Probability current, continuity equation, and the flow interpretation of the wavefunction

Intuition Beginner

The Born rule says $|\psi (x,t){|}^2$ is the probability density for finding a particle at position $x$ at time $t$. Total probability must always equal 1 — the particle has to be somewhere. If the probability increases in one region, it must decrease somewhere else. The probability flows.

This flow has a direction and a speed. The probability current $j(x,t)$ measures how much probability passes through a point per unit time. If $j$ is positive, probability flows to the right. If negative, it flows to the left. The relationship between probability density $\rho =|\psi {|}^2$ and current $j$ is the continuity equation: the rate at which $\rho$ changes at a point equals the negative of how fast $j$ increases away from that point. In symbols, the time derivative of $\rho$ plus the spatial derivative of $j$ is zero.

This means probability is neither created nor destroyed — it is conserved.

The continuity equation is not a new physical law. It follows directly from the time-dependent Schrodinger equation. It holds for any potential $V(x)$, any wavefunction, and any number of spatial dimensions.

Visual Beginner

Draw a one-dimensional probability density $|\psi (x){|}^2$ as a bell curve at two successive times, $t_1$ and $t_2>t_1$. The curve has shifted to the right and changed shape slightly. Annotate the graph with arrows indicating the probability current $j(x)$ at several points along the curve.

Where the density is building up (the leading edge of the moving packet), $j$ is large and positive — probability flows in. Where the density is thinning out (the trailing edge), $j$ exceeds what the local density can sustain — probability flows out. At the peak, $j$ matches the group velocity times the density.

A second panel shows the same idea in a pipe with fluid: the amount of fluid in a segment increases when more flows in from the left than flows out to the right. The probability current is the quantum analogue of fluid flux.

Worked example Beginner

A plane wave $\psi (x,t)=A{e}^{i(kx-\omega t)}$ has $|\psi {|}^2=|A{|}^2$ — constant everywhere. The probability density does not change in time, so there is no accumulation or depletion. But there is still a current.

The probability current in one dimension is defined as:

$$j=\frac{\hslash }{2mi}\left({\psi }^{\ast }\frac{d\psi }{dx}-\psi \frac{d{\psi }^{\ast }}{dx}\right).$$

For the plane wave $\psi =A{e}^{i(kx-\omega t)}$, the derivative is $d\psi /dx=ikA{e}^{i(kx-\omega t)}$, and the complex conjugate gives $d{\psi }^{\ast }/dx=-ik{A}^{\ast }{e}^{-i(kx-\omega t)}$. Substituting:

$$j=\frac{\hslash }{2mi}\left(|A{|}^{2}ik-|A{|}^2(-ik)\right)=\frac{\hslash }{2mi}\left(2ik|A{|}^2\right)=\frac{\hslash k}{m}|A{|}^2=\frac{p}{m}|A{|}^2.$$

Since $p/m=v$ is the classical velocity, $j=v|\psi {|}^2$. The current equals the density times the particle speed — exactly what you expect for a stream of particles all moving at velocity $v$. The plane wave carries probability at the classical speed.

Check your understanding Beginner

Formal definition Intermediate+

Start from the time-dependent Schrodinger equation in one dimension:

$$i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+V(x)\psi .$$

Take the complex conjugate:

$$-i\hslash \frac{\partial {\psi }^{\ast }}{\partial t}=-\frac{{\hslash }^2}{2m}\frac{{\partial }^2{\psi }^{\ast }}{\partial x^2}+V(x){\psi }^{\ast }.$$

Multiply the first equation by ${\psi }^{\ast }$ and the second by $\psi$, then subtract:

$$i\hslash \left({\psi }^{\ast }\frac{\partial \psi }{\partial t}+\psi \frac{\partial {\psi }^{\ast }}{\partial t}\right)=-\frac{{\hslash }^2}{2m}\left({\psi }^{\ast }\frac{{\partial }^2\psi }{\partial x^2}-\psi \frac{{\partial }^2{\psi }^{\ast }}{\partial x^2}\right).$$

The left side is $i\hslash \partial ({\psi }^{\ast }\psi )/\partial t=i\hslash \partial \rho /\partial t$. The right side simplifies via the product rule:

$${\psi }^{\ast }\frac{{\partial }^2\psi }{\partial x^2}-\psi \frac{{\partial }^2{\psi }^{\ast }}{\partial x^2}=\frac{\partial }{\partial x}\left({\psi }^{\ast }\frac{\partial \psi }{\partial x}-\psi \frac{\partial {\psi }^{\ast }}{\partial x}\right).$$

Define the probability current density:

$$\boxed{j(x,t)=\frac{\hslash }{2mi}\left({\psi }^{\ast }\frac{\partial \psi }{\partial x}-\psi \frac{\partial {\psi }^{\ast }}{\partial x}\right)=\frac{\hslash }{m}\mathrm{Im}\left({\psi }^{\ast }\frac{\partial \psi }{\partial x}\right).}$$

The equation becomes the continuity equation:

$$\boxed{\frac{\partial \rho }{\partial t}+\frac{\partial j}{\partial x}=0,\rho =|\psi {|}^2.}$$

The potential $V(x)$ cancelled in the subtraction. The continuity equation holds for any potential — it is a structural consequence of the Schrodinger equation, not a property of particular solutions.

Conservation of total probability. Integrating the continuity equation over all space:

$$\frac{d}{dt}{\int }_{-\infty }^{\infty }\rho dx=-{\int }_{-\infty }^{\infty }\frac{\partial j}{\partial x}dx=-[j{]}_{-\infty }^{\infty }=0,$$

where the boundary terms vanish because normalisable wavefunctions satisfy $\psi \to 0$ as $|x|\to \infty$. Hence $\int |\psi {|}^{2}dx=1$ for all $t$.

Three-dimensional generalisation. In three dimensions the Schrodinger equation is

$$i\hslash \frac{\partial \psi }{\partial t}=-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +V(\mathbf{r})\psi ,$$

and the same procedure yields

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

with the probability current vector

$$\boxed{\mathbf{j}(\mathbf{r},t)=\frac{\hslash }{2mi}\left({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast }\right)=\frac{\hslash }{m}\mathrm{Im}\left({\psi }^{\ast }\nabla \psi \right)=\frac{1}{m}\mathrm{Re}\left({\psi }^{\ast }\hat{\mathbf{p}}\psi \right).}$$

The last form shows that $\mathbf{j}=⟨\mathbf{r}|\hat{\mathbf{p}}|\psi ⟩|\psi (\mathbf{r}){|}^2/m$ up to a density factor — the current is the local expectation value of velocity times probability density.

Probability current for plane waves. For $\psi =A{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}$: $\mathbf{j}=(\hslash \mathbf{k}/m)|A{|}^2=\mathbf{v}|\psi {|}^2$.

Probability current for a Gaussian wave packet. The minimum-uncertainty Gaussian $\psi (x,t)=N\exp (-(x-v_{g}t{)}^2/(4{\sigma }_0\sigma (t))+i{k}_0(x-v_{g}t/2))$ yields

$$j(x,t)=v_g|\psi (x,t){|}^2+\text{correction terms from the spreading}.$$

At the centre of the packet, $j=v_g|\psi {|}^2$ exactly. Off-centre, the spreading introduces additional current that redistributes probability within the envelope.

Reflection and transmission coefficients. For a scattering problem with incident wave from the left on a step or barrier, the wavefunction has the asymptotic form

$$\psi (x)\to \{\begin{array}{ll}A{e}^{ikx}+B{e}^{-ikx}& x\to -\infty \\ C{e}^{i{k}^{\prime }x}& x\to +\infty \end{array}$$

The reflection coefficient $R$ and transmission coefficient $T$ are defined as ratios of probability currents:

$$R=\frac{|j_{\text{reflected}}|}{|j_{\text{incident}}|}=\frac{k|B{|}^2}{k|A{|}^2}=\frac{|B{|}^2}{|A{|}^2},$$ $$T=\frac{|j_{\text{transmitted}}|}{|j_{\text{incident}}|}=\frac{k^{\prime }|C{|}^2}{k|A{|}^2}.$$

Current conservation guarantees $R+T=1$ — the particle is either reflected or transmitted.

Counterexamples to common slips

• The continuity equation does not require $V(x)$ to be real. For a complex potential $V=V_0+i{V}_1$ (used to model absorption), the cancellation fails and probability is not conserved. This is by design: complex potentials describe open systems where particles can be lost.
• The probability current is not the charge current. For a charged particle the electric current density is $q\mathbf{j}$ where $q$ is the charge. The probability current is dimensionless in the sense that it tracks probability flow, not charge flow.
• A stationary state $\psi (x,t)=\varphi (x)e^{-iEt/\hslash }$ has $|\psi {|}^2=|\varphi (x){|}^2$ independent of $t$, so $\partial \rho /\partial t=0$. The continuity equation then gives $\partial j/\partial x=0$, meaning $j$ is constant in space. For a bound state with $\varphi \to 0$ at infinity, $j=0$ everywhere — bound states carry no current.

Key theorem with proof Intermediate+

Theorem (Conservation of probability). For any solution $\psi (\mathbf{r},t)$ of the time-dependent Schrodinger equation with real potential $V(\mathbf{r})$ and any volume $\Omega$ with boundary $\partial \Omega$,

$$\frac{d}{dt}{\int }_{\Omega }|\psi {|}^2{d}^{3}r=-{\oint }_{\partial \Omega }\mathbf{j}\cdot d\mathbf{S}.$$

In particular, for all of space ($\Omega ={\mathbb{R}}^3$) and normalisable $\psi$, total probability is conserved: $d/dt\int |\psi {|}^2{d}^{3}r=0$.

Proof. From the Schrodinger equation and its conjugate, derive the continuity equation $\partial \rho /\partial t+\nabla \cdot \mathbf{j}=0$ as above. Integrate over $\Omega$:

$${\int }_{\Omega }\frac{\partial \rho }{\partial t}{d}^{3}r=-{\int }_{\Omega }\nabla \cdot \mathbf{j}{d}^{3}r.$$

Move the time derivative outside the integral (justified for normalisable $\psi$ by dominated convergence) and apply the divergence theorem to the right side:

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

For $\Omega ={\mathbb{R}}^3$, the boundary is at infinity where $\psi \to 0$ (normalisability), so $\mathbf{j}\to 0$ and the surface integral vanishes. $\square$

Corollary (Reflection-transmission relation). For a one-dimensional scattering state with incident amplitude $A$, reflected amplitude $B$, and transmitted amplitude $C$, the probability current is constant in the asymptotic regions: $j_{\text{inc}}+j_{\text{ref}}=j_{\text{trans}}$. Writing $j_{\text{inc}}=\hslash k|A{|}^2/m$, $j_{\text{ref}}=-\hslash k|B{|}^2/m$, $j_{\text{trans}}=\hslash k^{\prime }|C{|}^2/m$ (the reflected current is negative because it flows in the $-x$ direction):

$$\frac{\hslash k}{m}|A{|}^2-\frac{\hslash k}{m}|B{|}^2=\frac{\hslash k^{\prime }}{m}|C{|}^2.$$

Dividing by $j_{\text{inc}}=\hslash k|A{|}^2/m$ gives $1-|B/A{|}^2=(k^{\prime }/k)|C/A{|}^2$, i.e. $1-R=T$, hence $R+T=1$.

Proof. The continuity equation in 1D gives $\partial j/\partial x=0$ for a stationary scattering state ($\partial \rho /\partial t=0$). Therefore $j$ is the same constant in every region. In the left asymptotic region, $j=j_{\text{inc}}+j_{\text{ref}}$ (incident and reflected waves superpose). In the right asymptotic region, $j=j_{\text{trans}}$. Equating and dividing by $j_{\text{inc}}$ yields $R+T=1$. $\square$

Bridge. This result connects to 12.04.03 where the step-potential and finite-square-well transmission coefficients are computed explicitly using the current-ratio definitions. The continuity equation also underpins 12.01.05 where the spreading Gaussian wave packet conserves total probability despite dramatic redistribution of the density.

Exercises Intermediate+

Gauge invariance and relativistic connections Master

Gauge invariance of the probability current. The probability current $\mathbf{j}=\frac{\hslash }{2mi}({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast })$ is invariant under global phase rotations $\psi \to e^{i\alpha }\psi$. Under a local gauge transformation $\psi \to e^{i\chi (\mathbf{r})}\psi$, the current transforms as

$$\mathbf{j}\to \mathbf{j}+\frac{\hslash }{m}|\psi {|}^2\nabla \chi .$$

In the presence of an electromagnetic vector potential $\mathbf{A}$, the Schrodinger equation requires the minimal coupling substitution $\hat{\mathbf{p}}\to \hat{\mathbf{p}}-q\mathbf{A}$, and the probability current becomes the gauge-invariant current:

$${\mathbf{j}}_{\text{em}}=\frac{1}{2m}\left[{\psi }^{\ast }(\hat{\mathbf{p}}-q\mathbf{A})\psi +\psi (\hat{\mathbf{p}}-q\mathbf{A}{)}^{\ast }{\psi }^{\ast }\right]=\frac{\hslash }{2mi}({\psi }^{\ast }\nabla \psi -\psi \nabla {\psi }^{\ast })-\frac{q}{m}\mathbf{A}|\psi {|}^2.$$

This is the physical current that couples to the electromagnetic field. The extra $-q\mathbf{A}|\psi {|}^2/m$ term cancels the gauge-dependent piece, making ${\mathbf{j}}_{\text{em}}$ gauge-invariant under $\mathbf{A}\to \mathbf{A}+\nabla \chi$ combined with $\psi \to e^{iq\chi /\hslash }\psi$.

Relation to the Klein-Gordon current. The relativistic Klein-Gordon equation for a spin-0 particle has the conserved current

$$j^{\mu }=\frac{i\hslash }{2m}({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })$$

where $\varphi$ is the scalar field. The spatial component $\mathbf{j}$ is the same as the non-relativistic probability current. The temporal component $j^0=\rho$ is not $|\varphi {|}^2$ but rather $(i\hslash /2m{c}^2)({\varphi }^{\ast }{\partial }_t\varphi -\varphi {\partial }_t{\varphi }^{\ast })$, which can be negative — the Klein-Gordon current is not a positive-definite probability density. This was one of the historical reasons for the development of the Dirac equation and, ultimately, quantum field theory, where $j^{\mu }$ is reinterpreted as a charge current rather than a probability current.

Relation to the Dirac current. The Dirac equation for a spin-1/2 particle has the conserved current

$$j^{\mu }=c\bar{\psi }{\gamma }^{\mu }\psi$$

where $\bar{\psi }={\psi }^{†}{\gamma }^0$ and ${\gamma }^{\mu }$ are the Dirac matrices. The charge density $j^0=c{\psi }^{†}\psi =c|\psi {|}^2$ is positive-definite, resolving the Klein-Gordon problem. The spatial current $\mathbf{j}=c{\psi }^{†}\mathbf{\alpha }\psi$ involves the Dirac alpha matrices and includes the spin contribution. In the non-relativistic limit, this reduces to the Schrodinger current plus a spin-magnetisation correction.

The Madelung formulation and de Broglie-Bohm theory Master

The polar decomposition $\psi =\sqrt{\rho }{e}^{iS/\hslash }$ transforms the Schrodinger equation into two coupled real equations:

1. Continuity equation: ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{v})=0$ where $\mathbf{v}=\nabla S/m$.

2. Quantum Hamilton-Jacobi equation: ${\partial }_{t}S+(\nabla S{)}^2/(2m)+V+Q=0$ where $Q=-\frac{{\hslash }^2}{2m}\frac{{\nabla }^2\sqrt{\rho }}{\sqrt{\rho }}$.

These are exactly the equations of a classical fluid with density $\rho$, velocity field $\mathbf{v}=\nabla S/m$, and an additional potential $Q$ — the quantum potential. This is the Madelung formulation (1927), a hydrodynamic representation of quantum mechanics.

The de Broglie-Bohm interpretation takes the Madelung equations literally: particles follow definite trajectories $\dot{\mathbf{r}}=\nabla S/m$ guided by the quantum potential. The probability current $\mathbf{j}=\rho \nabla S/m$ is the physical velocity field that governs these trajectories. The continuity equation guarantees that an initial $|\psi {|}^2$ distribution of particles remains $|\psi {|}^2$ distributed at all times — the Born rule is a consequence of the dynamics, not a separate postulate.

The Madelung formulation has a known mathematical difficulty: the velocity field $\nabla S/m$ is ill-defined at nodes of $\psi$ (where $\rho =0$ and the phase is singular). These are precisely the points where the quantum potential diverges. Recent work by Wallstrom (1994) showed that the Madelung equations alone do not uniquely recover the Schrodinger equation — an additional quantisation condition on the circulation $\oint \nabla S\cdot d\mathbf{l}=2\pi \hslash n$ must be imposed. This condition, equivalent to the single-valuedness of $\psi$, is the bridge between the hydrodynamic and Hilbert-space formulations.

Connections Master

• 12.01.01 Wave-particle duality. The probability current formalises the "flow" of probability that underlies the double-slit interference pattern. The current $\mathbf{j}$ is nonzero wherever the wavefunction has a spatially varying phase, and the interference pattern is built up by the accumulation of current at the detector.

• 12.02.01 Hilbert-space formalism. The current $\mathbf{j}=\mathrm{Re}({\psi }^{\ast }\hat{\mathbf{p}}\psi )/m$ is the expectation value of the velocity operator in the position representation. Its conservation follows from the self-adjointness of the Hamiltonian — the continuity equation is the local version of unitarity.

• 12.04.03 Finite square well and tunneling. The reflection and transmission coefficients are defined as current ratios. The condition $R+T=1$ follows from current conservation and holds for any one-dimensional scattering problem.

• 12.01.05 Spreading Gaussian wave packet. The spreading packet redistributes probability via the current $j$. The total current integrated over all space is zero (the centre moves but nothing enters or leaves at infinity), while local currents reshape the density profile.

• 12.10.01 Path integral formulation. The probability current has a semiclassical interpretation in terms of classical trajectories weighted by the action. In the Madelung picture, $\mathbf{j}=\rho \nabla S/m$ and $\nabla S$ is the classical momentum evaluated along the dominant path.

• 12.11.01 Dirac equation. The Dirac probability current $j^{\mu }=c\bar{\psi }{\gamma }^{\mu }\psi$ is the relativistic successor to the Schrodinger current. Its conservation follows from the Dirac equation, and its spatial component includes spin-dependent terms absent in the scalar theory.

• Electromagnetism. The gauge-invariant current ${\mathbf{j}}_{\text{em}}$ couples to the vector potential via minimal coupling. The probability current is the non-relativistic, spin-independent limit of the full electromagnetic current. In superconductivity the London equation relates the current to the vector potential through the same minimal-coupling structure.

Historical notes Master

The continuity equation for probability was implicit in Born's 1926 paper introducing the probabilistic interpretation of the wavefunction. Born recognised that $|\psi {|}^2$ must be conserved for the probability interpretation to be self-consistent, and the continuity equation provides the mechanism.

Madelung (1927) published the hydrodynamic reformulation of the Schrodinger equation, expressing the dynamics in terms of a probability fluid with density $\rho$ and velocity $\mathbf{v}=\nabla S/m$. The Madelung fluid is irrotational (except at nodes), compressible, and subject to the quantum potential $Q$. This was the first formulation that made the "flow" of probability explicit.

The de Broglie-Bohm theory (de Broglie 1927, Bohm 1952) took the Madelung formulation as its dynamical foundation. Bohm's key contribution was showing that the quantum potential $Q$ produces all the characteristic quantum effects (interference, tunneling, uncertainty) without modifying the equations of motion beyond adding $Q$ to the classical Hamilton-Jacobi equation.

The definition of reflection and transmission coefficients via probability current ratios became standard in the textbook literature through the work of Bohm (1951) and Landau and Lifshitz (1958). Earlier treatments used amplitude ratios, which give correct results only when the momenta are equal on both sides of the barrier. The current-ratio definition handles the general case correctly and makes $R+T=1$ transparent.

The relativistic generalisation via the Klein-Gordon current was problematic from the start (negative probability densities). Dirac (1928) resolved this with his first-order equation, which produces a positive-definite probability density. The modern interpretation (quantum field theory) treats both currents as charge currents rather than probability currents, and the conserved charge $Q=\int j^0{d}^{3}x$ is the particle number operator.

Bibliography Master

• Griffiths, D. J. & Schroeter, D. F., Introduction to Quantum Mechanics, 3e (Cambridge, 2018), Ch. 1.4. Standard textbook treatment of probability current and continuity.
• Sakurai, J. J. & Napolitano, J., Modern Quantum Mechanics, 3e (Cambridge, 2021), §2.1. Time evolution and probability conservation in the Hilbert-space formalism.
• Cohen-Tannoudji, C., Diu, B. & Laloë, F., Quantum Mechanics, Vol. 1 (Wiley, 1991), Complement B-I. Detailed treatment of probability current, gauge invariance, and the three-dimensional generalisation.
• Shankar, R., Principles of Quantum Mechanics, 2e (Plenum, 2012), Ch. 5. Probability current, reflection/transmission coefficients, and the step potential.
• Madelung, E., "Quantentheorie in hydrodynamischer Form," Z. Phys. 40, 322 (1927). The hydrodynamic formulation of quantum mechanics.
• Bohm, D., "A suggested interpretation of the quantum theory in terms of 'hidden' variables," Phys. Rev. 85, 166 (1952). The de Broglie-Bohm theory and the quantum potential.
• Wallstrom, T. C., "Inequivalence between the Schrodinger equation and the Madelung hydrodynamic equations," Phys. Rev. A 49, 1613 (1994). The quantisation condition required to recover the Schrodinger equation from the Madelung equations.
• Born, M., "Zur Quantenmechanik der Stossvorgange," Z. Phys. 37, 863 (1926). The Born rule and the probabilistic interpretation.