Probability current

Probability current
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics and electric currents in electromagnetism. As in those fields, the probability current (i.e. the probability current density) is related to the probability density function via a continuity equation. The probability current is invariant under gauge transformation.
The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in Brownian motion and the Fokker–Planck equation.
Definition (non-relativistic 3-current)
Free spin-0 particle
In non-relativistic quantum mechanics, the probability current of the wave function of a particle of mass in one dimension is defined as
where
is the reduced Planck constant;
denotes the complex conjugate of the wave function;
denotes the real part;
denotes the imaginary part.
Note that the probability current is proportional to a Wronskian
In three dimensions, this generalizes to
where denotes the del or gradient operator. This can be simplified in terms of the kinetic momentum operator,
to obtain
These definitions use the position basis (i.e. for a wavefunction in position space), but momentum space is possible.
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 and electric charge includes a term due to the interaction with the electromagnetic field;
where is the magnetic vector potential. The term 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 is the speed of light.
Spin-s particle in an electromagnetic field
If the particle has spin, it has a corresponding magnetic moment, so an extra term needs to be added incorporating the spin interaction with the electromagnetic field.
According to Landau-Lifschitz's Course of Theoretical Physics the electric current density is in Gaussian units:
And in SI units:
Hence the probability current (density) is in SI units:
where is the spin vector of the particle with corresponding spin magnetic moment and spin quantum number .
It is doubtful if this formula is vaild for particles with an interior structure. The neutron has zero charge but non-zero magnetic moment, so would be impossible (except would also be zero in this case). For composite particles with a non-zero charge – like the proton which has spin quantum number s=1/2 and µS= 2.7927·µN or the deuteron (H-2 nucleus) which has s=1 and µS=0.8574·µN – it is mathematically possible but doubtful.
Connection with classical mechanics
The wave function can also be written in the complex exponential (polar) form:
where are real functions of and .
Written this way, the probability density is and the probability current is:
The exponentials and terms cancel:
Finally, combining and cancelling the constants, and replacing with ,
If we take the familiar formula for the mass flux in hydrodynamics:
where is the mass density of the fluid and is its velocity (also the group velocity of the wave). In the classical limit, we can associate the velocity with which is the same as equating with the classical momentum . This interpretation fits with Hamilton–Jacobi theory, in which
in Cartesian coordinates is given by , where is Hamilton's principal function.
The de Broglie-Bohm theory equates the velocity with in general (not only in the classical limit) so it is always well defined. It is an interpretation of quantum mechanics.
Motivation
Continuity equation for quantum mechanics
The definition of probability current and Schrödinger's equation can be used to derive the continuity equation, which has exactly the same forms as those for hydrodynamics and electromagnetism:
where the probability density is defined as
If one were to integrate both sides of the continuity equation with respect to volume, so that
then the divergence theorem implies the continuity equation is equivalent to the integral equation
where is any volume and is the boundary of . This is the conservation law for probability in quantum mechanics.
In particular, if 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 when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume . Altogether the equation states that the time derivative of the probability of the particle being measured in is equal to the rate at which probability flows into .
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 and ; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy:
where and can be defined by:
where are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between and can be obtained from probability conservation:
In terms of a unit vector normal to the barrier, these are equivalently:
where the absolute values are required to prevent and being negative.
Examples
Plane wave
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 , confined to the region , the energy eigenstates are
and zero elsewhere. The associated probability currents are
since
Discrete definition
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:
References
Further reading
Quantum mechanics