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

→ Main article : Continuity equation

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

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

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

Web links

• Elektronium model Description of atoms with the help of the probability current density