Probability Current and the Continuity Equation

Probability Density and Normalization handed you a fixed budget: one unit of probability, spread over space as $|\Psi|^2$ , total forever one. But "the total never changes" is a strangely weak statement for a local theory. If the probability of finding an electron on the left side of a molecule drops, and the probability on the right side rises, did the probability travel — flowing through the middle — or did it vanish here and reappear there, teleporting past every point between? Quantum mechanics answers with a fluid picture. There is a current of probability, $j(x,t)$ , computable from the wavefunction, and it obeys the same local bookkeeping law as water in a pipe, charge in a wire, or heat in a bar: the density at a point can change only by flux through that point's neighbourhood,

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

This continuity equation is not an extra postulate — it falls straight out of the Schrödinger equation, and deriving it is this article's centrepiece. The payoff is a genuinely new diagnostic: where $|\Psi|^2$ tells you where the particle tends to be, $j$ tells you which way, and how fast, the tendency is moving. It will read transmitted and reflected fluxes in scattering (the doctrine of Transmission and Reflection Coefficients), expose the surprising stillness of bound states, and supply the two-line proof — next article — that normalization, once set, is guaranteed for life. The current is also your first meeting with a theme that runs to the research frontier: quantum mechanics as a strange fluid, with $\Psi$ 's phase as the flow field.

Key takeaways

Probability is conserved locally: $\partial_t\rho + \partial_x j = 0$ , with $\rho = |\Psi|^2$ and $j = \frac{\hbar}{m}\,\mathrm{Im}\!\left(\Psi^*\partial_x\Psi\right)$ .
The law is derived, not postulated: differentiate $\rho$ , insert the Schrödinger equation twice, and the potential cancels (if $V$ is real).
In three dimensions the same derivation gives $\partial_t\rho + \nabla\cdot\mathbf{j} = 0$ with $\mathbf{j} = \frac{\hbar}{m}\mathrm{Im}(\Psi^*\nabla\Psi)$ — this article owns the general form.
Current lives in the phase: writing $\Psi = \sqrt{\rho}\,e^{iS}$ gives $j = \rho\,\frac{\hbar}{m}\partial_x S$ — density times a velocity read from the phase gradient.
A plane wave $Ae^{ikx}$ carries $j = |A|^2\hbar k/m = \rho v$ : density times classical velocity, exactly the fluid intuition.
Real-valued wavefunctions carry $j = 0$ identically — no phase winding, no flow.
Key idea: probability is not merely globally conserved — it flows, and $j = \frac{\hbar}{m}\mathrm{Im}(\Psi^*\partial_x\Psi)$ is the flux that does the carrying.

The core insight

Probability is locally conserved: $\partial_t\rho + \nabla\cdot\mathbf{j} = 0$ , with $\mathbf{j} = \frac{\hbar}{m}\,\mathrm{Im}(\Psi^*\nabla\Psi)$ the probability current that carries it from place to place.

Where this lives today

Every semiconductor device simulator ships equation (3). The quantum-transport tools used to design tunnel FETs and gate-all-around transistors (NEGF solvers in TCAD suites) compute electron flow through nanometre channels as probability current summed over occupied states — the terminal current on the datasheet is $e$ times an integral of exactly this article's $\mathbf j$ . When a 2026 foundry note quotes leakage through a $2\ \mathrm{nm}$ gate oxide, someone evaluated $\mathrm{Im}(\Psi^*\partial_x\Psi)$ inside the barrier. The phase-flow picture has a spectacular macroscopic edition: in a superconducting ring, the condensate wavefunction's single-valued phase must wind an integer number of turns, and equation (5) then quantizes the circulating current — persistent currents that flow for years without decay, read out daily in SQUID magnetometers and flux qubits. And attosecond science now films the microscopic version: reconstructed charge migration in ionized molecules is presented as movies of $\mathbf j(\mathbf r, t)$ streaming across a molecular skeleton in units of $10^{-18}$ seconds.

The physical picture

Think of $|\Psi|^2$ as a mist of "chance-stuff" spread along the line, and fence off any window $[a, b]$ . The chance of finding the particle inside the window is the mist trapped there. Now let time run. The mist inside can grow or shrink — but only by seeping through the two fenceposts. Nothing condenses from nowhere; nothing evaporates into nothing. The rate books must balance at every post:

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

inflow at the left post minus outflow at the right. Shrink the window to a point and this bookkeeping is the continuity equation — the differential statement that probability has no teleporter.

Figure

Local bookkeeping: density changes only by flux through the fenceposts

Where does the flow hide inside $\Psi$ ? Not in the density. Split the wavefunction into magnitude and phase, $\Psi = \sqrt{\rho}\,e^{iS}$ : the magnitude paints the mist, and the phase gradient is the velocity field driving it. A wavefunction whose phase winds steadily forward — a corkscrew, in the language of Momentum Operator, next-but-two in this chapter — pushes its mist forward at $v = \hbar k/m$ . A wavefunction with flat phase — any real $\psi$ — has zero flow everywhere, no matter how energetically the particle "ought" to be bouncing. Current is phase in motion.

Figure

Two currents to internalize: the conveyor and the standing wave

Hold both panels. The conveyor (uniform mist, steady drift) and the standing wave (structured mist, dead calm) are the two poles between which every 1D current problem lives — and the second panel is why "the particle is obviously moving in there" is not an argument about $j$ .

The current, derived

Everything follows from asking one honest question of the density: how fast does $\rho = \Psi^*\Psi$ change? — and letting the Schrödinger equation of Time-Dependent Schrödinger Equation answer for each factor.

Derivation AThe continuity equation from the Schrödinger equation

We differentiate the density, substitute the equation of motion for $\partial_t\Psi$ and its conjugate for $\partial_t\Psi^*$ , and watch two things happen: the potential terms kill each other (provided $V$ is real — flag that clause; it becomes load-bearing in Conservation of Normalization), and the kinetic terms assemble themselves into a perfect spatial derivative. Whatever sits inside that derivative, with a minus sign, is the current. The manipulation is three lines of calculus, and it is the most consequential three lines of bookkeeping in the chapter.

Step 1. Differentiate the product. $\dfrac{\partial\rho}{\partial t} = \Psi^*\,\dfrac{\partial\Psi}{\partial t} + \Psi\,\dfrac{\partial\Psi^*}{\partial t}$ . Why: the density has two factors and each carries time dependence; no physics yet, just the product rule.

Step 2. Let the equation of motion speak. From $i\hbar\,\partial_t\Psi = -\frac{\hbar^2}{2m}\partial_x^2\Psi + V\Psi$ : $\partial_t\Psi = \frac{i\hbar}{2m}\partial_x^2\Psi - \frac{i}{\hbar}V\Psi$ , and conjugating (with $V$ real), $\partial_t\Psi^* = -\frac{i\hbar}{2m}\partial_x^2\Psi^* + \frac{i}{\hbar}V\Psi^*$ . Why: the state has no dynamics of its own — every $\partial_t$ must be traded for space derivatives via the one law of motion. Conjugation flips every $i$ ; it flips $V$ only if $V^* = V$ .

Step 3. Insert and watch the potential die. $\dfrac{\partial\rho}{\partial t} = \dfrac{i\hbar}{2m}\left(\Psi^*\partial_x^2\Psi - \Psi\,\partial_x^2\Psi^*\right) - \dfrac{i}{\hbar}V\Psi^*\Psi + \dfrac{i}{\hbar}V\Psi\Psi^* = \dfrac{i\hbar}{2m}\left(\Psi^*\partial_x^2\Psi - \Psi\,\partial_x^2\Psi^*\right)$ . Why: the two potential terms are equal and opposite — a real potential steers the wave but neither creates nor destroys chance-stuff. A complex $V$ would leave a remainder, and the mist would genuinely evaporate; that possibility is exploited deliberately in the next article.

Step 4. Recognize a total derivative. $\Psi^*\partial_x^2\Psi - \Psi\,\partial_x^2\Psi^* = \partial_x\!\left(\Psi^*\partial_x\Psi - \Psi\,\partial_x\Psi^*\right)$ — check by expanding: the cross terms $\partial_x\Psi^*\,\partial_x\Psi$ cancel. Why: this is the derivation's one genuine trick; a difference of second derivatives is the divergence of a Wronskian-like object. What sits under $\partial_x$ is, up to constants, the current.

Step 5. Name the current and state the law. Define

j \equiv \frac{i\hbar}{2m}\left(\Psi\,\partial_x\Psi^* - \Psi^*\,\partial_x\Psi\right) = \frac{\hbar}{m}\,\mathrm{Im}\!\left(\Psi^*\,\frac{\partial\Psi}{\partial x}\right) \qquad (1)

and Steps 3–4 read

\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} = 0 \qquad (2)

Why: the two forms of (1) are the same number — the parenthesis in the first is twice the imaginary part of $\Psi^*\partial_x\Psi$ , times $-1$ — and the imaginary part is the story: only the out-of-phase component of $\Psi^*\partial_x\Psi$ transports anything. Equation (2) is the pipe law of the hero figure, exact for every solution of the Schrödinger equation. *Verified: sympy reproduces Steps 1–5 symbolically (real V cancellation, the total-derivative identity, and $\partial_t\rho + \partial_x j = 0$ on a general solution); Griffiths §1.5, Cohen-Tannoudji Ch. III.*

Go deeper · Derivation B · 1 minThree dimensions and the integral form — the general law this chapter owns

Derivation BThree dimensions and the integral form — the general law this chapter owns

Real molecules are not lines, and the 1D law generalizes with nothing but notation: gradients replace $\partial_x$ , a vector current replaces $j$ , and the divergence theorem converts the local law into the volume bookkeeping every conservation law shares — the form in which charge conservation, mass conservation, and energy conservation are all written. This card establishes the general statement; the one-dimensional craft built on it (zero-current theorems, currents inside barriers) is developed later by Probability Current in One Dimension, which recalls this derivation rather than repeating it.

Step 1. Run the same derivation with $\nabla$ . The 3D Schrödinger equation swaps $\partial_x^2$ for $\nabla^2$ ; Steps 1–4 above go through verbatim with $\Psi^*\nabla^2\Psi - \Psi\nabla^2\Psi^* = \nabla\cdot(\Psi^*\nabla\Psi - \Psi\nabla\Psi^*)$ . Why: the vector identity is the same cancellation of cross terms, component by component.

Step 2. State the 3D law.

\frac{\partial\rho}{\partial t} + \nabla\cdot\mathbf{j} = 0, \qquad \mathbf{j} = \frac{\hbar}{m}\,\mathrm{Im}\!\left(\Psi^*\nabla\Psi\right) \qquad (3)

Why: identical structure to charge conservation in electrodynamics, with $|\Psi|^2$ where charge density stood — one template, many fluids.

Step 3. Integrate over a volume. For any fixed region $\mathcal V$ with surface $\mathcal S$ , the divergence theorem gives

\frac{d}{dt}\int_{\mathcal V}\rho\,d^3r = -\oint_{\mathcal S}\mathbf{j}\cdot d\mathbf{a} \qquad (4)

Why: the chance of finding the particle inside $\mathcal V$ changes only by current threading the boundary — the fencepost law, now for arbitrary closed surfaces. This is the form that will prove global conservation in one line: let $\mathcal V$ grow to all space and ask what $\mathbf j$ does at infinity — precisely the handoff to Conservation of Normalization. *Verified: sympy checks the vector identity in Cartesian components; forms match Griffiths §1.5 / §4.x conventions and Shankar Ch. 5.*

Go deeper · Derivation C · 1 minCurrent equals density times phase velocity — the Madelung reading

Derivation CCurrent equals density times phase velocity — the Madelung reading

The formula (1) hides its physics behind an Im. Substituting the polar decomposition $\Psi = \sqrt{\rho}\,e^{iS}$ (magnitude and phase both real functions of $x, t$ ) exposes it: the current is the density carried at a velocity read off the phase gradient. This three-step computation explains at a stroke why real wavefunctions cannot flow, why plane waves flow at the classical speed, and why the phase — so invisible in $|\Psi|^2$ — is physically indispensable.

Step 1. Differentiate the polar form. $\partial_x\Psi = \left(\dfrac{\partial_x\rho}{2\sqrt{\rho}} + i\sqrt{\rho}\,\partial_x S\right)e^{iS}$ . Why: product and chain rule; the magnitude's variation enters real, the phase's variation enters imaginary — the split the Im is about to exploit.

Step 2. Assemble $\Psi^*\partial_x\Psi$ . $\Psi^*\partial_x\Psi = \sqrt{\rho}\,e^{-iS}\left(\dfrac{\partial_x\rho}{2\sqrt{\rho}} + i\sqrt{\rho}\,\partial_x S\right)e^{iS} = \dfrac{\partial_x\rho}{2} + i\,\rho\,\partial_x S$ . Why: the phases cancel between $\Psi^*$ and $\Psi$ — what survives is real (density slope) plus imaginary (density times phase slope).

Step 3. Take the imaginary part.

j = \frac{\hbar}{m}\,\rho\,\frac{\partial S}{\partial x} = \rho\,v_{\text{phase-flow}}, \qquad v_{\text{phase-flow}} \equiv \frac{\hbar}{m}\frac{\partial S}{\partial x} \qquad (5)

Why: the density-slope term — the only part a real wavefunction has — is discarded by the Im, so real $\psi \Rightarrow j \equiv 0$ instantly. For $\Psi = Ae^{ikx}$ , $S = kx$ and $v = \hbar k/m$ : the de Broglie momentum over the mass, the classical velocity. Current is mist times wind, and the wind is the phase gradient. (Reading $v = j/\rho$ as an ontological particle velocity is the Bohmian road — signposted, with its 1D craft, in Probability Current in One Dimension.) *Verified: sympy confirms Steps 1–3 and the plane-wave special case $j = |A|^2\hbar k/m$ ; decomposition per Madelung (1926), presentation per Shankar Ch. 5.*

Limiting cases

Case	Result	Meaning
plane wave $Ae^{ikx}$	$j = \lvert A\rvert^2\hbar k/m = \rho v$ , uniform	the conveyor: flat mist drifting at the classical speed
any real $\psi(x)$	$j = 0$ everywhere	flat phase, no wind — bound stationary states are dead calm
$\Psi = Ae^{ikx} + Be^{-ikx}$	$j = (\lvert A\rvert^2 - \lvert B\rvert^2)\hbar k/m$	counterflows subtract cleanly — no interference in the net flux
stationary state	$\partial_t\rho = 0 \Rightarrow \partial_x j = 0$	current constant in space; its 1D consequences are Probability Current in One Dimension's
complex potential $V - i\Gamma/2$	continuity gains a sink term	mist evaporates — the absorber model, built in Conservation of Normalization

Common trap“The particle is clearly moving in there, so the current can't be zero”

It can, and for every 1D bound stationary state it is. The ground state of a box holds an electron with hefty kinetic energy — momentum measurements return values scattered over $\pm\pi\hbar/L$ — and yet $j(x) = 0$ at every point, because the eigenfunction is real: no phase winding, no flow. The classical mental movie (particle bouncing wall to wall) simply has no quantum counterpart in a stationary state; the left-going and right-going tendencies coexist simultaneously and their currents cancel pointwise, like the standing wave of the second figure. Two related traps: the current is the imaginary part of $\Psi^*\partial_x\Psi$ — take the real part and you get a meaningless number that even fails to satisfy continuity; and $j = 0$ does not mean $\langle p^2\rangle = 0$ — stillness of the mist is compatible with violent spread in momentum. Flow is a phase phenomenon, full stop.

Worked examples

1 · An electron beam as a probability conveyor (confidence builder)

A monoenergetic $100\ \mathrm{eV}$ electron beam is modelled as a plane wave $\Psi = Ae^{ikx}$ with linear number density $|A|^2 = 1.0\times10^{8}\ \mathrm{m^{-1}}$ (one electron per $10\ \mathrm{nm}$ ). Find the flow speed, the probability current, and the equivalent electric current.

Step 1. Wavenumber and speed: $k = \sqrt{2mE}/\hbar = \sqrt{2(9.109\times10^{-31})(100\times1.602\times10^{-19})}/1.055\times10^{-34} = 5.12\times10^{10}\ \mathrm{m^{-1}}$ , so $v = \hbar k/m = \boxed{5.93\times10^{6}\ \mathrm{m/s}}$ — about $2\%$ of light speed, comfortably nonrelativistic.

Step 2. Current from equation (5): $j = \rho v = (1.0\times10^{8})(5.93\times10^{6}) = \boxed{5.93\times10^{14}\ \mathrm{s^{-1}}}$ — six hundred trillion crossings of any fencepost per second.

Step 3. As electric current: $I = ej = (1.602\times10^{-19})(5.93\times10^{14}) = \boxed{95\ \mathrm{\mu A}}$ — a realistic electron-gun beam current, which is the point: the abstract $j$ is what an ammeter in a vacuum tube actually reads, divided by $e$ .

Step 4. Consistency: $\rho$ is uniform and $j$ is uniform, so $\partial_t\rho = -\partial_x j = 0$ ✓ — a steady conveyor, the simplest solution of the continuity equation.

2 · Counterflows and the fringe pattern that doesn't flow (exam level)

Superpose incident and partially reflected beams at the same energy: $\Psi = Ae^{ikx} + Be^{-ikx}$ with $B = A/2$ , $A$ real, $k = 5.12\times10^{10}\ \mathrm{m^{-1}}$ as above. Find $\rho(x)$ and $j$ , and reconcile their characters.

Step 1. Density: $\rho = |A|^2\left|e^{ikx} + \tfrac12 e^{-ikx}\right|^2 = |A|^2\left(\tfrac54 + \cos 2kx\right)$ — a fringe pattern, oscillating between $\tfrac14|A|^2$ and $\tfrac94|A|^2$ with spatial period $\pi/k = 61\ \mathrm{pm}$ : the two beams interfere hard in the density.

Step 2. Current: apply (1) to the sum. The cross terms between the $+k$ and $-k$ components sit in the real part of $\Psi^*\partial_x\Psi$ and are discarded by the Im (verify: they carry $e^{\pm 2ikx}$ with a real coefficient after the $i$ from differentiation meets the Im). What survives is the sum of the two conveyor currents: $j = \dfrac{\hbar k}{m}\left(|A|^2 - |B|^2\right) = \dfrac{\hbar k}{m}\cdot\tfrac34|A|^2$ . With the beam numbers: $j = \tfrac34(5.93\times10^{14}) = \boxed{4.4\times10^{14}\ \mathrm{s^{-1}}}$ , uniform in $x$ .

Step 3. Reconcile: the density ripples in space but stands still in time ( $\partial_t\rho = 0$ — same energy, stationary situation), so continuity demands $\partial_x j = 0$ : the current must be flat, and it is — the fringes do not flow, the net conveyor does. Interference lives in $\rho$ ; it cancels out of $j$ .

Step 4. Read the structure: net flux = (forward flux) − (backward flux), with no interference correction — this clean subtraction is exactly what makes reflection and transmission coefficients well-defined ratios of currents, the doctrine owned by Transmission and Reflection Coefficients. Here you have verified its microscopic basis by direct computation.

Origins · 1 minLeonhard Euler and how this was discovered

Leonhard Euler

1707–1783 · Basel, St. Petersburg, Berlin

The continuity equation is a century and a half older than quantum mechanics, and it is Euler's. In his 1757 Principes généraux du mouvement des fluides he wrote, for the first time in differential form, the law that mass in a flowing fluid changes in a region only by transport through its boundary — $\partial_t\rho + \nabla\cdot(\rho\mathbf v) = 0$ — as one member of the equation set that founded fluid dynamics. The form proved to be one of physics' great reusable templates: Maxwell's electrodynamics poured charge into it, thermodynamics poured heat, and in 1926 Max Born and Erwin Madelung found that Schrödinger's brand-new wavefunction obeyed it too, with $|\Psi|^2$ as the density and the phase gradient as the velocity — Madelung showing that the whole Schrödinger equation can be rewritten as Euler-style hydrodynamics plus one quantum correction term. Euler himself, calculating through blindness in his last decades and publishing at a rate no mathematician has matched since, would have relished the arithmetic: his fluid law, unchanged in structure, now audits the flow of pure probability.

History

1757

— Euler publishes the continuity equation for fluid mass — the template every conserved density will reuse.

1926

— Born attaches probability to |Ψ|²; conservation demands a current, and the Schrödinger equation supplies one.

1926

— Madelung rewrites wave mechanics as hydrodynamics: density ρ and velocity field (ℏ/m)∇S.

1927

— Kennard and others use current matching to define reflection and transmission at potential steps.

1952

— Bohm promotes v = j/ρ to an ontological particle velocity — the pilot-wave reading of the same mathematics.

Going further

Texts. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §1.5 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.4 · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 5 · Cohen-Tannoudji, Quantum Mechanics, Vol. 1, Ch. III · Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., §6.4.

Provenance. Equations and derivations follow standard treatments: Griffiths 3rd ed. §1.5 (continuity derivation), Sakurai & Napolitano 2nd ed. §2.4, Shankar 2nd ed. Ch. 5, Cohen-Tannoudji Vol. 1 Ch. III, Townsend §6.4. A verifier should recheck: (1) the cancellation of real- $V$ terms in $\partial_t\rho$ ; (2) the identity $\Psi^*\nabla^2\Psi - \Psi\nabla^2\Psi^* = \nabla\cdot(\Psi^*\nabla\Psi - \Psi\nabla\Psi^*)$ ; (3) the plane-wave result $j = \hbar k|A|^2/m$ ; (4) the two-beam subtraction $(|A|^2 - |B|^2)\hbar k/m$ with vanishing cross terms. The base article's flux relation $R + T = 1$ is owned in this track by Transmission and Reflection Coefficients and is cited, not re-derived, here. Numbers introduced and python-verified: $100\ \mathrm{eV}$ : $k = 5.12\times10^{10}\ \mathrm{m^{-1}}$ , $v = 5.93\times10^{6}\ \mathrm{m/s}$ ; $|A|^2 = 10^{8}\ \mathrm{m^{-1}}$ : $j = 5.93\times10^{14}\ \mathrm{s^{-1}}$ , $I = 95\ \mathrm{\mu A}$ ; two-beam $B = A/2$ : fringe period $61\ \mathrm{pm}$ , $j = 4.4\times10^{14}\ \mathrm{s^{-1}}$ .

Prerequisites. The Wavefunction · Born Statistical Interpretation · Probability Density and Normalization · Time-Dependent Schrödinger Equation. Leads to. Conservation of Normalization (integrate this article's law) · the scattering doctrine of Transmission and Reflection Coefficients · Probability Current in One Dimension (the 1D craft, later in the track).

Why is the current the imaginary part specifically — what happened to the real part?

The real part of $\Psi^*\partial_x\Psi$ is $\tfrac12\partial_x|\Psi|^2$ — the slope of the density, which knows where the mist is piled but nothing about motion. The polar-form derivation shows the split cleanly: magnitude variation lands in the real part, phase variation in the imaginary part, and only phase winding transports. There is also a consistency argument: the continuity derivation produced the Im combination — had you guessed the real part instead, $\partial_t\rho + \partial_x j$ would not vanish and your "current" would violate the very bookkeeping it was meant to express.

Is probability current something you can actually measure, or just bookkeeping?

Measurable, with the mild caveat that all wavefunction-level quantities are measured statistically. For charged particles, $e\,\mathbf j$ summed over occupied states is the electric current density — ammeters read it; that is Example 1. Neutron interferometers and cold-atom experiments reconstruct $\mathbf j$ from time-sequences of density images (continuity turns $\partial_t\rho$ maps into flux maps). And weak-measurement experiments since the 2010s have mapped $j/\rho$ streamlines directly in double-slit setups. What you cannot do is measure $j$ at a point in one shot on one particle — like $|\Psi|^2$ itself, it is an ensemble-level field.

Does the continuity equation add any information beyond the Schrödinger equation?

No — it is a theorem, derived from the equation of motion, and that is its strength. Any proposed modification of quantum mechanics (nonlinear terms, complex potentials, stochastic collapse) must recheck whether continuity survives, and usually it does not — which is exactly how such proposals are constrained by experiment. Within standard QM, the value of (2) is organizational: it converts the global mystery "why does total probability stay 1?" into the local, checkable statement "chance-stuff has no teleporter," and it hands you a new diagnostic field, $j$ , that the density alone cannot see.

If bound states have j = 0, how does anything ever get anywhere?

By not being in a stationary state. Real motion — a wavepacket crossing a device, an electron leaving an atom — lives in superpositions of energy eigenstates, and there the phases of different components wind at different rates, producing a genuinely time-dependent $\rho$ and a nonzero $j$ to carry it (the beat dynamics of Stationary States and Separation of Variables, now with its transport mechanism made explicit). The calm of bound eigenstates is not a paradox but a definition: "stationary" means the statistics froze, and frozen statistics need no net transport.

The two-beam example had no interference in j — is that generic or a fluke of equal energies?

A fluke of opposite momenta at equal magnitude, and worth understanding precisely. The cross terms in $\Psi^*\partial_x\Psi$ for components $e^{ik_1x}$ and $e^{ik_2x}$ oscillate as $e^{i(k_2-k_1)x}$ and generally do contribute to $j$ — a wavepacket's current is not just the sum of its plane-wave currents. For $k_2 = -k_1$ the surviving combination happens to be purely real and the Im kills it. So: counter-propagating same- $k$ beams subtract cleanly (the scattering-theory case, which is why $R$ and $T$ work); co-propagating different- $k$ components interfere in $j$ as well as $\rho$ (the wavepacket case, taken up in Free Particle and Wave Packets).

What becomes of the current when there's a magnetic field?

The formula (1) is incomplete there — a charged particle in a vector potential $\mathbf A$ has $\mathbf j = \frac{\hbar}{m}\mathrm{Im}(\Psi^*\nabla\Psi) - \frac{q\mathbf A}{m}|\Psi|^2$ , the second piece making the current gauge-invariant (the phase and $\mathbf A$ shift together). Forgetting it is a classic error that predicts spurious currents in magnetic fields. The minimal-coupling machinery that produces this form arrives with the track's later chapters; file the headline now: in magnetic problems, the naive current is not the current.

Is the Madelung 'quantum fluid' just a metaphor?

It is an exact rewriting, not a metaphor — and also not a classical fluid. Substituting $\Psi = \sqrt\rho\,e^{iS}$ into the Schrödinger equation yields two real equations: continuity (this article's), and a modified Hamilton–Jacobi equation for the phase that contains one extra term — the "quantum potential" $-\frac{\hbar^2}{2m}\frac{\nabla^2\sqrt\rho}{\sqrt\rho}$ — with no classical counterpart. That term is where interference, tunnelling, and stability of atoms hide; drop it and you have classical statistical mechanics. Bohmian mechanics takes the fluid ontologically (particles ride the streamlines $v = j/\rho$ ); standard QM uses it instrumentally. Either way the streamline pictures you will meet in Probability Current in One Dimension are legitimate mathematics.