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{{short description|Value for the flow of probability in quantum mechanics}}
{{quantum mechanics}}

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 number|real]] [[Euclidean vector|vector]] that changes with space and time. Probability currents are analogous to [[Mass flow (physics)|mass currents]] in [[hydrodynamic]]s 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 (mathematics)|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]].<ref>{{cite book |first=Wolfgang |last=Paul |first2=Jörg |last2=Baschnagel |title=Stochastic Processes : From Physics to Finance |location=Berlin |publisher=Springer |year=1999 |isbn=3-540-66560-9 |page=84 }}</ref>

The relativistic equivalent of the probability current is known as the [[probability four-current]].

== Definition (non-relativistic 3-current) ==

=== Free spin-0 particle ===

In non-relativistic quantum mechanics, the probability current {{math|'''j'''}} of the [[wave function]] {{math|Ψ}} of a particle of mass {{mvar|m}} in one dimension is defined as<ref>{{cite book |title=Quantum Field Theory |first=D. |last=McMahon |publisher=McGraw Hill |year=2008 |isbn=978-0-07-154382-8 }}</ref>
<math display="block"> j = \frac{\hbar}{2mi} \left(\Psi^* \frac{\partial \Psi }{\partial x}- \Psi \frac{\partial \Psi^*}{\partial x} \right) = \frac{\hbar}{m} \Re\left\{\Psi^* \frac{1}{i} \frac{\partial \Psi }{\partial x} \right\} = \frac{\hbar}{m} \Im\left\{\Psi^* \frac{\partial \Psi }{\partial x} \right\},</math>
where
*<math>\hbar</math> is the reduced [[Planck constant]];
*<math>\Psi^*</math> denotes the [[complex conjugate]] of the [[wave function]];
*<math>\Re</math> denotes the [[real part]];
*<math>\Im</math> denotes the [[imaginary part]]. 
Note that the probability current is proportional to a [[Wronskian]] <math>W(\Psi,\Psi^*).</math>

In three dimensions, this generalizes to
<math display="block">\mathbf j = \frac{\hbar}{2mi} \left( \Psi^* \mathbf \nabla \Psi - \Psi \mathbf \nabla \Psi^{*} \right) = \frac{\hbar}{m} \Re\left\{\Psi^* \frac{\nabla}{i} \Psi \right\} = \frac{\hbar}{m}\Im\left\{\Psi^* \nabla \Psi \right\} \,, </math>
where <math>\nabla</math> denotes the [[del]] or [[gradient]] [[operator (mathematics)|operator]]. This can be simplified in terms of the [[momentum operator|kinetic momentum operator]],
<math display="block">\mathbf{\hat{p}} = -i\hbar\nabla</math>
to obtain
<math display="block">\mathbf j = \frac{1}{2m} \left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right)\,. </math>

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 ===
{{Main|electromagnetic field|kinetic momentum}}

The above definition should be modified for a system in an external [[electromagnetic field]]. In [[SI units]], a [[charged particle]] of mass {{mvar|m}} and [[electric charge]] {{mvar|q}} includes a term due to the interaction with the electromagnetic field;<ref>{{cite book |title=Quantum Mechanics |last=Ballentine |first=Leslie E. |volume=280 |series=Prentice Hall Advanced Reference Series |location=Englewood Cliffs |publisher=Prentice Hall |year=1990 |isbn=0-13-747932-8 }}</ref>
<math display="block">\mathbf j = \frac{1}{2m}\left[\left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right) - 2q\mathbf{A} |\Psi|^2 \right]</math>
where {{math|1='''A''' = '''A'''('''r''', ''t'')}} is the [[magnetic vector potential]]. The term {{math|''q'''''A'''}} has dimensions of momentum. Note that <math>\mathbf{\hat{p}} = -i\hbar\nabla</math> used here is the [[canonical momentum]] and is not [[Gauge invariance|gauge invariant]], unlike the [[momentum operator|kinetic momentum operator]] <math>\mathbf{\hat{P}} = -i\hbar\nabla-q\mathbf{A}</math>.

In [[Gaussian units]]:
<math display="block">\mathbf j = \frac{1}{2m}\left[\left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right) - 2\frac{q}{c} \mathbf{A} |\Psi|^2 \right]</math>
where {{mvar|c}} is the [[speed of light]].

=== Spin-''s'' particle in an electromagnetic field ===

If the particle has [[Spin (physics)|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:<ref>see page 473, equation 115.4, {{cite web |author=L.D. Landau, E.M. Lifschitz |url=https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%203%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20mechanics%20-%20non-relativistic%20theory%20%283ed.%2C%201991%29.pdf |title=COURSE OF THEORETICAL PHYSICS Vol. 3 – Quantum Mechanics | edition = 3rd | website = ia803206.us.archive.org |access-date=29 April 2023}}</ref>
<math display="block">\mathbf{j}_e = \frac{q}{2m} \left[\left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right) - \frac{2q}{c} \mathbf{A} |\Psi|^2 \right] + \frac{\mu_S c}{s\hbar}\nabla\times(\Psi^* \mathbf{S}\Psi) </math>

And in SI units: <math display="block">\mathbf j_e = \frac{q}{2m}\left[\left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right) - 2q\mathbf{A} |\Psi|^2 \right] + \frac{\mu_S}{s\hbar}\nabla\times(\Psi^* \mathbf{S}\Psi) </math>

Hence the probability current (density) is in SI units: <math display="block">\mathbf{j} = \mathbf{j}_e/q = \frac{1}{2m}\left[\left(\Psi^* \mathbf{\hat{p}} \Psi - \Psi \mathbf{\hat{p}} \Psi^*\right) - 2q\mathbf{A} |\Psi|^2 \right] + \frac{\mu_S}{q s\hbar}\nabla\times(\Psi^* \mathbf{S}\Psi) </math>

where {{math|'''S'''}} is the [[Spin (physics)|spin]] vector of the particle with corresponding [[spin magnetic moment]] {{math|μ<sub>''S''</sub>}} and [[spin quantum number]] {{mvar|s}}.

It is doubtful if this formula is valid for particles with an interior structure.{{citation needed|date=May 2023}} The [[neutron]] has zero charge but non-zero magnetic moment, so <math>\frac{\mu_S}{q s\hbar}</math> would be impossible (except <math>\nabla \times (\Psi^* \mathbf{S}\Psi) </math> 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 μ<sub>S</sub>= 2.7927·[[nuclear magneton|μ<sub>N</sub>]] or the [[Deuterium#Nuclear properties (the deuteron)|deuteron]] (H-2 nucleus) which has s=1 and μ<sub>S</sub>=0.8574·μ<sub>N</sub> <ref>{{cite web |url=https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/spectrpy/nmr/nmr2.htm |title=Spin Properties of Nuclei |access-date=29 April 2023 |website=www2.chemistry.msu.edu}}</ref> – it is mathematically possible but doubtful.

== Connection with classical mechanics ==

The wave function can also be written in the [[complex number|complex]] [[Exponential function|exponential]] ([[polar coordinates#Complex numbers|polar]]) form:
<math display="block">\Psi = R e^{i S / \hbar }</math>
where {{mvar|R, S}} are real functions of {{math|'''r'''}} and {{math|t}}.

Written this way, the probability density is <math display="block"> \rho = \Psi^* \Psi = R^2</math> and the probability current is:
<math display="block">\begin{align}
  \mathbf{j} & = \frac{\hbar}{2mi}\left(\Psi^{*} \mathbf{\nabla} \Psi - \Psi \mathbf{\nabla}\Psi^{*} \right) \\[5pt]
    & = \frac{\hbar}{2mi}\left(R e^{-i S / \hbar} \mathbf{\nabla}R e^{i S / \hbar} - R e^{i S / \hbar} \mathbf{\nabla}R e^{-i S / \hbar}\right) \\[5pt]
    & = \frac{\hbar}{2mi}\left[ R e^{-i S / \hbar} \left( e^{i S / \hbar} \mathbf{\nabla}R + \frac{i}{\hbar}R e^{i S / \hbar} \mathbf{\nabla}S \right) - R e^{i S / \hbar} \left( e^{-i S / \hbar} \mathbf{\nabla}R - \frac{i}{\hbar} R e^{-i S / \hbar} \mathbf{\nabla} S \right)\right].
\end{align}</math>

The exponentials and {{math|''R''∇''R''}} terms cancel:
<math display="block"> \mathbf{j} = \frac{\hbar}{2mi}\left[\frac {i}{\hbar} R^2 \mathbf{\nabla} S + \frac {i}{\hbar} R^2 \mathbf{\nabla} S \right]. </math>

Finally, combining and cancelling the constants, and replacing {{math|''R''<sup>2</sup>}} with {{mvar|ρ}},
<math display="block">\mathbf{j} = \rho \frac{\mathbf{\nabla} S}{m}. </math>Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. If we take the familiar formula for the mass flux in hydrodynamics:
<math display="block">\mathbf{j} = \rho \mathbf{v},</math>

where <math>\rho</math> is the mass density of the fluid and {{math|'''v'''}} is its velocity (also the [[group velocity]] of the wave). In the classical limit, we can associate the velocity with <math>\tfrac{\nabla S}{m},</math> which is the same as equating {{math|∇''S''}} with the classical momentum {{math|1='''p''' = ''m'''''v'''}} however, it does not represent a physical velocity or momentum at a point since simultaneous measurement of position and velocity violates [[uncertainty principle]]. This interpretation fits with [[Hamilton–Jacobi theory]], in which
<math display="block"> \mathbf{p} = \nabla S </math>
in Cartesian coordinates is given by {{math|∇''S''}}, where {{mvar|S}} is [[Hamilton's principal function]].

The de Broglie-Bohm theory equates the velocity with <math>\tfrac{\nabla S}{m}</math> 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 ===

{{Main|continuity equation}}

The definition of probability current and Schrödinger's equation can be used to derive the [[continuity equation#Quantum mechanics|continuity equation]], which has ''exactly'' the same forms as those for [[hydrodynamics]] and [[electromagnetism]].<ref>Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, {{ISBN|978-0-13-146100-0}}</ref>

For some [[wave function]] {{math|Ψ}}, let:

<math display="block">\rho(\mathbf{r},t) = |\Psi|^2 = \Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t) .</math>be the [[Probability density function|probability density]] (probability per unit volume, {{math|*}} denotes [[complex conjugate]]). Then,

<math>\begin{align}
\frac{d}{dt}\int_\mathcal{V}dV\, \rho &= \int_\mathcal{V}dV\, \left(\frac {\partial \psi} {\partial t}\psi^*+ \psi\frac {\partial \psi^*} {\partial t}\right) \\ 
&=\int_\mathcal{V}dV\,\left(-\frac i\hbar\left( -\frac { \hbar^2 }{2m}\nabla^2 \psi + V \psi\right)\psi^*+\frac i\hbar\left( -\frac { \hbar^2 }{2m}\nabla^2 \psi^* +  V \psi^*\right)\psi\right)    \\
&=\int_\mathcal{V}dV\,\frac{i\hbar}{2m} [(\nabla^2\psi) \psi^* - \psi (\nabla^2 \psi^*)]\\ 

&=\int_\mathcal{V}dV\, \nabla\cdot\left(\frac{i\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*)\right)\\
&=\int_\mathcal{S}d\mathbf{a}\cdot\left(\frac{i\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*)\right)\\
\end{align}  </math>

where {{mvar|V}} is any volume and {{mvar|S}} is the boundary of {{mvar|V}}. 

This is the [[Conservation law (physics)|conservation law]] for probability in quantum mechanics. The integral form is stated as:

<math display="block">\int_V \left( \frac{\partial |\Psi|^2}{\partial t} \right) \mathrm{d}V + \int_V \left( \mathbf \nabla \cdot \mathbf j \right) \mathrm{d}V = 0</math>where<math display="block"> \mathbf{j} = \frac{1}{2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right) = -\frac{i\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*) = \frac \hbar m \operatorname{Im} (\psi^*\nabla \psi)     </math>is the [[probability current]] or probability flux (flow per unit area).

Here, equating the terms inside the integral gives the [[Continuity equation#Quantum mechanics|continuity equation]] for probability:<math display="block">\frac{\partial}{\partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, </math>and the integral equation can also be restated using the [[divergence theorem]] as:


{{oiint
| preintegral = <math>\frac{\partial}{\partial t} \int_V |\Psi|^2 \mathrm{d}V +</math>
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf j \cdot \mathrm{d}\mathbf{S} = 0</math>
}}.


In particular, if {{mvar|Ψ}} 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 {{mvar|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 {{mvar|V}}. Altogether the equation states that the time derivative of the probability of the particle being measured in {{mvar|V}} is equal to the rate at which probability flows into {{mvar|V}}.

By taking the limit of volume integral to include all regions of space, a well-behaved wavefunction that goes to zero at infinities in the surface integral term implies that the time derivative of total probability is zero ie. the normalization condition is conserved.<ref name=":1">{{Cite book |last=Sakurai |first=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref> This result is in agreement with the unitary nature of time evolution operators which preserve length of the vector by definition.

=== Transmission and reflection through potentials ===

{{hatnote|Main articles: [[Transmission coefficient#Quantum mechanics|Transmission coefficient]] and [[Reflection coefficient]].}}

In regions where a [[step potential]] or [[potential barrier]] occurs, the probability current is related to the transmission and reflection coefficients, respectively {{mvar|T}} and {{mvar|R}}; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy:
<math display="block">T + R = 1\,,</math>
where {{mvar|T}} and {{mvar|R}} can be defined by:
<math display="block"> T=  \frac{|\mathbf{j}_\mathrm{trans}|}{|\mathbf{j}_\mathrm{inc}|} \, , \quad  R = \frac{|\mathbf{j}_\mathrm{ref}|}{|\mathbf{j}_\mathrm{inc}|} \, , </math>
where {{Math|''j''<sub>inc</sub>, ''j''<sub>ref</sub>, ''j''<sub>trans</sub>}} are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the [[Norm (mathematics)|magnitudes]] of the current vectors. The relation between {{mvar|T}} and {{mvar|R}} can be obtained from probability conservation:
<math display="block">\mathbf{j}_\mathrm{trans} + \mathbf{j}_\mathrm{ref}=\mathbf{j}_\mathrm{inc}\,.</math>

In terms of a [[unit vector]] {{math|'''n'''}} [[Surface normal|normal]] to the barrier, these are equivalently:
<math display="block"> T= \left|\frac{\mathbf{j}_\mathrm{trans}\cdot\mathbf{n}}{\mathbf{j}_\mathrm{inc}\cdot\mathbf{n}}\right|\,, \qquad R= \left|\frac{\mathbf{j}_\mathrm{ref}\cdot\mathbf{n}}{\mathbf{j}_\mathrm{inc}\cdot\mathbf{n}} \right| \,, </math>
where the absolute values are required to prevent {{mvar|T}} and {{mvar|R}} being negative.

== Examples ==

=== Plane wave ===
{{Main|plane wave}}

For a [[plane wave]] propagating in space:
<math display="block"> \Psi(\mathbf{r},t) = \, A e^{ i (\mathbf{k}\cdot{\mathbf{r}} - \omega t)} </math>
the probability density is constant everywhere;
<math display="block"> \rho(\mathbf{r},t) = |A|^2 \rightarrow \frac{\partial |\Psi|^2}{\partial t} = 0 </math>
(that is, plane waves are [[stationary state]]s) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed;
<math display="block"> \mathbf{j}\left(\mathbf{r},t\right) = \left|A\right|^2 {\hbar \mathbf{k} \over m} = \rho \frac{\mathbf{p}}{m} = \rho \mathbf{v}</math>

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 {{mvar|L}}, confined to the region <math>0 < x < L</math>, the energy eigenstates are
<math display="block">\Psi_n = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi}{L} x \right)</math>
and zero elsewhere. The associated probability currents are
<math display="block">j_n = \frac{i\hbar}{2m}\left( \Psi_n^* \frac{\partial \Psi_n}{\partial x} - \Psi_n \frac{\partial \Psi_n^*}{\partial x} \right) = 0</math>
since <math display="block">\Psi_n = \Psi_n^*</math>

== Discrete definition ==
For a particle in one dimension on <math>\ell^2(\Z),</math> we have the Hamiltonian <math>H = -\Delta + V</math> where <math>-\Delta \equiv 2 I - S - S^\ast</math> is the discrete Laplacian, with {{mvar|S}} being the right shift operator on <math>\ell^2(\Z).</math> Then the probability current is defined as <math>j \equiv 2 \Im\left\{\bar{\Psi} i v \Psi\right\},</math> with {{mvar|v}} the velocity operator, equal to <math>v \equiv -i[X,\, H]</math> and {{mvar|X}} is the position operator on <math>\ell^2\left(\mathbb{Z}\right).</math> Since {{mvar|V}} is usually a multiplication operator on <math>\ell^2(\Z),</math> we get to safely write <math display="block">-i[X,\, H] = -i[X,\, -\Delta] = -i\left[X,\, -S - S^{\ast}\right] = i S - i S^{\ast}.</math>

As a result, we find: 
<math display="block">\begin{align}
j\left(x\right) \equiv 2 \Im\left\{\bar{\Psi}(x) i v \Psi(x)\right\}
    &= 2 \Im\left\{\bar{\Psi}(x) \left((-S\Psi)(x) + \left(S^\ast \Psi\right)(x)\right)\right\}\\
    &= 2 \Im\left\{\bar{\Psi}(x) \left(-\Psi(x-1) + \Psi(x + 1)\right)\right\}
\end{align}</math>

== References ==
{{reflist}}

== Further reading ==
*{{cite book |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles |edition=2nd |first=R. |last=Resnick |first2=R. |last2=Eisberg |publisher=John Wiley & Sons |year=1985 |isbn=0-471-87373-X }}

[[Category:Quantum mechanics]]