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, first we input the atomic number Z ( 1 = hydrogen H atom, 2 = helium He+ ion ) and whether you use reduced mass (= 0 ) or the original electron mass without reduced mass
From these values, it outputs theoretical values (= hydrogen solution of Bohr model or Schrodinger equation ) of total energy and (Bohr) radius of the hydrogen-like atoms.
Then we input the initial x-coordinate r1 (in MM = 1.0 × 10-14 meter ) of electron (= electron starts from x-axis to y-axis ), and the absolute value of the total energy E (in eV) of H or He+ atom.
From these inputted values of total energy of H or He+ atoms, this program outputs de Broglie wavelength (= WN ) of 1/4 orbit, x,y-components of the last velocity (= VX, VY ) of the electron after the electron has moved its 1/4 orbit from x-axis to y-axis.
↑ When the orbit is symmetrically circular, this last velocity's y-component (= VY ) becomes zero = the electron crosses y-axis perpendicularly after moving its 1/4 orbit.
Here we use new convenient unit 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 = 1 × 1032 m/s^2.
The initial x-coordinate r of the electron is automatically increased per calculation of 1/4 orbit until +100.
This method is the same as neutral helium program.
Theoretical values of energy and radius based on Bohr model or Schrodinger equation are:
In hydrogen atom (H), total energy = -13.60569 eV, and Bohr radius = 5291.77 MM.
Using reduced mass, total energy = -13.59829 eV, and radius = 5294.65 MM.
In helium ion (He+), total energy = -54.42277 eV, and radius = 2645.89 MM.
Using reduced mass, total energy = -54.41531 eV, and radius = 2646.25 MM.
In these values, one-orbit is just 1 × de Broglie wavelength, and last VY is just zero (= meaning this orbit becomes just symmetrically circular like the theoretical Bohr's hydrogen ).
↑ For example, after starting this program, when we input Z = 1 (= H atom ), reduced mass (= 0 ), initial r1 = 5294.65 (MM), total energy |E| of H atom = 13.59829 (eV), this program outputs the total de Broglie wavelength of 1/4 orbit equal to just 0.250000, and (last)VY velocity equal to just zero (= 0.000000 ), last-y-coordinate equal to the initial r1 (= meaning a symmetrical circular orbit ).
#include <stdio.h>
#include <math.h>
/* One-electron hydrogen, helium-ion */
int main(void)
{
int i;
double Z,whi,toenergy,radius,rm,r,E;
double vya,vyb,poten,VX,VY,prexx,preyy,WN,ra;
double xx,yy,vk,leng,wav,ac;
/* me = electron mass kg, pai = pi,
epsi = electric constant, h = Planck constant,
ele = electron 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;
/* mp = proton mass kg, alpha = helium nuclear mass */
double mp=1.67262171e-27;
double alph = 6.64465650e-27;
/* input Z = atomic number */
printf("Atomic number Z ( H = 1, He+ = 2 )？ ");
scanf("%lf",&Z);
/* choose reduced mass or usual electron mass */
printf("You use reduced mass (= 0 ) or usual electron mass (= 1 ) ?");
scanf("%lf", &whi);
/* rm = reduced mass */
rm=me;
if (Z < 1.5) { rm=(me*mp)/(me+mp); }
if (Z > 1.5 ) { rm=(me*alph)/(me+alph);}
if (whi > 0.5 ) { rm=me; }
/* toenergy = theoretical total energy of H atom or He+ ion */
toenergy = -(Z*Z*rm*ele*ele*ele*ele)/(8.0*epsi*epsi*h*h);
/* radius = thoretical radius of H atom or He+ ion */
radius = (epsi*h*h)/(pai*rm*Z*ele*ele);
/* change unit from J into eV */
toenergy = toenergy * 6.241509e18;
/* change unit from meter into MM (= 10^-14 meter ) */
radius = radius * 1.0e14;
printf("Theory-total-energy: %.5f ", toenergy);
printf("Theory-radius: %.2f \n", radius);
printf(" \n");
/* input electron-1's initial xc-coordinate r1 */
printf("r1 between nucleus and electron 1 (MM)？ ");
scanf("%lf",&r);
/* input absolute value of total energy |E| of H atom or He+ ion */
printf("absolute value of total energy |E| of H atom or He+ ion (eV) ? ");
scanf("%lf", &E);
for (i=1; i < 100 ;i++) {
/* repeat until r1 increases to r1+100 */
/* poten = initial Coulomb potential energy (J) */
poten=-(Z*ele*ele)/(4.0*pai*epsi*r*1.0e-14);
/* vya= total E - potential energy (J) */
vya=-(E*1.60217646e-19)-poten;
if (vya > 0) {
/* vyb=electron initial velocity (m/sec) */
vyb=sqrt((2*vya)/rm);
/* VX,VY = electron'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 = sum of de Broglie wavelength */
WN=0.0;
do {
/* xx,yy= electron's position after 1SS */
xx=prexx+VX; yy=preyy+VY;
vk=VX*VX+VY*VY;
/* leng = electron's moving length (= meter ) for 1SS */
leng=sqrt(vk)*1.0e-14;
/* wav = de Broglie wavelength (= meter ) */
wav=h/(rm*sqrt(vk)*1.0e9);
/* WN = sum of de Broglie wavelength */
WN=WN+leng/wav;
/* ra = distance between nucleus and electron */
ra=sqrt(prexx*prexx+preyy*preyy);
/* change unit of length from MM into meter */
ra=ra*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
ac=(ele*ele)/(4.0*pai*epsi*rm);
/* calculating acceleration (= MM/SS^2 = 10^32 m/s^2 ) from Coulomb force */
/* update electron's x,y-velocity = VX,VY from Coulomb force */
VX=VX+1.0e-32*ac*prexx*(-Z/(ra*ra*ra));
VY=VY+1.0e-32*ac*preyy*(-Z/(ra*ra*ra));
prexx=xx;preyy=yy;
} while (xx >= 0);
/* until electron has moved 1/4 orbit to reach y-axis */
/* display electron's initial x-coordinate r1 */
printf("r1= %.2f ", r );
/* display electron's last velocity VX, VY */
printf("VX= %.6f ", VX);
printf("VY= %.6f ", VY);
/* display electron's last y-coordinate */
printf("last y = %.2f ", yy);
/* display total de Broglie wavelength WN of 1/4 orbit */
printf("WN= %.6f\n", WN);
} r=r+1;
} return 0;
}