Probability Current in One Dimension

Every quantum transport calculation ships probability current at its core. The Landauer picture of nanoscale conduction expresses a device's conductance as $\frac{2e^2}{h}\sum T_n$ — charge times transmitted probability currents summed over channels, the direct descendant of this article's flux bookkeeping. Scanning tunnelling microscopes read a current that is equation (7) wearing laboratory units; time-dependent simulation codes monitor $j$ on grids and add complex absorbing potentials near the boundaries to remove outgoing waves. Because the potential is then no longer real, the continuity equation acquires a sink term, allowing probability to leave the computational domain without artificial reflections. And weak-measurement experiments now reconstruct averaged trajectories tracing the streamlines of $j/\rho$ in real interferometers — evidence that the probability current is far more than a bookkeeping device, being measured indirectly in transport experiments, computed routinely in simulations, and, as here, reconstructed experimentally through weak measurements.

Think of probability as an incompressible fluid smeared along the line, with density $\rho = |\psi|^2$. The current $j$ is how fast, and which way, that fluid streams past each point. Fence off any interval: if more flows in at the left than leaves at the right, the amount inside rises by exactly the difference. That bookkeeping is the continuity equation, and it is local — probability cannot teleport between regions; it must flow through every point in between, interval by interval, instant by instant.

Where does the flow hide in $\psi$? Not in its magnitude. In polar form $\psi = \sqrt{\rho}\,e^{iS/\hbar}$ the current is $j = \rho\,\partial_x S/m$: probability flows along the gradient of the phase. Flat phase — any real $\psi$, however dramatic its lumps — is a fluid at rest; a linearly climbing phase $e^{ikx}$ is uniform flow. This one idea organises everything: standing waves are still, travelling waves flow, and two real exponentials in a barrier carry current only because their coefficients hold a relative phase.

One more habit worth installing: $j$ has units of probability per unit time — what a particle counter reports. If $|\psi|^2$ is normalized as particles per unit length, then $j$ gives the number of particles crossing a point each second. Multiplying by the particle charge converts this directly into the measurable electric current. The current is the interface between wave mechanics and laboratory count rates.

Start from the two Born-rule objects. The density:

where: $\rho\,dx$ — the probability of finding the particle between $x$ and $x + dx$ at time $t$. For a one-particle wavefunction $\rho$ integrates to 1; for a beam it is calibrated to particles per metre. The density is only a snapshot of where the particle is likely to be. Different wavefunctions can have the same $\rho$ while carrying completely different probability currents. The companion, fixed uniquely by the conservation demand derived below, is the current:

where: the two forms are the same number. Read the structure: $j$ compares $\psi$ with its own derivative and keeps only the out-of-phase part — if the wavefunction has a spatially constant phase (so that $\psi$ and $\partial_x\psi$ are locally in phase), then $j = 0$. Units: $\mathrm{m}^{-2}$ times $\hbar/m$ ($\mathrm{m^2\,s^{-1}}$) gives $\mathrm{s^{-1}}$ — probability per second, a count rate. The two are yoked by the continuity equation — probability conserved locally:

where: At every point, the probability density changes only because probability flows into or out of an infinitesimal neighbourhood. Integrate (3) over $[a,b]$: $\frac{d}{dt}\int_a^b \rho\,dx = j(a,t) - j(b,t)$ — inflow minus outflow, the figure above. Send the fences to infinity for a normalizable state and both boundary currents die, recovering Conservation of Normalization: equation (3) is that theorem's local, sharper parent. The object is first introduced in Probability Current and the Continuity Equation, where it guarantees normalization survives time evolution. This article owns the one-dimensional craft: a clean derivation, the vanishing theorems, its value in every state this chapter uses, and the connection to what detectors count.

The polar decomposition makes the phase-flow doctrine exact. Write $\psi = \sqrt{\rho}\,e^{iS/\hbar}$ with $\rho$, $S$ real; substitute into (2) — the modulus factors are real and drop — leaving

where: $\partial_x S$ plays the role of a local momentum, making $j = \rho \times (\text{local velocity})$ — the exact analogue of the hydrodynamic flux $\rho v$. This is the Madelung form (1926): probability flows down the phase gradient, at a rate set by how steeply the phase climbs. Flat $S$ means still fluid, whatever $\rho$ looks like. Now evaluate it in the chapter's working states. For a plane wave $\psi = Ae^{ikx}$, $S = \hbar kx$ and (4) gives

where: $v = \hbar k/m$ is the classical velocity at momentum $\hbar k$ (the group velocity of a packet built around $k$ — Group Velocity versus Phase Velocity makes that identification honestly). A beam of $|A|^2$ particles per metre at speed $v$ delivers $v|A|^2$ per second — the reason this chapter's scattering coefficients carry velocity factors. Superpose both directions, $\psi = Ae^{ikx} + Be^{-ikx}$, and a small miracle cleans the bookkeeping:

where: the $A$–$B$ cross-terms — which ripple the density — cancel identically in the current: net flow is rightward flux minus leftward flux, as if the streams passed through each other. This cancellation is what lets scattering theory treat "incident" and "reflected" currents as separate numbers in the same region. (It is special to equal $|k|$: superpose two different wavenumbers and $j$ genuinely oscillates.) With (5) and (6), the definitions in Transmission and Reflection Coefficients — $R$, $T$ as current ratios, the velocity-factor law $T = (k_2/k_1)|t|^2$, the Wronskian proof of $R + T = 1$ — are fully underwritten. That article owns the ratio doctrine; here we add the object-level fact that makes a single transmitted number meaningful: constancy.

Example 1 — from wavefunction to ammeter (confidence builder). A monoenergetic electron beam of energy $E = 10$ eV is described by $\psi = Ae^{ikx}$ with $|A|^2 = 5\times10^{8}$ electrons per metre of beam. What current does a detector register?

Step 1. Wavenumber: $k = \sqrt{2m_eE}/\hbar = \sqrt{2(9.109\times10^{-31})(10)(1.602\times10^{-19})}/(1.055\times10^{-34}) = 1.62\times10^{10}\,\mathrm{m^{-1}}$.

Step 2. Velocity: $v = \hbar k/m_e = (1.055\times10^{-34})(1.62\times10^{10})/(9.109\times10^{-31}) = 1.88\times10^{6}\,\mathrm{m/s}$ — about $0.6\%$ of $c$, comfortably non-relativistic.

Step 3. Current: $j = v|A|^2 = (1.88\times10^{6})(5\times10^{8}) = \boxed{9.4\times10^{14}\ \text{electrons per second}}$ past any checkpoint.

Step 4. As electrical current: $I = ej = (1.602\times10^{-19})(9.4\times10^{14}) = 0.15$ mA — an ordinary bench-top beam current. The chain $\psi \to j \to$ ammeter is two multiplications long.

Example 2 — the sloshing box (exam level). An electron in a 1 nm infinite square well is prepared in the equal superposition $\Psi = \frac{1}{\sqrt2}\left(\psi_1 e^{-iE_1t/\hbar} + \psi_2 e^{-iE_2t/\hbar}\right)$ of the two lowest states. Find the current at the centre of the well and the period of the sloshing.

Step 1. At $x = a/2$: $\psi_1 = \sqrt{2/a}$ (crest), $\psi_2 = 0$ (node), $\psi_1' = 0$, $\psi_2' = -(2\pi/a)\sqrt{2/a}$. Each stationary eigenstate individually carries zero current. The entire oscillating current arises from the interference (cross) term between the two energy eigenstates.

Step 2. Assemble equation (2): $\Psi^*\partial_x\Psi\big|_{a/2} = \frac{1}{2}\,\psi_1(a/2)\,\psi_2'(a/2)\,e^{i(E_1 - E_2)t/\hbar} = -\frac{2\pi}{a^2}\,e^{-i\Delta E\,t/\hbar}$ with $\Delta E = E_2 - E_1$. Taking the imaginary part,

Step 3. Numbers: $E_1 = \pi^2\hbar^2/2m_ea^2 = 0.376$ eV at $a = 1$ nm, so $\Delta E = 3E_1 = 1.13$ eV, and $j_{\max} = 2\pi\hbar/(m_ea^2) = \boxed{7.3\times10^{14}\ \mathrm{s^{-1}}}$.

Step 4. Period: $T = 2\pi\hbar/\Delta E = \boxed{3.7\ \text{fs}}$. The lump sloshes across the nanometre box at petahertz rates, the centre current passing through zero exactly as it turns around. A femtosecond movie of charge sloshing in a quantum well is, almost literally, a plot of this $j$.

References. Griffiths, Introduction to Quantum Mechanics, 3rd ed., §1.5 (probability current) and §2.6 (scattering) · Cohen-Tannoudji, Quantum Mechanics, Vol. I, Complement D_III · Shankar, Principles of Quantum Mechanics, 2nd ed., Ch. 5 · Sakurai & Napolitano, Modern Quantum Mechanics, 2nd ed., §2.4 · Townsend, A Modern Approach to Quantum Mechanics, Ch. 6 Key results to keep in view: the plane-wave current $j=\hbar k|A|^2/m$; the $R+T=1$ identity relying on the $k_2/k_1$ flux factor; the cancellation of $V$ terms in the continuity derivation requiring real potential; the imaginary-part identity yielding the two equivalent current forms in equation (2); the barrier interior current (7); and the Ehrenfest flux identity $\dot{\langle x\rangle}=\langle p\rangle/m$.

Builds on: Time-Independent Schrödinger Equation in One Dimension · Free Particle and Plane Waves · Bound States versus Scattering States

Leads to: Finite Square Well · Wave Packets and Their Construction · Group Velocity versus Phase Velocity