# Probability current

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In quantum mechanics, the **probability current** (sometimes called **probability flux**) is a mathematical quantity describing the flow of probability (i.e. probability per unit time per unit area). Intuitively, if one pictures the probability density as an inhomogeneous 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 notion of a probability current is useful in some of the formalism in quantum mechanics.

## Definition (non-relativistic 3-current)

### Free spin-0 particle

In non-relativistic quantum mechanics, the probability current **j** of the wave function *Ψ* in one dimension is defined as ^{[1]}

in three dimensions, this generalizes to

where *ħ* is the reduced Planck constant, *m* is the particle's mass, *Ψ* is the wavefunction, and ∇ denotes the del or gradient operator.

This can be simplified in terms of the momentum operator,

to obtain

These definitions use the position basis (i.e. for a wavefunction in position space, momentum space is possible).

### Spin-0 particle in an electromagnetic field

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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;^{[2]}

where **A** = **A**(**r**, t) is the magnetic potential (aka "**A**-field"). The term *q***A** has dimensions of momentum.

In Gaussian units:

where *c* 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. In SI units:^{[3]}

where **S** is the spin vector of the particle with corresponding spin magnetic moment μ_{S} and spin quantum number *s*. In Gaussian units:

## Connection with classical mechanics

The wave function can also be written in the complex exponential (polar) form:^{[4]}

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 *R*^{2} 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** = *m***v**. This interpretation fits with Hamilton–Jacobi theory, in which

in Cartesian coordinates is given by ∇*S*, where *S* is Hamilton's principal function.

## Motivation

### Continuity equation for quantum mechanics

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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:^{[5]}

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 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 (without the 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 chance 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

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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 *j*_{inc}, *j*_{ref} and *j*_{trans} 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* is consistent with probability conservation:

In terms of a unit vector **n** normal to the barrier, these are equivalently:

where the absolute values are required to prevent *T* and *R* being negative.

## Examples

### Plane wave

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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

since

and we have used the identity

## References

- ↑ 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 Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6
- ↑
*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

- Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0