Chemistry - Nonlinear Coulomb field

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In theoretical physics, specifically in discussing, electromagnetism, the potential of a nonlinear Coulomb field

φ(g, x)

takes the dimensionless form

φ(g, x) = g φ(x)

$x={a \over R}$

where $g$ and $a$ are scales for potentials and lengths.

The Lagrangian for the Nonlinear Coulomb field in dimentionless units is

$L\sim {\dot \phi ^2\over 2}+k^2{\dot u_0^2\over 2}+\phi u_0\dot\phi^2+\gamma u_0^2\dot\phi^2.$

Maxwell's nonlinear equation for the Coulomb potential is

$\ddot\phi= u_0\dot \phi ^2$

or, equivalently,

$\dot\phi =\exp\int_{\phi_+}^{\phi}u_0(s)\,ds$ $={1\over 1-\int_{0}^{x} u_0(z)\,dz}$

φ + = φ(0)

where for simplicity $\dot \phi_+=1$ : this mean that states with positive electric gauge are considered.

In the physical region

$R\rightarrow \infty$ as $(x\rightarrow 0).$

The Coulomb potential for the vacuum state is

$\phi =\phi_+ +x + u_+ {x^2\over 2}.$

u + = u (0)

or, in dimensional form for a state with electric gauge $e = a g$ , in CGS units ,

$\phi (g,R)=g\phi_+ +{e\over R}+u_+{ea\over 2R^2}$

For the scalar velocity $u 0$ there are several equations. Here two of them are considered.

In vacuum state $\ddot u_0=0$ then $u 0 = u + + b x$ . Self energy any field must be finite so $b = 0$ . Coulomn potential is

$\phi =\phi_+ -{1\over u_+}\ln (1-xu_+)$ ;

u + (0)

Coherence condition for vacuum state is

$\int_{0}^{\infty}[\phi +2\gamma u_+]\dot \phi ^2\delta u_+\,dx =0$

$\int_{o}^{\infty}[\phi u_+ +\gamma u_+^2]\dot \phi\delta \dot\phi\,dx =0$

Hence potentials vacuum are connect between themselves

u + (φ + + 2γ u +) = 1

2 u + (φ + + γ u +) = 3

From physical reason in this vacuum state $u + = - 1$ then $φ + = - 2$ and interaction constant $\gamma=-{1\over 2}$ . Lorentz gauge condition $D\cdot A=0$ , fields bevector $F=D\land A=\vec E+i\vec H$ and linear Maxwells equations do not change at transformation $\phi\rightarrow\phi+const$ . In nonlinear model value potentials vacuum are determinate by coherence condition. There is analogy with relativistic machanics where energy free particle on infinity is fixid.

In coherent state equation for scalar velocity and solutions field equations are

$k^2\ddot u_0=\phi \dot\phi^2$

$v=-\tau +2\sum_{0}^N {1\over\tau-\tau_n}$

$\phi=\tau \sqrt{k}$

$v=u_0\sqrt{k}$

$\dot\tau={S(\tau)\over\sqrt kS(\tau_+)}$ ;

$S(\tau)=\prod_{0}^N (\tau -\tau_n)^2 e^{-\tau^2/ 2}$

$\tau_n =2\sum_{k}^N {1\over \tau_n -\tau_k}$ ; $k\ne n$

where N is integer number and $\phi_+^2>\phi_N^2$ because energy of free w-field must be finite. Coherence condition for this state is

$\int_{0}^{\infty}\left[\left(\phi u_0-{1\over 2}u_0^2\right)\dot\phi\delta\dot\phi-{1\over 2 }\dot\phi^2 u_0\delta u_0\right]\,dx =0$

$\delta u_0=(\partial_\phi u_0)\delta\phi_+$

$\delta\phi =\dot\phi (1-u_+ x)\delta\phi_+$

The state with N = 0 does not exist. States with N > 0 have electromagnetic structure because electric gauge density $\sim u_0\dot\phi^2$ change sign in intarnal area of field. Then there exist additional scale parameter, it is point corresponding $u 0 = 0$

Scale lengths $a$ and electric gauge $e$ is unknown parameter of field and they are need in defining from other sources. Electromagnetic interaction determinate value of Lamb shift Then for electron $a = 10^{-16}\ \mathrm{cm}.$

Linear Coulomb fields have infinite self-energy: this is the major difficulty of classical field theory. In nonlinear model all fields self energies are finite because dimensionless electrostatic force $\dot\phi$ speedily decrease when $x\rightarrow\infty$ .

Categories: Electromagnetism