Jamali equation

The Jamali equation is a nonlinear partial differential equation that governs the wave function of a quantum-mechanical system. It is a generalization of the Schrödinger equation. The equation is named after Alireza Jamali, who first proposed the equation in May 2021 .

The equation

Unlike Schrödinger who uses ${\hat {p}}_{\mu }=i\hbar \partial _{\mu }$ as the quantum mechanical definition of momentum, Jamali begins by the following definition for momentum,

$p_{\mu }:=i\hbar {\frac {\partial _{\mu }\psi }{\psi }}$

for a non-zero wavefunction $\psi :\mathbb {R} ^{3}\times \mathbb {R} \to \mathbb {C}$. Similar to the derivation of Schrödinger equation from the law of conservation of energy, one can substitute this new definition into conservation of energy and arrive at

Jamali equation

$i\hbar {\frac {\partial \psi }{\partial t}}=-{\frac {\hbar ^{2}}{2m}}{\frac {|\nabla \psi |^{2}}{\psi }}+V\psi$

Reduction to Schrödinger equation

The differing term in Jamali equation is ${\frac {|\nabla \psi |^{2}}{\psi }}$. One can readily see that this term is equal to the corresponding term in Schrödinger equation once the divergence of quantum-mechanical three-momentum (à la Jamali) is set equal to zero:

$\nabla \cdot {\textbf {p}}=\nabla \cdot ({\frac {\nabla \psi }{\psi }})={\frac {\psi \nabla ^{2}\psi -|\nabla \psi |^{2}}{\psi ^{2}}}=0$

$\Longrightarrow {\frac {|\nabla \psi |^{2}}{\psi }}=\nabla ^{2}\psi$

This observation is somewhat anticipated if one recalls from probability current of orthodox quantum mechanics that Schrödinger equation is associated with the motion of an incompressible fluid (without source or sinks), i.e. one with $\nabla \cdot {\textbf {v}}=0$, which is the case with $\nabla \cdot {\textbf {p}}=0$ if one recalls that in the regime of non-relativistic quantum mechanics ${\textbf {p}}=m{\textbf {v}}$.

References

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