Probability current

 ( 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 Euclidean vector that changes with space and time. Probability currents are analogous to mass currents in 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 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.

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

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Definition (non-relativistic 3-current)

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Free spin-0 particle In non-relativistic quantum mechanics, the probability current of the wave function of a particle of mass in one dimension is defined as

D. McMahon (2025). 9780071543828, McGraw Hill. ISBN 9780071543828

$$j\; =\; \backslash frac\{\backslash hbar\}\{2mi\}\; \backslash left(\backslash Psi^*\; \backslash frac\{\backslash partial\; \backslash Psi\; \}\{\backslash partial\; x\}-\; \backslash Psi\; \backslash frac\{\backslash partial\; \backslash Psi^*\}\{\backslash partial\; x\}\; \backslash right)\; =\; \backslash frac\{\backslash hbar\}\{m\}\; \backslash Re\backslash left\backslash \{\backslash Psi^*\; \backslash frac\{1\}\{i\}\; \backslash frac\{\backslash partial\; \backslash Psi\; \}\{\backslash partial\; x\}\; \backslash right\backslash \}\; =\; \backslash frac\{\backslash hbar\}\{m\}\; \backslash Im\backslash left\backslash \{\backslash Psi^*\; \backslash frac\{\backslash partial\; \backslash Psi\; \}\{\backslash partial\; x\}\; \backslash right\backslash \},$$ where

• $\backslash hbar$ is the reduced Planck constant;
• $\backslash Psi^*$ denotes the complex conjugate of the wave function;
• $\backslash Re$ denotes the real part;
• $\backslash Im$ denotes the imaginary part.

Note that the probability current is proportional to a Wronskian $W(\backslash Psi,\backslash Psi^*).$

In three dimensions, this generalizes to $$\backslash mathbf\; j\; =\; \backslash frac\{\backslash hbar\}\{2mi\}\; \backslash left(\; \backslash Psi^*\; \backslash mathbf\; \backslash nabla\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\; \backslash nabla\; \backslash Psi^\{*\}\; \backslash right)\; =\; \backslash frac\{\backslash hbar\}\{m\}\; \backslash Re\backslash left\backslash \{\backslash Psi^*\; \backslash frac\{\backslash nabla\}\{i\}\; \backslash Psi\; \backslash right\backslash \}\; =\; \backslash frac\{\backslash hbar\}\{m\}\backslash Im\backslash left\backslash \{\backslash Psi^*\; \backslash nabla\; \backslash Psi\; \backslash right\backslash \}\; \backslash ,,$$ where $\backslash nabla$ denotes the del or gradient operator. This can be simplified in terms of the kinetic momentum operator, $$\backslash mathbf\{\backslash hat\{p\}\}\; =\; -i\backslash hbar\backslash nabla$$ to obtain $$\backslash mathbf\; j\; =\; \backslash frac\{1\}\{2m\}\; \backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; +\; \backslash Psi\; \backslash left(\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; \backslash right)^*\backslash right)\backslash ,.$$

These definitions use the position basis (i.e. for a wavefunction in position space), but momentum space is possible. In fact, one can write the probability current operator as

|\mathbf{r}\rangle\langle\mathbf{r}|+|\mathbf{r}\rangle\langle\mathbf{r}|\mathbf{\hat{p}}}{2m}

which do not depend on a particular choice of basis. The probability current is then the expectation of this operator,

$\backslash mathbf\{j\}(\backslash mathbf\{r\},t)\; =\; \backslash langle\; \backslash Psi(t)|\backslash hat\{\backslash mathbf\{j\}\}(\backslash mathbf\{r\})|\; \backslash Psi(t)\backslash rangle.$

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Spin-0 particle in an electromagnetic field The above definition should be modified for a system in an external electromagnetic field. In SI units, a charged particle of mass and electric charge includes a term due to the interaction with the electromagnetic field;

Leslie E. Ballentine (1990). 9780137479320, Prentice Hall. ISBN 9780137479320

$$\backslash mathbf\; j\; =\; \backslash frac\{1\}\{2m\}\backslash left\backslash left(\backslash Psi^*$$ where is the magnetic vector potential. The term has dimensions of momentum. Note that $\backslash mathbf\{\backslash hat\{p\}\}\; =\; -i\backslash hbar\backslash nabla$ used here is the canonical momentum and is not Gauge invariance, unlike the kinetic momentum operator $\backslash mathbf\{\backslash hat\{P\}\}\; =\; -i\backslash hbar\backslash nabla-q\backslash mathbf\{A\}$.

In Gaussian units: $$\backslash mathbf\; j\; =\; \backslash frac\{1\}\{2m\}\backslash left\backslash left(\backslash Psi^*$$ where is the speed of light.

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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.

According to Landau-Lifschitz's Course of Theoretical Physics the electric current density is in Gaussian units:see page 473, equation 115.4, $$\backslash mathbf\{j\}\_e\; =\; \backslash frac\{q\}\{2m\}\; \backslash left\backslash left(\backslash Psi^*\; +\; \backslash frac\{\backslash mu\_S\; c\}\{s\backslash hbar\}\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\{S\}\backslash Psi)$$

And in SI units: $$\backslash mathbf\; j\_e\; =\; \backslash frac\{q\}\{2m\}\backslash left\backslash left(\backslash Psi^*\; +\; \backslash frac\{\backslash mu\_S\}\{s\backslash hbar\}\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\{S\}\backslash Psi)$$

Hence the probability current (density) is in SI units: $$\backslash mathbf\{j\}\; =\; \backslash mathbf\{j\}\_e/q\; =\; \backslash frac\{1\}\{2m\}\backslash left\backslash left(\backslash Psi^*\; +\; \backslash frac\{\backslash mu\_S\}\{q\; s\backslash hbar\}\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\{S\}\backslash Psi)$$

where is the spin vector of the particle with corresponding spin magnetic moment and spin quantum number .

It is doubtful if this formula is valid for particles with an interior structure. The neutron has zero charge but non-zero magnetic moment, so $\backslash frac\{\backslash mu\_S\}\{q\; s\backslash hbar\}$ would be impossible (except $\backslash nabla\; \backslash times\; (\backslash Psi^*\; \backslash mathbf\{S\}\backslash Psi)$ 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_S= 2.7927·nuclear magneton or the deuteron (H-2 nucleus) which has s=1 and μ_S=0.8574·μ_N – it is mathematically possible but doubtful.

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Connection with classical mechanics The wave function can also be written in the complex number exponential (polar) form: $$\backslash Psi\; =\; R\; e^\{i\; S\; /\; \backslash hbar\; \}$$ where are real functions of and .

Written this way, the probability density is $$\backslash rho\; =\; \backslash Psi^*\; \backslash Psi\; =\; R^2$$ and the probability current is: $$\backslash begin\{align\}$$

 \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}

The exponentials and terms cancel: $$\backslash mathbf\{j\}\; =\; \backslash frac\{\backslash hbar\}\{2mi\}\backslash left\backslash frac.$$

Finally, combining and cancelling the constants, and replacing with , $$\backslash mathbf\{j\}\; =\; \backslash rho\; \backslash frac\{\backslash mathbf\{\backslash nabla\}\; S\}\{m\}.$$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: $$\backslash mathbf\{j\}\; =\; \backslash rho\; \backslash mathbf\{v\},$$

where $\backslash rho$ is the mass density of the fluid and is its velocity (also the group velocity of the wave). In the classical limit, we can associate the velocity with $\backslash tfrac\{\backslash nabla\; S\}\{m\},$ which is the same as equating with the classical momentum 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 $$\backslash mathbf\{p\}\; =\; \backslash nabla\; S$$ in Cartesian coordinates is given by , where is Hamilton's principal function.

The de Broglie-Bohm theory equates the velocity with $\backslash tfrac\{\backslash nabla\; S\}\{m\}$ in general (not only in the classical limit) so it is always well defined. It is an interpretation of quantum mechanics.

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Motivation

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Continuity equation for quantum mechanics 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.Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,

For some wave function , let:

$$\backslash rho(\backslash mathbf\{r\},t)\; =\; |\backslash Psi|^2\; =\; \backslash Psi^*(\backslash mathbf\{r\},t)\backslash Psi(\backslash mathbf\{r\},t)\; .$$be the probability density (probability per unit volume, denotes complex conjugate). Then,

where is any volume and is the boundary of .

This is the conservation law for probability in quantum mechanics. The integral form is stated as:

$$\backslash int\_V\; \backslash left(\; \backslash frac\{\backslash partial\; |\backslash Psi|^2\}\{\backslash partial\; t\}\; \backslash right)\; \backslash mathrm\{d\}V\; +\; \backslash int\_V\; \backslash left(\; \backslash mathbf\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; j\; \backslash right)\; \backslash mathrm\{d\}V\; =\; 0$$where$$\backslash mathbf\{j\}\; =\; \backslash frac\{1\}\{2m\}\; \backslash left(\; \backslash Psi^*\backslash hat\{\backslash mathbf\{p\}\}\backslash Psi\; -\; \backslash Psi\backslash hat\{\backslash mathbf\{p\}\}\backslash Psi^*\; \backslash right)\; =\; -\backslash frac\{i\backslash hbar\}\{2m\}(\backslash psi^*\backslash nabla\backslash psi-\backslash psi\backslash nabla\backslash psi^*)\; =\; \backslash frac\; \backslash hbar\; m\; \backslash operatorname\{Im\}\; (\backslash psi^*\backslash nabla\; \backslash psi)$$is the probability current or probability flux (flow per unit area).

Here, equating the terms inside the integral gives the continuity equation for probability:$$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash rho\backslash left(\backslash mathbf\{r\},t\backslash right)\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{j\}\; =\; 0,$$and the integral equation can also be restated using the divergence theorem as:

In particular, if 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 when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume . Altogether the equation states that the time derivative of the probability of the particle being measured in is equal to the rate at which probability flows into .

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.

This result is in agreement with the unitary nature of time evolution operators which preserve length of the vector by definition.

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Conserved current for Klein–Gordon fields The probability (4-)current arises from Noether's theorem as applied to the Lagrangian the Klein-Gordon Lagrangian density

$$\backslash mathcal\{L\}=\backslash partial\_\backslash mu\backslash phi^*\backslash ,\backslash partial^\backslash mu\backslash phi\; +V(\backslash phi^*\backslash ,\backslash phi)$$ of the complex scalar field \backslash phi:\backslash mathbb\{R\}^\{n+1\}\backslash mapsto\backslash mathbb\{C\}. This is invariant under the symmetry transformation$$\backslash phi\backslash mapsto\backslash phi\text{'}=\backslash phi\backslash ,e^\{i\backslash alpha\}\backslash ,\; .$$ Defining \backslash delta\backslash phi=\backslash phi\text{'}-\backslash phi we find the Noether current $$j^\backslash mu:=\; \backslash frac\{d\backslash mathcal\{L\}\}\{d\; \backslash dot\{\backslash mathbf\{q\}\}\}\; \backslash cdot\; \backslash mathbf\{Q\}\_r\; =\; \backslash frac\{d\backslash mathcal\{L\}\}\{d(\backslash partial\_\backslash mu)\backslash phi\}\backslash ,\backslash frac\{d(\backslash delta\backslash phi)\}\{d\backslash alpha\}\backslash bigg|\_\{\backslash alpha=0\}+\backslash frac\{d\backslash mathcal\{L\}\}\{d(\backslash partial\_\backslash mu)\backslash phi^*\}\backslash ,\backslash frac\{d(\backslash delta\backslash phi^*)\}\{d\backslash alpha\}\backslash bigg|\_\{\backslash alpha=0\}=\; i\backslash ,\backslash phi\backslash ,(\backslash partial^\backslash mu\backslash phi^*)-i\backslash ,\backslash phi^*\backslash ,(\backslash partial^\backslash mu\backslash phi)$$which satisfies the continuity equation. Here $\backslash mathbf\{Q\}\_r$ is the generator of the symmetry, which is $\backslash frac\{d\; (\backslash delta\; \backslash mathbf\{q\})\}\{d\; \backslash alpha\_r\}$ in the case of a single parameter $\backslash alpha$.

The continuity equation $\backslash partial\_\backslash mu\; j^\backslash mu\; =\; 0$ is satisfied. However, note that now, the analog of the probability density is not $\backslash phi\; \backslash phi^*$ but rather $\backslash phi^*\; \backslash partial\_t\; \backslash phi\; -\; \backslash phi\; \backslash partial\_t\; \backslash phi^*$. As this quantity can now be negative, we must interpret it as a charge density, with an associated current density and 4-current.

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Transmission and reflection through potentials In regions where a step potential or potential barrier occurs, the probability current is related to the transmission and reflection coefficients, respectively and ; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy: $$T\; +\; R\; =\; 1\backslash ,,$$ where and can be defined by: $$T=\; \backslash frac$$ \, , \quad R = \frac \, , where are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between and can be obtained from probability conservation: $$\backslash mathbf\{j\}\_\backslash mathrm\{trans\}\; +\; \backslash mathbf\{j\}\_\backslash mathrm\{ref\}=\backslash mathbf\{j\}\_\backslash mathrm\{inc\}\backslash ,.$$

In terms of a unit vector Surface normal to the barrier, these are equivalently: $$T=\; \backslash left|\backslash frac\{\backslash mathbf\{j\}\_\backslash mathrm\{trans\}\backslash cdot\backslash mathbf\{n\}\}\{\backslash mathbf\{j\}\_\backslash mathrm\{inc\}\backslash cdot\backslash mathbf\{n\}\}\backslash right|\backslash ,,\; \backslash qquad\; R=\; \backslash left|\backslash frac\{\backslash mathbf\{j\}\_\backslash mathrm\{ref\}\backslash cdot\backslash mathbf\{n\}\}\{\backslash mathbf\{j\}\_\backslash mathrm\{inc\}\backslash cdot\backslash mathbf\{n\}\}\; \backslash right|\; \backslash ,,$$ where the absolute values are required to prevent and being negative.

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Examples

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

illustrating that the particle may be in motion even if its spatial probability density has no explicit time dependence.

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Particle in a box For a particle in a box, in one spatial dimension and of length , confined to the region , the energy eigenstates are $$\backslash Psi\_n\; =\; \backslash sqrt\{\backslash frac\{2\}\{L\}\}\; \backslash sin\; \backslash left(\; \backslash frac\{n\backslash pi\}\{L\}\; x\; \backslash right)$$ and zero elsewhere. The associated probability currents are $$j\_n\; =\; \backslash frac\{i\backslash hbar\}\{2m\}\backslash left(\; \backslash Psi\_n^*\; \backslash frac\{\backslash partial\; \backslash Psi\_n\}\{\backslash partial\; x\}\; -\; \backslash Psi\_n\; \backslash frac\{\backslash partial\; \backslash Psi\_n^*\}\{\backslash partial\; x\}\; \backslash right)\; =\; 0$$ since $$\backslash Psi\_n\; =\; \backslash Psi\_n^*$$

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Discrete definition For a particle in one dimension on $\backslash ell^2(\backslash Z),$ we have the Hamiltonian $H\; =\; -\backslash Delta\; +\; V$ where $-\backslash Delta\; \backslash equiv\; 2\; I\; -\; S\; -\; S^\backslash ast$ is the discrete Laplacian, with being the right shift operator on $\backslash ell^2(\backslash Z).$ Then the probability current is defined as $j\; \backslash equiv\; 2\; \backslash Im\backslash left\backslash \{\backslash bar\{\backslash Psi\}\; i\; v\; \backslash Psi\backslash right\backslash \},$ with the velocity operator, equal to $v\; \backslash equiv\; -iX,\backslash ,$ and is the position operator on $\backslash ell^2\backslash left(\backslash mathbb\{Z\}\backslash right).$ Since is usually a multiplication operator on $\backslash ell^2(\backslash Z),$ we get to safely write $$-iX,\backslash ,\; =\; -iX,\backslash ,\; =\; -i\backslash leftX,\backslash ,\; =\; i\; S\; -\; i\; S^\{\backslash ast\}.$$

As a result, we find: $$\backslash begin\{align\}\; j\backslash left(x\backslash right)\; \backslash equiv\; 2\; \backslash Im\backslash left\backslash \{\backslash bar\{\backslash Psi\}(x)\; i\; v\; \backslash Psi(x)\backslash right\backslash \}$$

   &= 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}

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Further reading

• R. Resnick (1985). 047187373X, John Wiley & Sons. 047187373X

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