Probability Current

This is a lookup projection. Probability Current owns the physical interpretation, Hamiltonian dependence, gauge analysis, examples, limitations, and exercises. Continuity Equation owns the full local and integral derivation.

Helpful background. Probability Current supplies the Hamiltonian-specific definition when the model is not the constant-mass scalar Schrödinger operator assumed below.

Scalar Schrödinger formula

For one spinless particle with $H=-\hbar^2\nabla^2/(2m)+V$ and real local $V$,

$$\rho=|\psi|^2, \qquad \mathbf j = \frac{\hbar}{2mi} (\psi^*\nabla\psi-\psi\nabla\psi^*) = \frac{\hbar}{m}\operatorname{Im}(\psi^*\nabla\psi).$$

In one dimension,

$$j=\frac{\hbar}{m}\operatorname{Im}(\psi^*\partial_x\psi).$$

If $\psi=Re^{iS/\hbar}$ away from nodes,

$$\mathbf j=\rho\frac{\nabla S}{m}.$$

Electromagnetic formula

For charge $q$ and vector potential $\mathbf A$,

$$\mathbf j = \frac1m\operatorname{Re} \left[ \psi^*(-i\hbar\nabla-q\mathbf A)\psi \right] = \frac{\rho}{m}(\nabla S-q\mathbf A).$$

The kinetic derivative of $\psi$ is gauge covariant; the bilinear current is gauge invariant under

$$\mathbf A' = \mathbf A+\nabla\chi, \qquad \Phi'=\Phi-\partial_t\chi, \qquad \psi'=e^{iq\chi/\hbar}\psi.$$

Plane waves and scattering

For $\psi=Ae^{ikx}$,

$$j=\frac{\hbar k}{m}|A|^2.$$

For incident, reflected, and transmitted signed currents,

$$j_{\mathrm{inc}}+j_{\mathrm{ref}}=j_{\mathrm{trans}}, \qquad j_{\mathrm{ref}}<0.$$

The reflection and transmission probabilities are

$$R=\frac{|j_{\mathrm{ref}}|}{j_{\mathrm{inc}}}, \qquad T=\frac{j_{\mathrm{trans}}}{j_{\mathrm{inc}}}.$$

For incident amplitude one and transmitted amplitude $t$,

$$T=\frac{v_R}{v_L}|t|^2, \qquad v=\frac1\hbar\frac{dE}{dk}.$$

An asymptotically evanescent channel carries no transmitted flux.

Regional flux

With outward-oriented $d\mathbf S$,

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

Positive outward flux decreases the probability stored in $\Omega$.

Symbols and units

Symbol	Meaning
$\rho$	probability density
$\mathbf j$	probability current density
$m$	particle mass in the scalar formula
$q$	particle charge
$S$	wavefunction phase action, undefined at nodes
$v_L,v_R$	positive asymptotic channel group velocities

In $d$ spatial dimensions,

$$[\rho]=L^{-d}, \qquad [\mathbf j]=L^{1-d}T^{-1}.$$

Assumptions and limits

• Exact plane waves are generalized states; state box, delta, energy, or flux normalization before interpreting $A$.
• The displayed current changes for position-dependent mass, nonlocal kernels, spin–orbit or Pauli terms, lattice Hamiltonians, and reduced many-particle densities.
• Continuity determines the divergence of a current, not a unique current in dimensions where divergence-free additions are possible.
• $R+T=1$ assumes conservative stationary single-channel scattering with all open channels included.
• In the stated spinless model, charge current is $q\mathbf j$; spinful models may add magnetization terms.

Calculation checks

• Reversing $k$ reverses $j$.
• A real stationary scalar wavefunction gives $j=0$.
• Unequal channel velocities appear in $T$.
• The electromagnetic current is unchanged by a gauge transformation.
• Boundary flux agrees with the time derivative of regional probability.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.