Madelung equations

The Madelung equations are an alternative to the Schrödinger equation formulated by Erwin Madelung (1881–1972) .

If one replaces the complex function there by its absolute value and its phase accordingly , one obtains the Madelung equations: ${\displaystyle \psi }$ ${\displaystyle \rho }$ ${\displaystyle S}$ ${\displaystyle \psi ={\sqrt {\rho }}e^{{\frac {i}{\hbar }}S}}$

${\displaystyle \partial _{t}\rho +{\frac {1}{m}}\nabla (\rho \nabla S)=0}$

${\displaystyle \partial _{t}S+{\frac {1}{2m}}(\nabla S)^{2}+V(x)-{\frac {\hbar ^{2}}{2m}}{\frac {\Delta {\sqrt {\rho }}}{\sqrt {\rho }}}=0.}$

The first is in the form of a continuity equation ,

the second is a Hamilton-Jacobi equation (see canonical equations ).

${\displaystyle S}$is interpreted as an effect , as an impulse . ${\displaystyle \nabla S}$

Because of their non-linearity , the Madelung equations are difficult to work with, but they show that there are non- linear equations that can be reduced to linear equations .