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. 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 j of the wave function Ψ of a particle of mass m in one dimension is defined as
j = ℏ 2 m i ( Ψ ∗ ∂ Ψ ∂ x − Ψ ∂ Ψ ∗ ∂ x ) = ℏ m ℜ { Ψ ∗ 1 i ∂ Ψ ∂ x } = ℏ m ℑ { Ψ ∗ ∂ Ψ ∂ x } , {\displaystyle 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\},}
where
ℏ {\displaystyle \hbar } is the reduced Planck constant;
Ψ ∗ {\displaystyle \Psi ^{*}} denotes the complex conjugate of the wave function;
ℜ {\displaystyle \Re } denotes the real part;
ℑ {\displaystyle \Im } denotes the imaginary part. Note that the probability current is proportional to a Wronskian W ( Ψ , Ψ ∗ ) . {\displaystyle W(\Psi ,\Psi ^{*}).}
In three dimensions, this generalizes to
j = ℏ 2 m i ( Ψ ∗ ∇ Ψ − Ψ ∇ Ψ ∗ ) = ℏ m ℜ { Ψ ∗ ∇ i Ψ } = ℏ m ℑ { Ψ ∗ ∇ Ψ } , {\displaystyle \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\}\,,}
where ∇ {\displaystyle \nabla } denotes the del or gradient operator. This can be simplified in terms of the kinetic momentum operator,
p ^ = − i ℏ ∇ {\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla }
to obtain
j = 1 2 m ( Ψ ∗ p ^ Ψ + Ψ ( p ^ Ψ ) ∗ ) . {\displaystyle \mathbf {j} ={\frac {1}{2m}}\left(\Psi ^{*}\mathbf {\hat {p}} \Psi +\Psi \left(\mathbf {\hat {p}} \Psi \right)^{*}\right)\,.}
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
j ^ ( r ) = p ^ | r ⟩ ⟨ r | + | r ⟩ ⟨ r | p ^ 2 m {\displaystyle \mathbf {\hat {j}} (\mathbf {r} )={\frac {\mathbf {\hat {p}} |\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,
j ( r , t ) = ⟨ Ψ ( t ) | j ^ ( r ) | Ψ ( t ) ⟩ . {\displaystyle \mathbf {j} (\mathbf {r} ,t)=\langle \Psi (t)|{\hat {\mathbf {j} }}(\mathbf {r} )|\Psi (t)\rangle .}
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;
j = 1 2 m [ ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) − 2 q A | Ψ | 2 ] {\displaystyle \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 vector potential. The term qA has dimensions of momentum. Note that p ^ = − i ℏ ∇ {\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla } used here is the canonical momentum and is not gauge invariant, unlike the kinetic momentum operator P ^ = − i ℏ ∇ − q A {\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} } . In Gaussian units:
j = 1 2 m [ ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) − 2 q c A | Ψ | 2 ] {\displaystyle \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. According to Landau-Lifschitz's Course of Theoretical Physics the electric current density is in Gaussian units:
j e = q 2 m [ ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) − 2 q c A | Ψ | 2 ] + μ S c s ℏ ∇ × ( Ψ ∗ S Ψ ) {\displaystyle \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 )}
And in SI units: j e = q 2 m [ ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) − 2 q A | Ψ | 2 ] + μ S s ℏ ∇ × ( Ψ ∗ S Ψ ) {\displaystyle \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 )}
Hence the probability current (density) is in SI units: j = j e / q = 1 2 m [ ( Ψ ∗ p ^ Ψ − Ψ p ^ Ψ ∗ ) − 2 q A | Ψ | 2 ] + μ S q s ℏ ∇ × ( Ψ ∗ S Ψ ) {\displaystyle \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}}{qs\hbar }}\nabla \times (\Psi ^{*}\mathbf {S} \Psi )}
where S is the spin vector of the particle with corresponding spin magnetic moment μS and spin quantum number s. 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 μ S q s ℏ {\displaystyle {\frac {\mu _{S}}{qs\hbar }}} would be impossible (except ∇ × ( Ψ ∗ S Ψ ) {\displaystyle \nabla \times (\Psi ^{*}\mathbf {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 μS= 2.7927·μN or the deuteron (H-2 nucleus) which has s=1 and μS=0.8574·μN – it is mathematically possible but doubtful.
Connection with classical mechanics
The wave function can also be written in the complex exponential (polar) form:
Ψ = R e i S / ℏ {\displaystyle \Psi =Re^{iS/\hbar }}
where R, S are real functions of r and t. Written this way, the probability density is ρ = Ψ ∗ Ψ = R 2 {\displaystyle \rho =\Psi ^{*}\Psi =R^{2}} and the probability current is:
j = ℏ 2 m i ( Ψ ∗ ∇ Ψ − Ψ ∇ Ψ ∗ ) = ℏ 2 m i ( R e − i S / ℏ ∇ R e i S / ℏ − R e i S / ℏ ∇ R e − i S / ℏ ) = ℏ 2 m i [ R e − i S / ℏ ( e i S / ℏ ∇ R + i ℏ R e i S / ℏ ∇ S ) − R e i S / ℏ ( e − i S / ℏ ∇ R − i ℏ R e − i S / ℏ ∇ S ) ] . {\displaystyle {\begin{aligned}\mathbf {j} &={\frac {\hbar }{2mi}}\left(\Psi ^{*}\mathbf {\nabla } \Psi -\Psi \mathbf {\nabla } \Psi ^{*}\right)\\[5pt]&={\frac {\hbar }{2mi}}\left(Re^{-iS/\hbar }\mathbf {\nabla } Re^{iS/\hbar }-Re^{iS/\hbar }\mathbf {\nabla } Re^{-iS/\hbar }\right)\\[5pt]&={\frac {\hbar }{2mi}}\left[Re^{-iS/\hbar }\left(e^{iS/\hbar }\mathbf {\nabla } R+{\frac {i}{\hbar }}Re^{iS/\hbar }\mathbf {\nabla } S\right)-Re^{iS/\hbar }\left(e^{-iS/\hbar }\mathbf {\nabla } R-{\frac {i}{\hbar }}Re^{-iS/\hbar }\mathbf {\nabla } S\right)\right].\end{aligned}}}
The exponentials and R∇R terms cancel:
j = ℏ 2 m i [ i ℏ R 2 ∇ S + i ℏ R 2 ∇ S ] . {\displaystyle \mathbf {j} ={\frac {\hbar }{2mi}}\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,
j = ρ ∇ S m . {\displaystyle \mathbf {j} =\rho {\frac {\mathbf {\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:
j = ρ v , {\displaystyle \mathbf {j} =\rho \mathbf {v} ,}
where ρ {\displaystyle \rho } is the mass density of the fluid and v is its velocity (also the group velocity of the wave). In the classical limit, we can associate the velocity with ∇ S m , {\displaystyle {\tfrac {\nabla S}{m}},} which is the same as equating ∇S with the classical momentum p = mv 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
p = ∇ S {\displaystyle \mathbf {p} =\nabla S}
in Cartesian coordinates is given by ∇S, where S is Hamilton's principal function. The de Broglie-Bohm theory equates the velocity with ∇ S m {\displaystyle {\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
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. For some wave function Ψ, let:
ρ ( r , t ) = | Ψ | 2 = Ψ ∗ ( r , t ) Ψ ( r , t ) . {\displaystyle \rho (\mathbf {r} ,t)=|\Psi |^{2}=\Psi ^{*}(\mathbf {r} ,t)\Psi (\mathbf {r} ,t).} be the probability density (probability per unit volume, * denotes complex conjugate). Then,
d d t ∫ V d V ρ = ∫ V d V ( ∂ ψ ∂ t ψ ∗ + ψ ∂ ψ ∗ ∂ t ) = ∫ V d V [ − i ℏ ( − ℏ 2 2 m ∇ 2 ψ + V ψ ) ψ ∗ + i ℏ ( − ℏ 2 2 m ∇ 2 ψ ∗ + V ψ ∗ ) ψ ] = ∫ V d V i ℏ 2 m [ ( ∇ 2 ψ ) ψ ∗ − ψ ( ∇ 2 ψ ∗ ) ] = ∫ V d V ∇ ⋅ ( i ℏ 2 m ( ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ) ) = ∫ S d a