This Quantum World/Implications and applications/Probability flux

Probability flux

The time rate of change of the probability density $\rho (t,\mathbf {r} )=|\psi (t,\mathbf {r} )|^{2}$ (at a fixed location $\mathbf {r}$) is given by

${\frac {\partial \rho }{\partial t}}=\psi ^{*}{\frac {\partial \psi }{\partial t}}+\psi {\frac {\partial \psi ^{*}}{\partial t}}.$

With the help of the Schrödinger equation and its complex conjugate,

$i\hbar {\frac {\partial \psi }{\partial t}}={\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi +V\psi ,$

${\hbar \over i}{\partial \psi ^{*} \over \partial t}={\frac {1}{2m}}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}+V\psi ^{*},$

one obtains

${\frac {\partial \rho }{\partial t}}=-{\frac {i}{\hbar }}\psi ^{*}\left[{\frac {1}{2m}}\left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left({\frac {\hbar }{i}}{\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi +V\psi \right]$

$+{\frac {i}{\hbar }}\psi \left[{\frac {1}{2m}}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}+V\psi ^{*}\right].$

The terms containing $V$ cancel out, so we are left with

${\frac {\partial \rho }{\partial t}}=-{\frac {i}{2m\hbar }}\left[\psi ^{*}\left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}+\mathbf {A} \right)\psi -\psi \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\cdot \left(i\hbar {\frac {\partial }{\partial \mathbf {r} }}-\mathbf {A} \right)\psi ^{*}\right]$

$=\dots =-{\frac {\hbar }{2mi}}\left({\frac {\partial ^{2}\psi }{\partial \mathbf {r} ^{2}}}\psi ^{*}-\psi {\frac {\partial ^{2}\psi ^{*}}{\partial \mathbf {r} ^{2}}}\right)+{\frac {1}{m}}\left(\psi \psi ^{*}{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {A} +\mathbf {A} {\frac {\partial \psi }{\partial \mathbf {r} }}\psi ^{*}+\mathbf {A} \psi {\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\right).$

Next, we calculate the divergence of $\mathbf {j} ={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi }{\partial \mathbf {r} }}-{\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\psi \right)-{\frac {1}{m}}\mathbf {A} \psi ^{*}\psi$:

${\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {j} ={\frac {\hbar }{2mi}}\left({\frac {\partial ^{2}\psi }{\partial \mathbf {r} ^{2}}}\psi ^{*}-\psi {\frac {\partial ^{2}\psi ^{*}}{\partial \mathbf {r} ^{2}}}\right)-{\frac {1}{m}}\left(\psi \psi ^{*}{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {A} +\mathbf {A} {\frac {\partial \psi }{\partial \mathbf {r} }}\psi ^{*}+\mathbf {A} \psi {\frac {\partial \psi ^{*}}{\partial \mathbf {r} }}\right).$

The upshot:

${\frac {\partial \rho }{\partial t}}=-{\frac {\partial }{\partial \mathbf {r} }}\cdot \mathbf {j} .$

Integrated over a spatial region $R$ with unchanging boundary $\partial R:$

${\partial \over \partial t}\int _{R}\rho \,d^{3}r=-\int _{R}{\partial \over \partial \mathbf {r} }\cdot \mathbf {j} \,d^{3}r.$

According to Gauss's law, the outward flux of $\mathbf {j}$ through $\partial R$ equals the integral of the divergence of $\mathbf {j}$ over $R:$

$\oint _{\partial R}\mathbf {j} \cdot d\Sigma =\int _{R}{\partial \over \partial \mathbf {r} }\cdot \mathbf {j} \,d^{3}r.$

We thus have that

${\partial \over \partial t}\int _{R}\rho \,d^{3}r=-\oint _{\partial R}\mathbf {j} \cdot d\Sigma .$

If $\rho$ is the continuous density of some kind of stuff (stuff per unit volume) and $\mathbf {j}$ is its flux (stuff per unit area per unit time), then on the left-hand side we have the rate at which the stuff inside $R$ increases, and on the right-hand side we have the rate at which stuff enters through the surface of $R.$ So if some stuff moves from place A to place B, it crosses the boundary of any region that contains either A or B. This is why the framed equation is known as a continuity equation.

In the quantum world, however, there is no such thing as continuously distributed and/or continuously moving stuff. $\rho$ and $\mathbf {j} ,$ respectively, are a density (something per unit volume) and a flux (something per unit area per unit time) only in a formal sense. If $\psi$ is the wave function associated with a particle, then the integral $\int _{R}\rho \,d^{3}r=\int _{R}|\psi |^{2}\,d^{3}r$ gives the probability of finding the particle in $R$ if the appropriate measurement is made, and the framed equation tells us this: if the probability of finding the particle inside $R,$ as a function of the time at which the measurement is made, increases, then the probability of finding the particle outside $R,$ as a function of the same time, decreases by the same amount. (Much the same holds if $\psi$ is associated with a system having $n$ degrees of freedom and $R$ is a region of the system's configuration space.) This is sometimes expressed by saying that "probability is (locally) conserved." When you hear this, then remember that the probability for something to happen in a given place at a given time isn't anything that is situated at that place or that exists at that time.