# Probability current density

The quantum mechanical probability current density (more precisely: residence probability current density ) is a current density that is associated with the quantum mechanical residence probability density within the framework of the quantum mechanical continuity equation . It is determined by the wave function in spatial space and, in the absence of magnetic fields, has the form (for more precise details see below): ${\ displaystyle \ psi ({\ vec {x}}, t)}$

${\ displaystyle {\ vec {j}} = {\ frac {i \ hbar} {2m}} [\ psi {\ vec {\ nabla}} \ psi ^ {*} - \ psi ^ {*} {\ vec {\ nabla}} \ psi].}$

## background

In physical field theories , conserved quantities appear as integrals over certain densities. Such densities, which belong to the conservation quantities, then suffice continuity equations , which are a special form of a balance equation .

In general, equations of continuity contain a density and a current and link them in shape ${\ displaystyle \ rho}$${\ displaystyle {\ vec {j}}}$

${\ displaystyle \ partial _ {t} \ rho + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0}$

or in an integral formulation using the Gaussian integral theorem :

${\ displaystyle \ partial _ {t} \ int _ {V} \ rho \, \ mathrm {d} V = - \ int _ {\ partial _ {V}} {\ vec {j}} \ cdot \ mathrm { d} {\ vec {A}}}$

The continuity equations gain clear meaning from the integral formulation, since the change in density over time within a volume element is equal to the flow beyond the boundaries of the volume element ( points out of the volume element). ${\ displaystyle {\ vec {j}}}$

Since only the divergence of the current density occurs in the continuity equation , a term proportional to the rotation of an arbitrary vector-valued (sufficiently smooth) function can always be added to this , since according to Schwarz's theorem it holds. ${\ displaystyle {\ vec {f}}}$ ${\ displaystyle {\ vec {\ nabla}} \ cdot ({\ vec {\ nabla}} \ times {\ vec {f}}) = 0}$

## Non-relativistic quantum mechanics

In quantum mechanics, as in statistical mechanics, the probability of presence is a conserved quantity. This probability, if you look at the entire space, is equal to one: the individual particle must be found somewhere in space. In quantum mechanics, the probability density is given by the square of the magnitude of the wave function : ${\ displaystyle \ psi}$

${\ displaystyle \ rho = \ psi ^ {*} \ psi}$

Since the wave function of quantum mechanics represents a complete description of the physical state of the system, it is initially unclear what the associated current density of the probability density could look like, since, unlike in continuum mechanics , no additional velocity field has been given a priori . Rather, the current density must be a function of the wave function.

### Probability current density without an external electromagnetic field

The probability density can be reformulated using the Schrödinger equation :

${\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho = \ psi ^ {*} {\ frac {\ partial} {\ partial t}} \ psi + \ psi {\ frac {\ partial} {\ partial t}} \ psi ^ {*} = {\ frac {1} {\ mathrm {i} \ hbar}} [\ psi ^ {*} {\ hat {\ mathcal {H}}} \ psi - \ psi {\ hat {\ mathcal {H}}} \ psi ^ {*}],}$

where is the Hamilton operator . If you use the explicit form of the Hamilton operator, you can see that the potential falls out of the equation. It remains a term that you can still put into the form ${\ displaystyle {\ hat {\ mathcal {H}}} = - {\ frac {\ hbar ^ {2}} {2m}} \ Delta + V ({\ vec {x}})}$${\ displaystyle V}$

${\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho = {\ frac {i \ hbar} {2m}} {\ vec {\ nabla}} \ cdot [\ psi ^ {*} {\ vec {\ nabla}} \ psi - \ psi {\ vec {\ nabla}} \ psi ^ {*}]}$

can bring. A comparison with the continuity equation gives the following form of the probability current density:

${\ displaystyle {\ vec {j}} = {\ frac {i \ hbar} {2m}} [\ psi {\ vec {\ nabla}} \ psi ^ {*} - \ psi ^ {*} {\ vec {\ nabla}} \ psi],}$

as described at the beginning of the article.

Alternative formulations:

${\ displaystyle {\ vec {j}} = {\ frac {\ hbar} {m}} \ operatorname {Im} \ left (\ psi ^ {*} {\ vec {\ nabla}} \ psi \ right) = {\ frac {1} {m}} \ operatorname {Re} \ left (\ psi ^ {*} {\ hat {\ vec {p}}} \ psi \ right),}$

where is the canonical momentum operator. ${\ displaystyle {\ hat {\ vec {p}}} = {\ frac {\ hbar} {i}} {\ vec {\ nabla}}}$

### Probability current density with an external electromagnetic field

The wave function in the external electromagnetic field obeys the Pauli equation . The following substitutions are made in the Schrödinger equation:

• For a correct description of the spin, the wave function of the scalar Schrödinger equation is replaced by the two-component spinor .${\ displaystyle \ psi}$${\ displaystyle \ Psi}$
• The momentum operator is replaced by , where is the vector potential and the electric charge of the particle. In contrast to the canonical , the term (without the “operator hat”) is also called kinetic momentum.${\ displaystyle {\ hat {\ vec {p}}} = - \ mathrm {i} \ hbar {\ vec {\ nabla}}}$${\ displaystyle {\ hat {\ vec {P}}} = - \ mathrm {i} \ hbar {\ vec {\ nabla}} - q {\ vec {A}}}$${\ displaystyle {\ vec {A}}}$${\ displaystyle q}$${\ displaystyle {\ vec {p}} - q {\ vec {A}}}$
• The Hamilton operator receives an additional term for the electrostatic energy with the electrical potential and becomes .${\ displaystyle {\ hat {\ mathcal {H}}} = \ mathrm {i} \ hbar \ partial _ {t}}$ ${\ displaystyle \ phi}$${\ displaystyle {\ hat {\ mathcal {H}}} = \ mathrm {i} \ hbar \ partial _ {t} -q \ phi}$

This results in a possible probability current density

${\ displaystyle {\ vec {j}} _ {\ text {naive}} = - {\ frac {\ mathrm {i} \ hbar} {2m}} \ left (\ Psi ^ {\ dagger} {\ vec { \ nabla}} \ Psi - ({\ vec {\ nabla}} \ Psi ^ {\ dagger}) \ Psi \ right) - {\ frac {q} {m}} \ Psi ^ {\ dagger} \ Psi { \ vec {A}} = {\ frac {1} {m}} \ operatorname {Re} \ left (\ Psi ^ {\ dagger} ({\ hat {\ vec {P}}}) \ Psi \ right) = {\ frac {1} {m}} \ operatorname {Re} \ left (\ Psi ^ {\ dagger} ({\ hat {\ vec {p}}} - q {\ vec {A}}) \ Psi \ right)}$.

This probability flux density is invariant under the gauge transformations due to the replacement of the canonical by the kinetic momentum

${\ displaystyle \ Psi \ to \ Psi \ exp \ left ({\ mathrm {i} q {\ frac {f ({\ vec {x}}, t)} {\ hbar}}} \ right), \ quad {\ vec {A}} \ to {\ vec {A}} + {\ vec {\ nabla}} f ({\ vec {x}}, t), \ quad \ phi \ to \ phi - \ partial _ {t} f}$

with any real function . ${\ displaystyle f ({\ vec {x}}, t)}$

It turns out, however, that a term can be added proportionally to this naive probability current density , which is also gauge invariant (and as a rotation term does not violate the continuity equation). In fact, the Dirac equation in the non-relativistic limit case with the Pauli matrices results : ${\ displaystyle {\ vec {\ nabla}} \ times (\ Psi ^ {\ dagger} {\ vec {\ sigma}} \ Psi)}$ ${\ displaystyle \ sigma}$

${\ displaystyle {\ vec {j}} = - {\ frac {\ mathrm {i} \ hbar} {2m}} \ left (\ Psi ^ {\ dagger} {\ vec {\ nabla}} \ Psi - ( {\ vec {\ nabla}} \ Psi ^ {\ dagger}) \ Psi \ right) - {\ frac {q} {m}} \ Psi ^ {\ dagger} \ Psi {\ vec {A}} + { \ frac {\ hbar} {2m}} {\ vec {\ nabla}} \ times (\ Psi ^ {\ dagger} {\ vec {\ sigma}} \ Psi) + {\ mathcal {O}} (v ^ {2} / c ^ {2})}$

## References and footnotes

1. ^ Marek Nowakowski: The Quantum Mechanical Current of the Pauli Equation. American Journal of Physics 67, 916 (1999). Article on arxiv.org