Probability current explained

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 vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics 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.^[1]

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 of the wave function of a particle of mass in one dimension is defined as^[2] $$j\; =\; \backslash frac\; \backslash left(\backslash Psi^*\; \backslash frac-\; \backslash Psi\; \backslash frac\; \backslash right)\; =\; \backslash frac\; \backslash Re\backslash left\backslash \; =\; \backslash frac\; \backslash Im\backslash left\backslash ,$$where

\hbar is the reduced Planck constant;\Psi ^{*} denotes the complex conjugate of the wave function;\Re denotes the real part;\Im denotes the imaginary part. Note that the probability current is proportional to a Wronskian W(\Psi,\Psi ^{*).}

In three dimensions, this generalizes to$$\backslash mathbf\; j\; =\; \backslash frac\; \backslash left(\backslash Psi^*\; \backslash mathbf\; \backslash nabla\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\; \backslash nabla\; \backslash Psi^\; \backslash right)\; =\; \backslash frac\; \backslash Re\backslash left\backslash \; =\; \backslash frac\backslash Im\backslash left\backslash \; \backslash ,,$$where

\nabla denotes the del or gradient operator. This can be simplified in terms of the kinetic momentum operator,$$\backslash mathbf\; =\; -i\backslash hbar\backslash nabla$$to obtain$$\backslash mathbf\; j\; =\; \backslash frac\; \backslash left(\backslash Psi^*\; \backslash mathbf\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\; \backslash Psi^*\backslash right)\backslash ,.$$

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

See main article: 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;^[3] $$\backslash mathbf\; j\; =\; \backslash frac\backslash left[\backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi^*\backslash right)\; -\; 2q\backslash mathbf\{A\}\; |\backslash Psi|^2\; \backslash right]$$where is the magnetic vector potential. The term has dimensions of momentum. Note that

\hat{p } = -i\hbar\nabla used here is the canonical momentum and is not gauge invariant, unlike the kinetic momentum operator \hat{P } = -i\hbar\nabla-q\mathbf.

In Gaussian units:$$\backslash mathbf\; j\; =\; \backslash frac\backslash left[\backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi^*\backslash right)\; -\; 2\backslash frac\{q\}\{c\}\; \backslash mathbf\{A\}\; |\backslash Psi|^2\; \backslash right]$$where is the speed of light.

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:^[4] $$\backslash mathbf\_e\; =\; \backslash frac\; \backslash left[\backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi^*\backslash right)\; -\; \backslash frac\{2q\}\{c\}\; \backslash mathbf\{A\}\; |\backslash Psi|^2\; \backslash right]\; +\; \backslash frac\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\backslash Psi)$$

And in SI units: $$\backslash mathbf\; j\_e\; =\; \backslash frac\backslash left[\backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi^*\backslash right)\; -\; 2q\backslash mathbf\{A\}\; |\backslash Psi|^2\; \backslash right]\; +\; \backslash frac\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\backslash Psi)$$

Hence the probability current (density) is in SI units: $$\backslash mathbf\; =\; \backslash mathbf\_e/q\; =\; \backslash frac\backslash left[\backslash left(\backslash Psi^*\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi\; -\; \backslash Psi\; \backslash mathbf\{\backslash hat\{p\}\}\; \backslash Psi^*\backslash right)\; -\; 2q\backslash mathbf\{A\}\; |\backslash Psi|^2\; \backslash right]\; +\; \backslash frac\backslash nabla\backslash times(\backslash Psi^*\; \backslash mathbf\backslash Psi)$$

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

(\mu _{S})/(q s\hbar) would be impossible (except \nabla x (\Psi ^{*} S \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·μ_N or the deuteron (H-2 nucleus) which has s=1 and μ_S=0.8574·μ_N ^[5] – it is mathematically possible but doubtful.

Connection with classical mechanics

The wave function can also be written in the complex exponential (polar) form:^[6] $$\backslash Psi\; =\; R\; e^$$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:

The exponentials and terms cancel:$$\backslash mathbf\; =\; \backslash frac\backslash left[\backslash frac\; \{i\}\{\backslash hbar\}\; R^2\; \backslash mathbf\{\backslash nabla\}\; S\; +\; \backslash frac\; \{i\}\{\backslash hbar\}\; R^2\; \backslash mathbf\{\backslash nabla\}\; S\; \backslash right].$$

Finally, combining and cancelling the constants, and replacing with,$$\backslash mathbf\; =\; \backslash rho\; \backslash frac.$$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:$$\backslash mathbf\; =\; \backslash rho\; \backslash mathbf,$$

where

\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 \tfrac{\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$$\backslash mathbf\; =\; \backslash nabla\; S$$in Cartesian coordinates is given by, where is Hamilton's principal function.

The de Broglie-Bohm theory equates the velocity with

\tfrac{\nabla S}{m} 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

See main article: continuity equation.

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.^[7]

For some wave function, let:

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

\begin{align} (d)/(dt) \int _{l{V}dV\rho} & = \int ^{*\psi +} _{l{V}dV (\psi'} \psi ^{*\psi')} \ & = \int ^{2} _{l{V}dV\left(- (i\hbar\left()/(- (\hbar ^{2})/(2m)) \nabla} \psi + V \psi\right)\psi ^{* + (i\hbar\left()/(- (\hbar ^{2})/(2 m))} \nabla ^{2} \psi ^{*} + V \psi ^{*\right)\psi\right)} \\ & = \int _{l{V}dV (i\hbar)/(2m)} (\nabla ^{2\psi} \psi ^{*} - \psi \nabla ^{2} \psi ^{*)\ & = \int} _{l{V}dV \nabla ⋅ \left( (i\hbar)/(2 m)} (\psi ^{*\nabla\psi-\psi\nabla\psi} ^{*\nabla\psi - \psi\nabla\psi} ^{l{S}d a ⋅ \left( (i\hbar)/(2m) (\psi} ^{*)\right)\\ \end{align}}

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:

$$\backslash int\_V\; \backslash left(\backslash frac\; \backslash right)\; \backslash mathrmV\; +\; \backslash int\_V\; \backslash left(\backslash mathbf\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; j\; \backslash right)\; \backslash mathrmV\; =\; 0$$where$$\backslash mathbf\; =\; \backslash frac\; \backslash left(\backslash Psi^*\backslash hat\backslash Psi\; -\; \backslash Psi\backslash hat\backslash Psi^*\; \backslash right)\; =\; -\backslash frac(\backslash psi^*\backslash nabla\backslash psi-\backslash psi\backslash nabla\backslash psi^*)\; =\; \backslash frac\; \backslash hbar\; m\; \backslash operatorname\; (\backslash psi^*\backslash nabla\; \backslash psi)$$is the probability current or probability flux (flow per unit area).

Here, equating the terms inside the integral gives the continuity equation for probability:$$\backslash frac\; \backslash rho\backslash left(\backslash mathbf,t\backslash right)\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0,$$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.^[8] 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

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:$$\backslash mathbf\_\backslash mathrm\; +\; \backslash mathbf\_\backslash mathrm=\backslash mathbf\_\backslash mathrm\backslash ,.$$

In terms of a unit vector normal to the barrier, these are equivalently:$$T=\; \backslash left|\backslash frac\backslash right|\backslash ,,\; \backslash qquad\; R=\; \backslash left|\backslash frac\; \backslash right|\; \backslash ,,$$where the absolute values are required to prevent and being negative.

Examples

Plane wave

See main article: plane wave.

For a plane wave propagating in space:$$\backslash Psi(\backslash mathbf,t)\; =\; \backslash ,\; A\; e^$$the probability density is constant everywhere;$$\backslash rho(\backslash mathbf,t)\; =\; |A|^2\; \backslash rightarrow\; \backslash frac\; =\; 0$$(that is, plane waves are stationary states) but the probability current is nonzero – the square of the absolute amplitude of the wave times the particle's speed;$$\backslash mathbf\backslash left(\backslash mathbf,t\backslash right)\; =\; \backslash left|A\backslash right|^2\; =\; \backslash rho\; \backslash frac\; =\; \backslash rho\; \backslash mathbf$$

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, confined to the region

0 < x < L , the energy eigenstates are$$\backslash Psi\_n\; =\; \backslash sqrt\; \backslash sin\; \backslash left(\backslash frac\; x\; \backslash right)$$and zero elsewhere. The associated probability currents are$$j\_n\; =\; \backslash frac\backslash left(\backslash Psi\_n^*\; \backslash frac\; -\; \backslash Psi\_n\; \backslash frac\; \backslash right)\; =\; 0$$since $$\backslash Psi\_n\; =\; \backslash Psi\_n^*$$

Discrete definition

For a particle in one dimension on

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

As a result, we find:

Further reading

• Book: Resnick, R. . Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles . 2nd . R. . Eisberg . John Wiley & Sons . 1985 . 0-471-87373-X .

Notes and References

1. Book: Paul, Wolfgang . Jörg . Baschnagel . Stochastic Processes : From Physics to Finance . Berlin . Springer . 1999 . 3-540-66560-9 . 84 .
2. Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008,
3. Quantum mechanics, Ballentine, Leslie E, Vol. 280, Englewood Cliffs: Prentice Hall, 1990.
4. see page 473, equation 115.4, Web site: L.D. Landau, E.M. Lifschitz . COURSE OF THEORETICAL PHYSICS Vol. 3 – Quantum Mechanics . 3rd . ia803206.us.archive.org . 29 April 2023.
5. Web site: Spin Properties of Nuclei . 29 April 2023 . www2.chemistry.msu.edu.
6. Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008,
7. Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,
8. Book: Sakurai, Jun John . Modern quantum mechanics . Napolitano . Jim . 2021 . Cambridge University Press . 978-1-108-47322-4 . 3rd . Cambridge.