Editing Schrödinger equation (section)

=== Probability current ===
{{Main|Probability current|Continuity equation}}

The Schrödinger equation is consistent with [[conservation of probability|local probability conservation]].<ref name = "Cohen-Tannoudji"/>{{rp|238}} It also ensures that a normalized wavefunction remains normalized after time evolution. In matrix mechanics, this means that the [[Time evolution|time evolution operator]] is a [[unitary operator]].<ref name=":1">{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref> In contrast to, for example, the Klein Gordon equation, although a redefined inner product of a wavefunction can be time independent, the total volume integral of modulus square of the wavefunction need not be time independent.<ref>{{Cite journal |last=Mostafazadeh |first=Ali |date=2003-01-07 |title=Hilbert Space Structures on the Solution Space of Klein-Gordon Type Evolution Equations |journal=Classical and Quantum Gravity |volume=20 |issue=1 |pages=155–171 |doi=10.1088/0264-9381/20/1/312 |arxiv=math-ph/0209014 |bibcode=2003CQGra..20..155M |issn=0264-9381}}</ref>

The continuity equation for probability in non relativistic quantum mechanics is stated as: 
<math display="block">\frac{\partial}{\partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, </math>where
<math display="block"> \mathbf{j} = \frac{1}{2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right) = -\frac{i\hbar}{2m}(\psi^*\nabla\psi-\psi\nabla\psi^*) = \frac \hbar m \operatorname{Im} (\psi^*\nabla \psi)     </math>
is the [[probability current]] or probability flux (flow per unit area).

If the wavefunction is represented as <math display="inline">\psi( {\bf x},t)=\sqrt{\rho({\bf x},t)}\exp\left(\frac{i S({\bf x},t)}{\hbar}\right), </math> where <math>S(\mathbf x,t) </math> is a real function which represents the complex phase of the wavefunction, then the probability flux is calculated as:<math display="block"> \mathbf{j} = \frac{\rho \nabla S}  {m} </math>Hence, the spatial variation of the phase of a wavefunction is said to characterize the probability flux of the wavefunction. Although the <math display="inline"> \frac{ \nabla S}  {m} </math> term appears to play the role of velocity, it does not represent velocity at a point since simultaneous measurement of position and velocity violates [[uncertainty principle]].<ref name=":1" />