Probability current

Quantum mechanics
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In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is equivalent to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

Definition (non-relativistic 3-current)

Free spin-0 particle

In non-relativistic quantum mechanics, the probability current j of the wave function $\Psi$ in one dimension is defined as ^[1]

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

where $\Psi ^{*}$ denotes the complex conjugate of the wave function. In three dimensions, this generalizes to

\mathbf {j} ={\frac {\hbar }{2mi}}\left(\Psi ^{*}\mathbf {\nabla } \Psi -\Psi \mathbf {\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.

This can be simplified in terms of the momentum operator,

\mathbf {\hat {p}} =-i\hbar \nabla

to obtain

{\mathbf j}={\frac {1}{2m}}\left(\Psi ^{*}{\mathbf {{\hat {p}}}}\Psi -\Psi {\mathbf {{\hat {p}}}}\Psi ^{*}\right)\,.

These definitions use the position basis (i.e. for a wavefunction in position space, momentum space is possible).

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 m and electric charge q includes a term due to the interaction with the electromagnetic field;^[2]

{\mathbf j}={\frac {1}{2m}}\left[\left(\Psi ^{*}{\mathbf {{\hat {p}}}}\Psi -\Psi {\mathbf {{\hat {p}}}}\Psi ^{*}\right)-2q{\mathbf {A}}|\Psi |^{2}\right]\,\!

where A = A(r, t) is the magnetic potential (aka "A-field"). The term qA has dimensions of momentum.

In Gaussian units:

{\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]\,\!

where c 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. In SI units:^[3]

{\mathbf j}={\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}}{s}}\nabla \times (\Psi ^{*}{\mathbf {S}}\Psi )\,\!

where S is the spin vector of the particle with corresponding spin magnetic moment μ_S and spin quantum number s. In Gaussian units:

{\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]+{\frac {\mu _{S}c}{s}}\nabla \times (\Psi ^{*}{\mathbf {S}}\Psi )\,\!

Connection with classical mechanics

The wave function can also be written in the complex exponential (polar) form:^[4]

\Psi =Re^{{iS/\hbar }}

where R and S are real functions of r and t.

Written this way, the probability density is

\rho =\Psi ^{*}\Psi =R^{2}

and the probability current is:

{\begin{aligned}\mathbf {j} &={\frac {-i\hbar }{2m}}\left(\Psi ^{*}\mathbf {\nabla } \Psi -\Psi \mathbf {\nabla } \Psi ^{*}\right)\\&={\frac {-i\hbar }{2m}}\left(Re^{-iS/\hbar }\mathbf {\nabla } Re^{iS/\hbar }-Re^{iS/\hbar }\mathbf {\nabla } Re^{-iS/\hbar }\right)\\&={\frac {-i\hbar }{2m}}\left[Re^{-iS/\hbar }(e^{iS/\hbar }\mathbf {\nabla } R+{\frac {i}{\hbar }}Re^{iS/\hbar }\mathbf {\nabla } S)-Re^{iS/\hbar }(e^{-iS/\hbar }\mathbf {\nabla } R-{\frac {i}{\hbar }}Re^{-iS/\hbar }\mathbf {\nabla } S)\right]\end{aligned}}

The exponentials and R∇R terms cancel:

={\frac {-i\hbar }{2m}}\left[{\frac {i}{\hbar }}R^{2}\mathbf {\nabla } S+{\frac {i}{\hbar }}R^{2}\mathbf {\nabla } S\right]

Finally, combining and cancelling the constants, and replacing R² with ρ,

{\mathbf {j}}=\rho {\frac {{\mathbf {\nabla }}S}{m}}

If we take the familiar formula for the current:

{\mathbf {j}}=\rho {\mathbf {v}}

where v is the velocity of the particle (also the group velocity of the wave), we can associate the velocity with ∇S/m, which is the same as equating ∇S with the classical momentum p = mv. This interpretation fits with Hamilton–Jacobi theory, in which

{\mathbf {p}}=\nabla S

in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function.

Motivation

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:^[5]

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

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

\rho ({\mathbf {r}},t)=|\Psi |^{2}=\Psi ^{*}({\mathbf {r}},t)\Psi ({\mathbf {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({\mathbf \nabla }\cdot {\mathbf 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+

$\oiint$

\scriptstyle S

{\mathbf j}\cdot {\mathrm {d}}{\mathbf {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, sans 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 probability of the particle being measured in V is equal to the rate at which probability flows into V.

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

T+R=1\,,

where T and R can be defined by:

T={\frac {|{\mathbf {j}}_{{\mathrm {trans}}}|}{|{\mathbf {j}}_{{\mathrm {inc}}}|}}\,,\quad R={\frac {|{\mathbf {j}}_{{\mathrm {ref}}}|}{|{\mathbf {j}}_{{\mathrm {inc}}}|}}\,,

where j_inc, j_ref and j_trans are the incident, reflected and transmitted probability currents respectively, and the vertical bars indicate the magnitudes of the current vectors. The relation between T and R is consistent with probability conservation:

{\mathbf {j}}_{{\mathrm {trans}}}+{\mathbf {j}}_{{\mathrm {ref}}}={\mathbf {j}}_{{\mathrm {inc}}}\,.

In terms of a unit vector n normal to the barrier, these are equivalently:

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|\,,

where the absolute values are required to prevent T and R being negative.

Examples

Plane wave

For a plane wave propagating in space:

\Psi ({\mathbf {r}},t)=\,Ae^{{i({\mathbf {k}}\cdot {{\mathbf {r}}}-\omega t)}}

the probability density is constant everywhere;

\rho ({\mathbf {r}},t)=|A|^{2}\rightarrow {\frac {\partial |\Psi |^{2}}{\partial t}}=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;

{\mathbf {j}}\left({\mathbf {r}},t\right)=\left|A\right|^{2}{\hbar {\mathbf {k}} \over m}=\rho {\frac {{\mathbf {p}}}{m}}=\rho {\mathbf {v}}

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 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 {i\hbar }{2m}}\left(\Psi _{n}^{*}{\frac {\partial \Psi _{n}}{\partial x}}-\Psi _{n}{\frac {\partial \Psi _{n}^{*}}{\partial x}}\right)=0

since

\Psi _{n}=\Psi _{n}^{*}

Discrete Definition

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

As a result we find: $j\left(x\right)\equiv 2\Im \{{\bar {\Psi (x)}}iv\Psi (x)\}=2\Im \{{\bar {\Psi (x)}}\left(\left(-S\Psi \right)(x)+\left(S^{\ast }\Psi \right)(x)\right)\}=2\Im \{{\bar {\Psi (x)}}\left(-\Psi (x-1)+\Psi (x+1)\right)\}$

References

↑ Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008, ISBN 978-0-07-154382-8
↑ Quantum mechanics, Ballentine, Leslie E, Vol. 280, Englewood Cliffs: Prentice Hall, 1990.
↑ Quantum mechanics, E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN 978-0-07-145533-6
↑ Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
↑ Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0

Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0

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