particle contributions.

At a finite material or potential interface with no declared surface source or accumulation, integrating the law across a thin pillbox makes the normal current continuous. At singular interfaces the statement is distributional. Continuity of $\psi'$ is correct for a constant-mass finite jump; position-dependent effective-mass models require the current associated with their actual self-adjoint kinetic operator.

Limits of the local formula

A self-adjoint nonlocal kernel can conserve the global norm without producing the displayed local differential current; probability may be transferred between separated points by a nonlocal term. Position-dependent mass, spin–orbit and Pauli Hamiltonians, lattice dynamics, curved measures, and reduced many-particle densities likewise require Hamiltonian-specific currents. Local conservation should be derived from the actual generator rather than assumed from the word “Hermitian.”

Common Mistakes

• Treating global normalization conservation as if it automatically gives the local current.
• Forgetting the minus sign in the integral flux law.
• Ignoring boundary currents on finite intervals.
• Assuming stationary density always means zero current.
• Applying $R+T=1$ in a model with absorption, gain, time dependence, or hidden channels.
• Using the free-particle current formula unchanged after introducing a vector potential.

Exercises

1. Derive the one-dimensional interval form of the continuity equation on $[a,b]$.

Solution

Start with

$$\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} =0.$$

Integrate from $a$ to $b$:

$$\frac{d}{dt} \int_a^b\rho\,dx + \int_a^b\frac{\partial j}{\partial x}\,dx =0.$$

The second integral is $j(b,t)-j(a,t)$, so

$$\frac{d}{dt} \int_a^b\rho\,dx = j(a,t)-j(b,t).$$

2. On $[0,L]$, suppose $\psi(L)=e^{i\theta}\psi(0)$ and $\psi'(L)=e^{i\theta}\psi'(0)$. Verify norm balance at the endpoints.

Solution

The scalar current is

$$j=\frac{\hbar}{m}\operatorname{Im}(\psi^*\psi').$$

At $L$, the two phase factors cancel:

$$\psi^*(L)\psi'(L) =e^{-i\theta}\psi^*(0)e^{i\theta}\psi'(0) =\psi^*(0)\psi'(0).$$

Hence $j(L)=j(0)$ and $d\int_0^L\rho\,dx/dt=j(0)-j(L)=0$.

3. Suppose a normalized wavefunction on the full line obeys the continuity equation and $j(x,t)\to0$ as $x\to\pm\infty$. Show that its norm is time independent.

Solution

Use the interval result on $[-L,L]$:

$$\frac{d}{dt} \int_{-L}^{L}\rho\,dx = j(-L,t)-j(L,t).$$

Take $L\to\infty$. If both boundary currents vanish, then

$$\frac{d}{dt} \int_{-\infty}^{\infty}\rho\,dx =0.$$

Thus the total probability stays fixed.

4. Let $V_{\mathrm{eff}}=V_R-i\Gamma/2$ with $\Gamma$ constant and no spatial current through the boundary. If $\rho$ satisfies $\partial_t\rho=-(\Gamma/\hbar)\rho$, find the time dependence of the total norm.

Solution

Let

$$N(t)=\int\rho(\mathbf r,t)\,d^3r.$$

With no boundary flux,

$$\frac{dN}{dt} = -\frac{\Gamma}{\hbar}N.$$

Solving gives

$$N(t)=N(0)e^{-\Gamma t/\hbar}.$$

The state loses norm because the effective Hamiltonian contains an absorptive imaginary potential.

5. For a two-site Hamiltonian $H=-J(|1\rangle\langle2|+|2\rangle\langle1|)$ with real $J$, derive the current from site $1$ to site $2$.

Solution

With the convention above,

$$J_{1\to2} =\frac{2}{\hbar}\operatorname{Im}(c_2^*H_{21}c_1) =-\frac{2J}{\hbar}\operatorname{Im}(c_2^*c_1).$$

The reverse current is its negative, so $d|c_1|^2/dt=-J_{1\to2}$ and $d|c_2|^2/dt=J_{1\to2}$. Their sum is conserved.

Where This Is Used

• Probability Current defines the current entering the continuity equation.
• Time-Dependent Schrödinger Equation in Coordinate Space supplies the equation of motion.
• Boundary Conditions explains how domains enforce probability conservation.
• Normalization Conventions compares square normalization, delta normalization, and flux normalization.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. G. Muga, J. P. Palao, B. Navarro, and I. L. Egusquiza, “Complex absorbing potentials,” Physics Reports 395, 357–426 (2004).
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.