The continuity equation & probability current

Statement

The Schrödinger equation conserves total probability locally, and the flow is described by a current whose form follows without any additional assumption.

Why it matters

Normalisation is not an extra postulate bolted onto quantum mechanics — it is preserved automatically by the dynamics, and this derivation proves it. The current is then the tool that makes transmission and reflection coefficients meaningful in D-011.

Assumptions

Derivation

Result

Limiting cases

• Plane wave ψ = Ae^ikx: j = ħk|A|²/m = v|A|² — density times velocity, exactly as a classical flux would be.
• Real wavefunction: j = 0 — bound stationary states carry no current, which is why they are stationary.

Breaks when

Failure modes

• Writing j = ħ/m · ψ^*∂ψ/∂x without taking the imaginary part — the result is then complex and not a current.
• Assuming a non-zero j means the particle is definitely moving. It is a probability flux, not a trajectory.

Worked number

For a free electron plane wave with k fixed, the current reduces to j = v|A|²: evaluate the group velocity v = ħk/m and multiply by the density |A|² to confirm the flux units come out as s⁻¹.

Run the check →

Discussion

The result is a continuity equation, and its power lies in being local. A statement like d/dt∫|ψ|² dx = 0 only fixes the grand total; on its own it would even permit probability to vanish here and reappear elsewhere. The differential form ∂ρ/∂t + ∂j/∂x = 0 forbids exactly that: whatever probability leaves a small interval must physically cross its endpoints, carried by the current j. Probability behaves like an incompressible, indestructible fluid — never created, never annihilated, only transported. The global conservation law is then a corollary, recovered by integrating over all space and discarding the surface term.

Notice where the current actually comes from. Writing ψ = √ρ e^iφ in amplitude–phase form, one line gives j = (ℏ/m) ρ ∂φ/∂x. The current is the density times a velocity field v = (ℏ/m) ∂φ/∂x set entirely by the gradient of the phase. This is why the imaginary part in the boxed result is not a technicality: a real wavefunction has constant phase, zero phase gradient, and hence zero current — bound stationary states carry no flux, which is precisely what “stationary” means. All flow in quantum mechanics is stored in how fast the phase winds through space, not in the shape of the amplitude.

This phase-gradient reading — the Madelung, or hydrodynamic, form of quantum mechanics — exposes the deep origin of the law. Probability conservation is Noether’s theorem applied to the global U(1) phase symmetry ψ → e^iαψ of the Schrödinger Lagrangian: a symmetry that costs nothing yields a conserved charge (total probability) and its current j. This is also why the assumption that decides everything is V real: only then is the Hamiltonian Hermitian, and Hermiticity is what guarantees unitary, norm-preserving evolution. Promote α to a local, position-dependent phase and consistency forces a gauge field — the electromagnetic potential — correcting the current to j = (ℏ/m) Im(ψ* ∇ψ) − (q/m) A|ψ|². Only this gauge-covariant combination is gauge invariant; the naive expression is not, which is the “breaks when” magnetic warning made precise.

The same skeleton recurs everywhere: charge conservation ∂ρ/∂t + ∇·J = 0 in electromagnetism, mass conservation in fluids, energy–momentum conservation via the stress tensor. Each is the local shadow of a global symmetry. What is special to the Schrödinger case is that ρ = |ψ|² is automatically non-negative, so it can be read as a probability. That positivity fails relativistically: the Klein–Gordon density is not positive definite, and rescuing a sensible interpretation is one of the pressures that forces the move from a single-particle wavefunction to quantum field theory in PU-402.

Common misconceptions. (i) A non-zero j does not mean the particle is definitely moving along a path — it is a probability flux, not a trajectory. (ii) Dropping the Im and writing j = (ℏ/m) ψ*∂ψ/∂x leaves a complex number, which cannot be a physical current. (iii) Probability conservation is not an extra postulate; it is a theorem whose only input is a real potential.

Worked examples

Example 1 — Incident and reflected flux of a stationary scattering state. To the left of a barrier the energy eigenstate is a superposition of a right-moving incident wave and a left-moving reflected wave, ψ = A e^ikx + B e^−ikx. Find the current and the reflection coefficient for a 5 eV electron with |A|² = 1, |B|² = 0.36.

Example 2 — Current of a moving Gaussian wave packet. A normalised electron packet is ψ(x) = (2πσ²)^−1/4 e^{−x²/(4σ²)} e^ik₀x with width σ = 1 nm and mean wavenumber that of a 5 eV electron, k₀ = 1.14×10¹⁰ m⁻¹. Find the probability current at the peak, x = 0.

Problems

1. Plane-wave current. For a free-particle plane wave ψ = A e^ikx show that j = ℏk|A|²/m = v|A|². Evaluate for an electron with k = 5×10⁹ m⁻¹ and |A|² = 1.
   Solution

   ∂ψ/∂x = ikψ, so ψ*∂ψ/∂x = ik|A|², whose imaginary part is k|A|². Hence j = (ℏ/m)k|A|² = v|A|² with v = ℏk/m. Numerically v = (1.055×10⁻³⁴ × 5×10⁹)/(9.109×10⁻³¹) = 5.79×10⁵ m s⁻¹, so j = 5.79×10⁵|A|² (in units of |A|² m s⁻¹). Density × velocity, as expected.

2. Standing wave vs. partial reflection. (a) Show that the standing wave ψ = A(e^ikx + e^−ikx) carries zero net current. (b) Now take ψ = A e^ikx + B e^−ikx with |A| = 1, |B| = 0.5; find the reflection coefficient and the net flux as a fraction of the incident flux.
   Solution

   (a) ψ = 2A cos kx is real (up to the constant phase of A), so Im(ψ*∂ψ/∂x) = 0 and j = 0: total reflection, equal counter-propagating fluxes cancel. (b) From the boxed result / Example 1, j = v(|A|² − |B|²) = v(1 − 0.25) = 0.75v. The reflection coefficient is R = |B|²/|A|² = 0.25; the net transmitted flux is 75% of the incident flux.

3. Three dimensions. Starting from the 3D Schrödinger equation with real V, derive ∂ρ/∂t + ∇·j = 0 with j = (ℏ/m) Im(ψ* ∇ψ).
   Solution

   ∂ρ/∂t = ψ*∂ψ/∂t + ψ∂ψ*/∂t. Insert iℏ∂ψ/∂t = −(ℏ²/2m)∇²ψ + Vψ and its conjugate. The V|ψ|² terms cancel (V real), leaving ∂ρ/∂t = (iℏ/2m)(ψ*∇²ψ − ψ∇²ψ*). Use the identity ψ*∇²ψ − ψ∇²ψ* = ∇·(ψ*∇ψ − ψ∇ψ*) (the ∇ψ*·∇ψ cross terms cancel). Thus ∂ρ/∂t = −∇·j with j = −(iℏ/2m)(ψ*∇ψ − ψ∇ψ*) = (ℏ/m) Im(ψ*∇ψ), using z − z* = 2i Im z.

4. Complex (optical) potential. Let V = V₀ − iΓ/2 with Γ > 0 constant. Show that d/dt∫|ψ|² dx = −(Γ/ℏ)∫|ψ|² dx, so the norm decays as e^−Γt/ℏ with lifetime τ = ℏ/Γ. Evaluate τ for Γ = 0.1 eV.
   Solution

   Repeating the density calculation but keeping V complex, the potential terms no longer cancel: ∂ρ/∂t = −∂j/∂x + (i/ℏ)(V* − V)|ψ|². Here V* − V = 2i Im(V)·(−1)... explicitly V* − V = (V₀ + iΓ/2) − (V₀ − iΓ/2) = iΓ, so (i/ℏ)(iΓ)|ψ|² = −(Γ/ℏ)|ψ|². Integrate over all space; the current’s divergence integrates to the vanishing surface term, giving d/dt∫|ψ|² dx = −(Γ/ℏ)∫|ψ|² dx. Hence N(t) = N(0) e^−Γt/ℏ and τ = ℏ/Γ = (1.055×10⁻³⁴)/(0.1×1.602×10⁻¹⁹) = 6.6×10⁻¹⁵ s. A negative imaginary potential removes probability — the standard model of absorption and decay.

5. Phase-gradient (Madelung) form and the magnetic current. (a) Writing ψ = √ρ e^iS/ℏ, show j = ρ ∇S/m, i.e. a velocity field v = ∇S/m. (b) With minimal coupling (p → p − qA) the conserved current becomes j = (ℏ/m) Im(ψ*∇ψ) − (q/m)A|ψ|². Show it is invariant under the gauge transformation A → A + ∇χ, ψ → e^iqχ/ℏψ, whereas the first term alone is not.
   Solution

   (a) ∇ψ = (∇√ρ + i√ρ ∇S/ℏ) e^iS/ℏ, so ψ*∇ψ = √ρ ∇√ρ + iρ ∇S/ℏ = ½∇ρ + iρ∇S/ℏ. The first term is real; Im(ψ*∇ψ) = ρ∇S/ℏ, giving j = (ℏ/m)(ρ∇S/ℏ) = ρ∇S/m. So v = ∇S/m: the local velocity is the phase gradient. (b) In this notation the gauge current is j = (ρ/m)(∇S − qA). Under the transformation S → S + qχ (the phase shift) and A → A + ∇χ, so ∇S − qA → ∇S + q∇χ − q(A + ∇χ) = ∇S − qA, unchanged, and ρ = |ψ|² is manifestly invariant. Thus j is gauge invariant. The bare term (ℏ/m)Im(ψ*∇ψ) = ρ∇S/m alone picks up the spurious piece ρq∇χ/m and is therefore unphysical in a magnetic field.