Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability density. Intuitively; if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid (change in probability per unit time). This is analogous to hydrodynamic mass currents and electromagnetic charge currents. It is a vector quantity, but is complex-valued. Therefore it is not a physical property that can be measured like mass density or electric current - the notion of a probability current is a theoretical abstraction, useful in some of the formalism in quantum mechanics.

Definition: non-relativistic 3-current

In non-relativistic quantum mechanics, the probability current j of the wave function Ψ in one dimension is defined as

$j = \frac{\hbar}{2mi}\left(\Psi^* \frac{\partial \Psi }{\partial x}- \Psi \frac{\partial \Psi^* }{\partial x}\right) ,$

in three dimensions, this generalizes to

$\bold j = \frac{\hbar}{2mi}\left(\Psi^* \bold \nabla \Psi - \Psi \bold \nabla \Psi^*\right),$

where ħ is the reduced Planck constant, m is the particle's mass, Ψ is the wavefunction, and ∇ denotes the del or gradient operator.

These definitions utilize the position basis (i.e. for a wavefunction in position space, momentum space is possible). The 3-d form in terms of the real and imaginary parts are:

$\bold j = \frac\hbar m \mathrm{Im}\left ( \Psi^*\bold\nabla\Psi \right )=\mathrm{Re}\left ( \Psi^* \frac{\hbar}{im} \bold\nabla \Psi \right ) .$

In terms of the momentum operator,

$\bold{\hat{p}} = -i\hbar\nabla$

the simplist one to use is

$\bold j = \frac{1}{m} \mathrm{Re}\left ( \Psi^*\bold{\hat{p}}\Psi \right )$

The above definition should be modified for a system in an external electromagnetic field. For a charged particle of mass m and charge q, the probability current now is similar to the previous definition, up to a correction term;

$\bold j = \frac{\hbar}{2mi}\left(\Psi^* \bold \nabla \Psi - \Psi \bold \nabla \Psi^*\right) - \frac{q}{mc} \bold{A} |\Psi|^2 \,\!$

where A = A(r, t) is the magnetic potential (aka "A-field").

Motivation

Continuity equation for quantum mechanics

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:

$\frac{\partial \rho}{\partial t} + \bold \nabla \cdot \bold j = 0$

where the probability density $\rho\,$ is defined as

$\rho(\bold{r},t) = |\Psi|^2 = \Psi^*(\bold{r},t)\Psi(\bold{r},t) \,$.

If one were to integrate both sides of the continuity equation with respect to volume, so that

$\int_V \left( \frac{\partial |\Psi|^2}{\partial t} \right) \mathrm{d}V + \int_V \left( \bold \nabla \cdot \bold j \right) \mathrm{d}V = 0$

then the divergence theorem implies the continuity equation is equivalent to the integral equation

$\frac{\partial}{\partial t} \int_V |\Psi|^2 \mathrm{d}V + \int_S \bold j \cdot \mathrm{d}\bold{S} = 0$

where the V is any volume and S is the boundary of V. This is the conservation law for probability in quantum mechanics.

In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within 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 V. Altogether the equation states that the time derivative of the chance of the probability of the particle being measured in V is equal to the rate at which probability flows into V.

Transmission and reflection through potentials

Main articles: 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 T and R; they measure the extent the particles reflect from the potential barrier or are transmitted through it. Both satisfy:

$T+R=1.\,\!$

T and R can be defined by :

$T=\left | \frac{\bold{j}_\mathrm{trans}}{\bold{j}_\mathrm{inc}} \right | , \qquad R=\left | \frac{\bold{j}_\mathrm{ref}}{\bold{j}_\mathrm{inc}} \right | ,$

or equivalently:

$T= \frac{\bold{j}_\mathrm{trans}\cdot\bold{n}}{\bold{j}_\mathrm{inc}\cdot\bold{n}}, \qquad R= \frac{\bold{j}_\mathrm{ref}\cdot\bold{n}}{\bold{j}_\mathrm{inc}\cdot\bold{n}} ,$

where j_inc, j_ref and j_trans are the incident, reflected and transmitted probability currents respectively, and n is a unit vector normal to the barrier. The vertical bars denote absolute values, here complex moduli, since the magnitudes of the current vectors are complex-valued.

Substituting these definitions into the first equation gives the intuitive relation, a statement of probability conservation:

$\bold{j}_\mathrm{trans} + \bold{j}_\mathrm{ref}=\bold{j}_\mathrm{inc}.$

Examples

Plane wave

For the (three dimensional) plane wave

$\Psi = A e^{i\bold k \cdot \bold r} e^{-i \omega t}$

the associated probability current is

$\bold j = \frac{\hbar}{2mi} |A|^2 \left( e^{-i\bold k \cdot \bold r} \bold \nabla e^{i\bold k \cdot \bold r} - e^{i\bold k \cdot \bold r} \bold \nabla e^{-i\bold k \cdot \bold r} \right) = |A|^2 \frac{\hbar \bold k}{m}.$

This is just the square of the amplitude of the wave times the particle's velocity,

$\bold v = \frac{\bold p}{m} = \frac{\hbar \bold k}{m}$.

Note that the probability current is nonzero despite the fact that plane waves are stationary states and hence

$\frac{\mathrm{d}|\Psi|^2}{\mathrm{d}t} = 0\,$

everywhere. This demonstrates that a 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 L, confined to the region;

$0 < x < L\,\!$

the energy eigenstates are

$\Psi_n = \sqrt{\frac{2}{L}} \sin \left( \frac{n\pi}{L} x \right)$

and zero elsewhere. The associated probability currents are

$j_n = \frac{\hbar}{2mi}\left( \Psi_n^* \frac{\partial \Psi_n}{\partial x} - \Psi_n \frac{\partial \Psi_n^*}{\partial x} \right) = 0$

since

$\Psi_n = \Psi_n^*$

and we have used the identity

$\sin(x)=\frac{i}{2} \left(e^{-ix}-e^{ix} \right)$.

Definition (relativistic 4-current)

Above, the components of the probability 3-current are;

$\begin{align} J_x & = -\frac{i\hbar}{2m}\left(\psi^*\frac{\partial \psi}{\partial x} - \psi\frac{\partial \psi^*}{\partial x}\right), \quad J_y & = -\frac{i\hbar}{2m}\left(\psi^*\frac{\partial \psi}{\partial y} - \psi\frac{\partial \psi^*}{\partial y}\right), \quad J_z & = -\frac{i\hbar}{2m}\left(\psi^*\frac{\partial \psi}{\partial z} - \psi\frac{\partial \psi^*}{\partial z}\right) \, .\\ \end{align}$

The fact that the density is positive definite and convected according to continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace by probability density by the symmetrically formed expression

$\rho = \frac{i\hbar}{2m}\left(\psi^*\frac{\partial \psi}{\partial t} - \psi \frac{\partial \psi^*}{\partial t}\right)\, ,$

which now becomes the 4th component of a space-time vector, and the entire 4-current density has the relativistically covariant expression

$J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*) \, .$

where (translating usual cartesian-subscript notation into vector indicies):

$\begin{align} & J^\mu = (J^0, J^1, J^2, J^3) = (\rho, J_x, J_y, J_z) \\ & (x^0, x^1, x^2, x^3) = (t, x, y, z) \\ \end{align} \,$

and the partial derivatives are

$\partial^\mu = g^{\mu\eta} \partial_\eta = g^{\mu\eta} \frac{\partial }{\partial x^\eta} \, .$

where $\scriptstyle g^{\mu\eta}$ is the metric tensor. Using the (+−−−) metric signature and contracting on like indicies implies $\scriptstyle \partial^\mu = \partial_\mu = \partial /\partial x^\mu$ for $\scriptstyle \mu = 0$ (the probability density component), and $\scriptstyle \partial^\mu = -\partial_\mu = - \partial /\partial x^\mu$ for $\scriptstyle \mu$ = 1, 2, 3 (the probability current components). Setting the index $\scriptstyle \mu$ to these values returns the initial expressions for the current and density components. The continuity equation is as before:

$\partial_\mu J^\mu = 0 \Rightarrow \frac{\partial \rho}{\partial t} = \nabla\cdot\bold{J} \,$

This is compatible with relativity, though the expression for the density is no longer positive definite - the initial values of both ψ and $\scriptstyle \partial \psi/\partial t$ may be freely chosen, and the density may thus become negative.

References

• Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-873730