Equation

Standard Description

Velocity Field Description

Klein-Gord equation

$\frac{{\partial }^2\psi }{\partial t^2}-\frac{{\partial }^2\psi }{\partial x^2}+m^2\psi =0$

$\text{div}\left(\psi u\right)=\pm iL\psi$

Schrödinger equation

$i\frac{\partial \psi }{\partial t}=-\frac{1}{2m}\frac{{\partial }^2\psi }{\partial x^2}+V\left(x\right)\psi$

$\text{div}\left(\psi v\right)=iL\psi$

if $V\left(x\right)=\alpha x^2+\beta x+\gamma$

cl./electrom.wave equation

$\frac{{\partial }^2\psi }{\partial t^2}-v^2\Delta \psi =0$

$\text{div}\left(\psi v\right)=\pm iL\psi$

non-rel. pres.

equation

$\frac{\partial {\psi }^{\mathrm{<mo>\; *\; </mo>}}\psi }{\partial t}+\frac{\partial s}{\partial x}=0$

$\text{div}\left(v\sqrt{{\psi }^{\mathrm{<mo>\; *\; </mo>}}\psi }\right)=0$

rel. pres. equation

none in general

$\text{div}\left(u\sqrt{{\psi }^{\mathrm{<mo>\; *\; </mo>}}\psi }\right)=0$

cl. vac. feq.

$\Delta V=0$

$v\left[\text{div}\left(v\right)\right]+\text{div}\left(v,v\right)=0$

rel. vac. feq.

$\text{Ric}\left(v,w\right)=0$

$u\left[\text{div}\left(u\right)\right]+\text{div}\left(u,u\right)=0$

cl. mat. feq.

$\Delta V=4\pi \rho$

$v\left[\text{div}\left(v\right)\right]+\text{div}\left(v,v\right)=-4\pi \rho$

rel. mat. feq. (free part.)

$\text{Ric}-\left(R/2\right)g=8\pi T$

$w\left[\text{div}\left(w\right)\right]+\text{div}\left(w,w\right)=-4\pi \rho$

for $w=u,u_{\perp },\left(u+u_{\perp }\right)/\sqrt{2}$

pres. en.-mo.

$\text{div}T=0$

$\text{div}\left(\rho u\right)=0$