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If you copy and paste the program source code below into a text editor, you can easily compile and run this.
(This program is simple C language, so save this text editor as "filename.c", and compile it.)
In this program, we first input atomic number Z = 2, 3, 4, 5, 6, 7, 8, 9, or 10.
Next we input the initial x-coordinate r1 (in MM) of electron 1, and the absolute value of the total energy E (in eV) of two-electron atoms (ions).
From the inputted values, this program outputs the y component of electron 1 velocity after a quarter of its orbit, and WN (the number of de Broglie's waves included in one quarter of the orbital).
Here 1 SS = 1 × 10-23 second and 1 MM = 1 × 10-14 meter.
The initial x-coodinate is automatically increased per calculation until +100.
#include <stdio.h>
#include <math.h>
int main(void)
{
int i;
double Z,r,E,nucle,nuclee,rm,Ene;
double vya,vyb,poten,VX,VY,prexx,preyy,WN,ra,rb;
double xx,yy,vk,preVY,preWN,midWN,leng,wav,ac;
double me=9.1093826e-31;
double pai=3.141592653589793;
double epsi=8.85418781787346e-12;
double h=6.62606896e-34;
double ele=1.60217653e-19;
/* input Z, r1 and |E| */
printf("Atomic number Z ? (He=2, Li+=3, Be2+=4, B3+=5, C4+=6, N5+=7...) ");
scanf("%lf",&Z);
printf("r1 between nucleus and electron 1 (MM)？ ");
scanf("%lf",&r);
printf("total energy |E| of two-electron atom (eV) ? ");
scanf("%lf", &E);
nucle=2.0*me; nuclee=0.0;
if (Z == 2 ) { nucle = 6.64465650e-27; nuclee=nucle;} /* He alpha particle */
if (Z == 3 ) { nucle = 1.1646e-26; nuclee=nucle;} /* Li7 nucleus */
rm=(2.0*me*nucle)/(2.0*me+nuclee);
rm=rm*0.5; /* reduced mass ( He, Li ) */
Ene=-((4.0*Z-1)*(4.0*Z-1)*ele*ele*ele*ele*me)/(64.0*epsi*epsi*h*h);
Ene=Ene*6.241509e18;
printf("Wrong Ene = %.3f\n", Ene); /* Energy of circular orbit */
for (i=1; i < 100 ;i++) { /* repeat until r1=initial r1+100 */
/* poten = potential energy */
poten=-(2.0*Z*ele*ele)/(4.0*pai*epsi*r)+(ele*ele)/(4.0*pai*epsi*2.0*r);
/* vya= total E-potential energy */
vya=-(E*1.60217646e-19)-poten*1.0e14;
if (vya > 0) {
/* vyb=electron initial velocity (m/sec) */
vyb=sqrt(vya/me);
VY=vyb*1.0e-9; /* change m/sec to MM/SS */
prexx=r; VX=0.0; WN=0.0; preyy=0.0;
do {
xx=prexx+VX; yy=preyy+VY; /* electron 1 position after 1SS */
preVY=VY;preWN=WN ;
vk=VX*VX+VY*VY;
leng=sqrt(vk)*1.0e-14; /* moving length (m) for 1 SS */
wav=h/(rm*sqrt(vk)*1.0e9); /* de Broglie wavelength (m) */
WN=WN+leng/wav; /* add de Broglie wavelength */
/* calculation of VX,VY from Coulomb force */
ra=sqrt(prexx*prexx+preyy*preyy); /* between nucleus and electron */
rb=sqrt(4.0*prexx*prexx+2.0*preyy*preyy); /* between two electrons */
ra=ra*1.0e-14; rb=rb*1.0e-14; /* change MM to meter */
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
ac=(ele*ele)/(4.0*pai*epsi*rm);
/* acceleration (MM/SS^2) */
VX=VX+1.0e-32*ac*prexx*(-Z/(ra*ra*ra)+2.0/(rb*rb*rb));
VY=VY+1.0e-32*ac*preyy*(-Z/(ra*ra*ra)+1.0/(rb*rb*rb));
prexx=xx;preyy=yy;
} while (xx >= 0); /* electron has moved one quater of an orbit? */
if (VY > -0.0001 && VY < 0.0001) { /* last VY condition */
printf("r1= %.2f ", r );
printf("VX= %.6f ", VX);
printf("VY= %.6f ", VY);
printf("preVY= %.6f ", preVY);
midWN=(preWN+WN)/2.0; printf("midWN= %.6f\n", midWN);
}
} r=r+1;
} return 0;
}