Probability Current Density

Equivalent forms

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

\vec{j} = \frac{\hbar}{m}\,\mathrm{Im}(\psi^{*}\nabla\psi)

Probability obeys the same continuity equation as charge and mass — nothing is created or destroyed, only transported.

Symbol	Name	SI unit	Dimension	Range
$j$	Probability currentoutput Flow of probability per unit time	1/s	T^-1	-5 – 5
$k$	Wavenumber Spatial frequency of the plane-wave component	1/m	L^-1	-5 – 5
$rho$	Probability density Local probability density \|psi\|^2	1/m	L^-1	0 – 5

Unit systems

Symbol	SI	CGS	Imperial
$j$	1/s	1/s	1/s
$k$	1/m	1/cm	1/ft
$rho$	1/m	1/cm	1/ft

Where it holds

Valid for any single-particle Schrodinger wavefunction; gauge-covariant form adds the vector potential for charged particles in a field.

Dimensional analysis

j in 1D has units 1/s

(hbar/m) Im(psi* dpsi/

dx) = (J s/kg)(m^-1/2)(m^-3/2)

(J s/kg)(m^-2) = (kg m^2\,\mathrm{s}^-1/kg)(m^-2) = s^-1

.

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Discovery

Erwin Schrodinger · 1926

Deriving his equation in 1926, Schrodinger noted that |psi|^2 obeys a conservation law of the same form as the continuity equation in electromagnetism, with a current j that carries probability from place to place.

Try this

If probability can move, where does it flow?

A wavepacket drifts, a tunneling electron leaks through a barrier — probability itself is in motion. The probability current is the flow that, together with the density, obeys a continuity equation just like charge or mass conservation.

Research status: stable

Real-world applications

Defining transmission coefficients for tunneling diodes and STM
Persistent currents in superconducting and mesoscopic rings
Flux conservation checks in time-dependent quantum simulations

Common misconceptions

A real-valued wavefunction carries zero current, even if |psi|^2 is nonzero
Current is not the classical velocity times density unless the phase is a simple plane wave
j can be nonzero even where the particle spends little time — it measures flow, not occupancy

Experimental verification

Transmission and reflection coefficients in tunneling experiments are ratios of probability currents, verified in scanning tunneling microscopy and resonant-tunneling diodes. Persistent currents in mesoscopic rings are direct measurements of a non-zero quantum j.

Derivation

Start from

\rho = |\psi |^2

and differentiate:

d \rho /dt = \psi * d \psi /dt + \psi d \psi */dt

.

Use the Schrodinger equation

d \psi /dt = (i

hbar/2m) d^2 psi/dx^2 - (i/hbar)V psi and its conjugate.

The potential terms cancel (V real) and what remains is

d \rho /dt =

-(hbar/

2mi) d/dx (\psi * d \psi /dx - \psi d \psi */dx) = -d j/dx

, with

j =

(hbar/m) Im(psi* d psi/dx).

This is the continuity equation

d \rho /dt + d j/dx = 0

— local conservation of probability.

Limiting cases

plane wave e^{ikx}⟶

j =

(hbar k/m)|psi|^2Current equals density times group velocity hbar k/m.

real wavefunction⟶

j = 0

A purely real psi carries no net probability flow — standing state.

bound stationary state⟶

j = 0

everywhereNo net transport; density is time-independent.

What if…

What if the particle is charged in a magnetic field?

Replace the gradient with the covariant derivative (nabla - iqA/hbar); the current gains a term from the vector potential.

What if probability were not conserved?

The continuity equation would fail and normalization would drift in time — but Hermiticity of H guarantees it holds.

1

Current of a plane wave

Given ·

psi:: A e^{ikx}

Find · j

Steps

$d \psi /dx = ik$ A e^{ikx}
$\psi * d \psi /dx = ik$ |A|^2
Im gives k|A|^2, so $j =$ hbar k|A|^2/m

Result ·

j =

(hbar/m) Im(A* e^{-ikx} * ik A

e^{ikx}) =

(hbar k/m)|A|^2 — density times velocity.

2

Why a bound state has no current

Given ·

psi:: real

Find · j

Steps

$\psi = \psi$ * (real)
psi* dpsi/dx is real
Im(real $) = 0$ so $j = 0$

Result · For real psi, psi* d psi/dx is real, so its imaginary part — and hence j — is zero.

Born Rule→Wavefunction Normalization→Quantum Tunneling Probability→Time-Dependent Schrödinger Equation→