Editing Wave function (section)

==== Physical significance of phase ====
In non-relativistic quantum mechanics, it can be shown using Schrodinger's time dependent wave equation that the equation:

<math display="block">\frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf J = 0  </math>is satisfied, where <math display="inline">\rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2   </math> is the probability density and <math display="inline">\mathbf J(\mathbf x,t) = \frac{\hbar}{2im}(\psi^* \nabla\psi-\psi\nabla\psi^*) = \frac{\hbar}{m} \text{Im}(\psi^* \nabla\psi)  </math>, is known as the [[Probability current|probability flux]] in accordance with the continuity equation form of the above equation.

Using the following expression for wavefunction:<math display="block">\psi(\mathbf x,t)= \sqrt{\rho(\mathbf x,t)}\exp{\frac{iS(\mathbf x,t )}{\hbar}} </math>where <math display="inline">\rho(\mathbf x,t) = | \psi(\mathbf x,t)|^2   </math> is the probability density and <math display="inline">S(\mathbf x,t) </math> is the phase of the wavefunction, it can be shown that:

<math display="block">\mathbf J(\mathbf x,t) = \frac{\rho \nabla S}{m} </math>

Hence the spacial variation of phase characterizes the [[Probability current|probability flux]].

In classical analogy, for <math display="inline">\mathbf J = \rho \mathbf v </math>, the quantity <math display="inline"> \frac{\nabla S}{m} </math> is analogous with velocity. Note that this does not imply a literal interpretation of <math display="inline"> \frac{\nabla S}{m} </math> as velocity since velocity and position cannot be simultaneously determined as per the [[uncertainty principle]]. Substituting the form of wavefunction in Schrodinger's time dependent wave equation, and taking the classical limit, <math display="inline"> \hbar |\nabla^2 S| \ll |\nabla S|^2 </math>:

<math display="block">\frac{1}{2m} |\nabla S(\mathbf x, t)|^2 + V(\mathbf x) + \frac{\partial S}{\partial t} = 0 </math>

Which is analogous to [[Hamilton–Jacobi equation|Hamilton-Jacobi equation]] from classical mechanics. This interpretation fits with [[Hamilton–Jacobi theory]], in which <math display="inline"> \mathbf{P}_{\text{class.}} = \nabla S </math>, where ''{{mvar|S}}'' is [[Hamilton's principal function]].<ref>{{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 |pages=94–97}}</ref>