# Hydrogen atom -e decay orbit - Lorentz/Dirac equation + random Zero Po

hi everyone, I'm making a research work to my master's degree and I have a problem I will appreciate your help.

I want simulate classically (not quantum mechanics) a electron orbiting a proton (H atom) immerse in a random distribution electromagnetic field. This theory is known as SED http://www.bu.edu/simulation/publications/dcole/PD...

For now we will forget the random EM. Since I'm trying first simulate the electron orbit without influence of EM.

As you now classically the electron feel the Coulomb force and the reaction term. The reaction term make the electron spin down and shock with the nucleus, the proton.

The non-relativistic Lorentz/Dirac equation to the movement is:

m*z'' = (-e^2)*z/|z|^3 + REAC - e*(EM)

The EM are the E and B random equations but for now we will forget this term.

REAC is the reaction term =>

2/3 * e^2 /c^3 * z''' (so third order differential eq)

but REAC can be approximated to

2/3 * e^2 /c^3 * d/dt( (-e^2)/m * z / z^3 ) ---> first order equation

the problem is in 2D x-y, so the vector z have two components, the z is the radius = sqrt(x^2 + y^2)

I can plot the electron orbit only with the Coulomb term:

((-e^2)*z/|z|^3)

but I cannot plot correctly with the REAC term. I already try use the ode45, the ode23 and the ode23t

I also tried use the REAC in the first form (third order ode) and in the second form

I also tried use polar coordinates, I'm lost I need help the following code is the my successful implementation of ode without the REAC

function mov = semZPF2(t,y)

r=(sqrt(y(1)^2+y(2)^2));

e = 1.602177e-19;

m=9.10938291*10^-31;

mov(1)=y(3);

mov(2)=y(4);

mov(3)=(-e^2*y(1)/r^3)/m;

mov(4)=(-e^2*y(2)/r^3)/m;

return;

------------------------------------------------------

yi=[0 5.3*10^-11]; % initial position

vi=[23.0583421525753 0]; % initial velocity

[t,y]= ode45(@(t,y) semZPF2(t,y,), [0 1E-12] , [yi vi] );

plot (y(i,1), y(i,2))