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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 choose a two-electron atom or ion whose energy we want to calculate, and input the chosen two-electron ion's atomic number Z = 2 (= He = helium ), 3 (= Li+ = singly-ionized lithium ), 4 (= Be2+ = doubly-ionized beryllium ), 5 (= B3+ ), 6 (= C4+ ), 7 (= N5+ ), 8 (= O6+ ), 9 (= F7+ ), 10 (= Ne8+ ), 11 (= Na9+ ), 12 (= Mg10+ ), 13 (= Al11+ ). 14 (= Si12+ ), 15 (= P13+ ), or 16 (= S14+ )...
Next we input the initial x-coordinate r1 (in unit MM = 1.0 × 10-14 meter ) of electron 1, and the absolute value of the total energy |E| (in eV) of the two-electron atoms (ions) that we want to calculate.
From the inputted values of atomic energy and initial coordinate, this program outputs the total de Broglie wavelength (= WN ) of 1/4 orbit, and the last x,y-velocity (= VX, VY ) of the electron-1, after the electron-1 has moved its 1/4 orbit from the initial x-axis to y-axis, by updating electrons' coordinate and velocity at short time intervals.
Here we use new convenient units of time = 1 SS = 1 × 10-23 second, length = 1 MM = 1 × 10-14 meter, velocity = 1MM/SS = 1 × 109 m/s, acceleration 1MM/SS^2 = 1032 m/s^2.
The initial electron-1's x-coordinate (= r1 ) is automatically increased per calculation until r1 +100 (= there is only one r1 which gives the last VY velocity being zero in each chosen atomic energy |E| ).
#include <stdio.h>
#include <math.h>
int main(void)
{
/* all two-electron atoms, ions */
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,leng,wav,ac;
/* me = electron mass (kg), pai = pi,
epsi = electric constant,
h = Planck constant,
ele = electron's charge
*/
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 atomic number Z */
printf("Atomic number Z ? (He=2, Li+=3, Be2+=4, B3+=5, C4+=6, N5+=7...) ");
scanf("%lf",&Z);
/* input electron-1's initial x-coordinate r1 */
printf("r1 between nucleus and electron 1 (MM)？ ");
scanf("%lf",&r);
/* inutput absolute value of total energy of a two-electron atom or ion */
printf("absolute value of total energy |E| of two-electron atom (eV) ? ");
scanf("%lf", &E);
/* rm = reduced mass is used only in light helium He and lithium ion Li+ with their nuclear mass = nucle */
nucle=2.0*me; nuclee=0.0;
if (Z < 2.5 ) { nucle = 6.64465650e-27; nuclee=nucle;} /* He alpha particle */
if (Z > 2.5 && Z < 3.5 ) { nucle = 1.1646e-26; nuclee=nucle;} /* Li7 nucleus */
rm=(2.0*me*nucle)/(2.0*me+nuclee);
rm=rm*0.5;
/* all two-electron ions except for He and Li+ use ordinary electron mass instead of reduced mass */
/* display old wrong Bohr's two-electron atomic energy (= Ene ) with one circular orbit */
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("Old wrong helium model's energy (eV) = %.3f\n", Ene);
printf(" \n");
for (i=1; i < 100 ;i++) {
/* repeat until electron-1's initial r1 x-coordinate increases to r1+100 */
/* poten = initial Coulomb potential energy (J) */
poten=-(2.0*Z*ele*ele)/(4.0*pai*epsi*r*1.0e-14)+(ele*ele)/(4.0*pai*epsi*2.0*r*1.0e-14);
/* vya = total energy - potential energy (J) */
vya=-(E*1.60217646e-19)-poten;
if (vya > 0) {
/* vyb=electron initial velocity (m/sec) */
vyb=sqrt(vya/me);
/* VX,VY = electron-1's x,y-velocity (= MM/SS = 10^9 m/s )*/
VY=vyb*1.0e-9; VX=0.0;
/* prexx,preyy = electron-1's x,y-coordinate */
prexx=r; preyy=0.0;
/* WN = number of de Broglie wavelength */
WN=0.0;
do {
/* xx,yy = electron-1's x,y-coordinate after 1SS (= 1x10^-23 second ) */
xx=prexx+VX; yy=preyy+VY;
vk=VX*VX+VY*VY;
/* leng = electron's moving distance (= meter) for 1SS */
leng=sqrt(vk)*1.0e-14;
/* wave = de Broglie wavelength = h/mv (meter) */
wav=h/(rm*sqrt(vk)*1.0e9);
/* WN = sum of de Broglie wavelength */
WN=WN+leng/wav;
/* ra = distance between electron-1 and nucleus */
ra=sqrt(prexx*prexx+preyy*preyy);
/* rb = distance between two electrons */
rb=sqrt(4.0*prexx*prexx+2.0*preyy*preyy);
/* change unit of length from MM into meter */
ra=ra*1.0e-14; rb=rb*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
ac=(ele*ele)/(4.0*pai*epsi*rm);
/* calculating acceleration (= 1MM/SS^2 = 10^32 m/s^2 ) */
/* update electron-1's x,y-velocity VX,VY from Coulomb force */
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);
/* until electron-1 has moved its 1/4 orbit to reach y-axis */
/* display electron-1's initial x-coordinate r1 */
printf("r1= %.2f ", r );
/* display electron-1's last velocity VX,VY */
printf("VX= %.6f ", VX);
printf("VY= %.6f ", VY);
/* display the total de Broglie wavelength (= WN ) of 1/4 orbit */
printf("WN= %.6f\n", WN);
} r=r+1;
} return 0;
}