Editing Continuity equation (section)

== Quantum mechanics ==
<!-- This section is linked from [[Conservation law (physics)|Conservation law]] -->
{{see also|Madelung equations}}

[[Quantum mechanics]] is another domain where there is a continuity equation related to ''conservation of probability''. The terms in the equation require the following definitions, and are slightly less obvious than the other examples above, so they are outlined here:

* The [[wavefunction]] {{math|Ψ}} for a single [[particle]] in [[position and momentum space|position space]] (rather than [[position and momentum space|momentum space]]), that is, a function of position {{math|'''r'''}} and time {{math|''t''}}, {{math|1=Ψ = Ψ('''r''', ''t'')}}.
* The [[probability density function]] is <math display="block">\rho(\mathbf{r}, t) = \Psi^{*}(\mathbf{r}, t)\Psi(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2. </math>
* The [[probability]] of finding the particle within {{mvar|V}} at {{mvar|t}} is denoted and defined by <math display="block">P = P_{\mathbf{r} \in V}(t) = \int_V \Psi^*\Psi dV = \int_V |\Psi|^2 dV.</math>
* The [[probability current]] (probability flux) is <math display="block">\mathbf{j}(\mathbf{r}, t) = \frac{\hbar}{2mi} \left[ \Psi^{*} \left( \nabla\Psi \right) - \Psi \left( \nabla\Psi^{*} \right) \right].</math>

With these definitions the continuity equation reads:
<math display="block">\nabla \cdot \mathbf{j} + \frac{\partial\rho}{\partial t} = 0 \mathrel{\rightleftharpoons} \nabla \cdot \mathbf{j} + \frac{\partial |\Psi|^2}{\partial t} = 0.</math>

Either form may be quoted. Intuitively, the above quantities indicate this represents the flow of probability. The ''chance'' of finding the particle at some position {{math|'''r'''}} and time {{mvar|t}} flows like a [[fluid]]; hence the term ''probability current'', a [[vector field]]. The particle itself does ''not'' flow [[Deterministic system|deterministically]] in this [[vector field]].

{{math proof|title=Consistency with Schrödinger equation|proof=
The time dependent [[Schrödinger equation]] and its [[complex conjugate]] ({{math|''i'' → −''i''}} throughout) are respectively:<ref>For this derivation see for example {{cite book |title=Quantum Mechanics Demystified |first=D. |last=McMahon |publisher=McGraw Hill |year=2006 |isbn=0-07-145546-9 }}</ref>
<math display="block">\begin{align}
         -\frac{\hbar^2}{2m}\nabla^2\Psi + U\Psi &=  i\hbar\frac{\partial\Psi}{\partial t}, \\
 -\frac{\hbar^2}{2m}\nabla^2\Psi^{*} + U\Psi^{*} &= -i\hbar\frac{\partial\Psi^{*}}{\partial t}, \\
\end{align}</math>
where {{math|''U''}} is the [[Potential|potential function]]. The [[partial derivative]] of {{math|''ρ''}} with respect to {{math|''t''}} is:
<math display="block">
\frac{\partial \rho}{\partial t}
= \frac{\partial | \Psi |^2}{\partial t }
= \frac{\partial}{\partial t} \left( \Psi^{*} \Psi \right)
= \Psi^{*} \frac{\partial\Psi}{\partial t} + \Psi\frac{\partial\Psi^{*}}{\partial t}.
</math>

Multiplying the Schrödinger equation by {{math|Ψ*}} then solving for {{math|Ψ* {{sfrac|∂Ψ|∂''t''}}}}, and similarly multiplying the complex conjugated Schrödinger equation by {{math|Ψ}} then solving for {{math|Ψ {{sfrac|∂Ψ*|∂''t''}}}};
<math display="block">\begin{align}
\Psi^*\frac{\partial\Psi}{\partial t} &=  \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^*}{2m}\nabla^2\Psi + U\Psi^*\Psi \right], \\
\Psi\frac{\partial\Psi^*}{\partial t} &= -\frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^* + U\Psi\Psi^* \right], \\
\end{align}</math>

substituting into the time derivative of {{math|''ρ''}}:
<math display="block">\begin{align}
 \frac{\partial \rho}{\partial t}
    &= \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2\Psi + U\Psi^{*}\Psi \right] - \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^{*} + U\Psi\Psi^{*} \right] \\
    &= \frac{1}{i\hbar} \left[ -\frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + U\Psi^{*}\Psi \right] + \frac{1}{i\hbar} \left[ +\frac{\hbar^2\Psi}{2m}\nabla^2\Psi^{*} - U\Psi^{*}\Psi \right] \\[2pt]
    &= -\frac{1}{i\hbar} \frac{\hbar^2\Psi^{*}}{2m}\nabla^2 \Psi + \frac{1}{i\hbar} \frac{\hbar^2\Psi}{2m}\nabla^2 \Psi^{*} \\[2pt]
    &= \frac{\hbar}{2im} \left[ \Psi\nabla^2\Psi^{*} - \Psi^{*}\nabla^2\Psi \right] \\
\end{align} </math>

The [[Laplace operator|Laplacian]] [[Operator (mathematics)|operators]] ({{math|∇<sup>2</sup>}}) in the above result suggest that the right hand side is the divergence of {{math|'''j'''}}, and the reversed order of terms imply this is the negative of {{math|'''j'''}}, altogether:
<math display="block">\begin{align}
\nabla \cdot \mathbf{j}
&=  \nabla \cdot \left[ \frac{\hbar}{2mi} \left( \Psi^{*} \left( \nabla \Psi \right) - \Psi \left( \nabla \Psi^{*} \right) \right) \right] \\
&=  \frac{\hbar}{2mi} \left[ \Psi^{*} \left( \nabla^2 \Psi \right) - \Psi \left( \nabla^2\Psi^{*} \right) \right] \\
&= -\frac{\hbar}{2mi} \left[ \Psi \left( \nabla^2\Psi^{*} \right) - \Psi^{*} \left( \nabla^2 \Psi \right) \right] \\
\end{align}</math>
so the continuity equation is:
<math display="block">\begin{align}
                  &\frac{\partial \rho}{\partial t} = -\nabla \cdot \mathbf{j} \\[3pt]
  {}\Rightarrow{} &\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 \\
\end{align}</math>

The integral form follows as for the general equation.
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