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):
![\ psi ({\ vec {x}}, t)](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a93597f6de9f63c844b93fa513ff5cd1b372694)
![{\ displaystyle {\ vec {j}} = {\ frac {i \ hbar} {2m}} [\ psi {\ vec {\ nabla}} \ psi ^ {*} - \ psi ^ {*} {\ vec {\ nabla}} \ psi].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dba37e4e271853e8fce61f07ebe1b952206260ed)
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
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![{\ displaystyle {\ vec {j}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1)
![{\ displaystyle \ partial _ {t} \ rho + {\ vec {\ nabla}} \ cdot {\ vec {j}} = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6fec8ed503448bedeef0cd2bf3f77a4a623e19)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c69da1065a1208482e96f1c2170a2c88d16c18)
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).
![{\ vec j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce1ed1de8493f7cc7d856ca5427cf311b1597f1)
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 {\ nabla}} \ cdot ({\ vec {\ nabla}} \ times {\ vec {f}}) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd208b0b79d12f7124a4d2c44dfb6ed1afba8c58)
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 :
![\ psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![{\ displaystyle \ rho = \ psi ^ {*} \ psi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/611559323025bd6c43e66beba6054b6e093d6864)
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 ^ {*}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87ac5c173e1c238dfe22b508c014491d1a3b16d4)
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}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9e8a283e7d0ee7df1e0e34a4a6bf07df04b950)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho = {\ frac {i \ hbar} {2m}} {\ vec {\ nabla}} \ cdot [\ psi ^ {*} {\ vec {\ nabla}} \ psi - \ psi {\ vec {\ nabla}} \ psi ^ {*}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1819b71510ca06c003ff89a6c0220496aef06950)
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],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2830e412c5b231c7ec1ce764692b39ca95f19b8a)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d989881cbedcbe8d49039d35d5325b1a2d99e1fb)
where is the canonical momentum operator.
![{\ displaystyle {\ hat {\ vec {p}}} = {\ frac {\ hbar} {i}} {\ vec {\ nabla}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb866abe3e3aa0100d9e22c4d139d0f80dccf78a)
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 .
![\ psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a)
![\ Psi](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5471531a3fe80741a839bc98d49fae862a6439a)
- 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b593ea577bddb8faccaf1344a29528d4f89cca9)
![{\ displaystyle {\ hat {\ vec {P}}} = - \ mathrm {i} \ hbar {\ vec {\ nabla}} - q {\ vec {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/827c8a454720d10dfcb7ffd59b803b5a78628939)
![{\ vec A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95)
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![{\ displaystyle {\ vec {p}} - q {\ vec {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/551f93a67f293c0c872ea079654d6447efdea5eb)
- The Hamilton operator receives an additional term for the electrostatic energy with the electrical potential and becomes .
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![{\ displaystyle {\ hat {\ mathcal {H}}} = \ mathrm {i} \ hbar \ partial _ {t} -q \ phi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/517e4a1bf18efb84ef6a87c9a37c1a6d6e87e7a6)
This results in a possible probability current density
-
.
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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66b85f61d3aa28f4f8f8ead63e9d8e18e271f241)
with any real function .
![{\ displaystyle f ({\ vec {x}}, t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4aca1de51a9bf7a133343ca72c6db43de2cbb671)
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 :
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
![{\ 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58ef06f56f6d24c7ccb251149f3b42eb003696c5)
References and footnotes
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^ Marek Nowakowski: The Quantum Mechanical Current of the Pauli Equation. American Journal of Physics 67, 916 (1999). Article on arxiv.org
Web links
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Elektronium model Description of atoms with the help of the probability current density