In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics. The probability current is invariant under gauge transformation.
Definition (non-relativistic 3-current)
Free spin-0 particle
In three dimensions, this generalizes to
This can be simplified in terms of the kinetic momentum operator,
Spin-0 particle in an electromagnetic field
The above definition should be modified for a system in an external electromagnetic field. In SI units, a charged particle of mass m and electric charge q includes a term due to the interaction with the electromagnetic field;
where A = A(r, t) is the magnetic potential (aka "A-field"). The term qA has dimensions of momentum. Note that used here is the canonical momentum and is not gauge invariant, unlike the kinetic momentum operator .
In Gaussian units:
where c is the speed of light.
Spin-s particle in an electromagnetic field
Connection with classical mechanics
where R and S are real functions of r and t.
Written this way, the probability density is
and the probability current is:
The exponentials and R∇R terms cancel:
Finally, combining and cancelling the constants, and replacing R2 with ρ,
If we take the familiar formula for the current:
where v is the velocity of the particle (also the group velocity of the wave), we can associate the velocity with ∇S/m, which is the same as equating ∇S with the classical momentum p = mv. This interpretation fits with Hamilton–Jacobi theory, in which
in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function.
Continuity equation for quantum mechanics
where the probability density is defined as
If one were to integrate both sides of the continuity equation with respect to volume, so that
where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.
In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation, sans time derivative, is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V. Altogether the equation states that the time derivative of the probability of the particle being measured in V is equal to the rate at which probability flows into V.
Transmission and reflection through potentials
In regions where a step potential or potential barrier occurs, the probability current is related to the transmission and reflection coefficients, respectively T and R; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy:
where T and R can be defined by:
where jinc, jref and jtrans are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between T and R can be obtained from probability conservation:
where the absolute values are required to prevent T and R being negative.
For a plane wave propagating in space:
the probability density is constant everywhere;
(that is, plane waves are stationary states) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed;
illustrating that the particle may be in motion even if its spatial probability density has no explicit time dependence.
Particle in a box
For a particle in a box, in one spatial dimension and of length L, confined to the region;
the energy eigenstates are
and zero elsewhere. The associated probability currents are
For a particle in one dimension on , we have the Hamiltonian where is the discrete Laplacian, with being the right shift operator on . Then the probability current is defined as , with the velocity operator, equal to and is the position operator on . Since is usually a multiplication operator on , we get to safely write .
As a result, we find:
- Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
- Quantum mechanics, Ballentine, Leslie E, Vol. 280, Englewood Cliffs: Prentice Hall, 1990.
- Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Outlines Crash Course, Mc Graw Hill (USA), 2006, ISBN 978-0-07-145533-6
- Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
- Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0