If you copy and paste the program source code below into a text editor, you can easily compile and run this.
(This class file name is MathMethod, so save this text editor as "MathMethod.java", 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 ). Try them.
↑ 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 ).
import java.util.Scanner;
class MathMethod {
public static void main(String[] args) {
// hydrogen like atoms (= H or He+ )
// input atomic number Z
Scanner stdIn=new Scanner(System.in);
System.out.println("Atomic number Z ( H = 1, He+ = 2 ) ? ");
double Z=stdIn.nextDouble();
// input whether use reduced mass (= 0 )
// or the original electron mass (= 1 )
System.out.println("You use reduced mass (= 0 ) or usual electron mass (= 1 ) ? ");
double whi=stdIn.nextDouble();
// me = electron mass
double me=9.1093826e-31;
// epsi = electric constant, pai = pi
double epsi=8.85418781787346e-12;
double pai=3.141592653589793;
// h = Planck constant
// ele = electron's charge
double h=6.62606896e-34;
double ele=1.60217653e-19;
// mp = mass of a proton (= H atomic nucleus )
double mp=1.67262171e-27;
// alph = mass of helium mass
double alph = 6.64465650e-27;
// rm = reduced mass
double 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 energy of H or He+
double toenergy = -(Z*Z*rm*ele*ele*ele*ele)/(8.0*epsi*epsi*h*h);
// radius = theoretical radius of H or He
double radius = (epsi*h*h)/(pai*rm*Z*ele*ele);
// change energy unit from J to eV
toenergy = toenergy * 6.241509e18;
// change length unit from meter to MM
radius = radius * 1.0e14;
// display theoretical energy, radius
System.out.printf("Theory-total energy: %.5f ", toenergy);
System.out.printf("Theory-radius: %.2f \n", radius);
System.out.println(" \n");
// input the initial electron's x-coordinate r
System.out.println("initial r between nucleus and electron (MM)？ ");
double r=stdIn.nextDouble();
// input total energy |E| of H or He+
System.out.println("absolute value of total energy |E| of H or He+ atom (eV) ? ");
double E=stdIn.nextDouble();
// increase r per calculation until r+100
for (int i=1;i < 100;i++) {
// poten = initial Coulomb potential energy
double poten=-(Z*ele*ele)/(4.0*pai*epsi*r*1.0e-14);
//vya= total E-potential energy (J)
double vya=-(E*1.60217646e-19)-poten;
if (vya > 0) {
// vyb=electron initial velocity (m/sec)
double vyb=Math.sqrt((2*vya)/rm);
// change velocity unit from m/s to MM/SS
// VX, VY = x, y-velocity of the electron.
double VY=vyb*1.0e-9; double VX = 0.0;
// prexx, preyy = electron's x,y coordinate
double prexx=r; double preyy=0.0;
// WN = sum of de Broglie wavelength
double WN=0.0;
double yy,vk,leng,wav; double xx=0.0;
do {
// xx, yy = electron's position after 1SS
xx=prexx+VX; yy=preyy+VY;
vk=VX*VX+VY*VY;
// leng = moving length (= m ) for 1SS
leng=Math.sqrt(vk)*1.0e-14;
// wav = de Broglie wavelength (= meter )
wav=h/(rm*Math.sqrt(vk)*1.0e9);
// WN = sum of de Broglie wavelength
WN=WN+leng/wav;
// ra = distance between electron and nucleus
double ra=Math.sqrt(prexx*prexx+preyy*preyy);
// change unit from MM to meter
ra=ra*1.0e-14;
prexx=prexx*1.0e-14; preyy=preyy*1.0e-14;
double ac=(ele*ele)/(4.0*pai*epsi*rm);
// update electron's x,y velocity = VX, VY
// from Coulomb force
// acceleration (= MM/SS^2) of electron
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);
// electron has moved 1/4 orbit
System.out.print("r1: "+r+" ");
// display electron's last velocity VX, VY
System.out.printf("VX:%.6f ", VX);
System.out.printf("VY:%.6f ", VY);
// display electron's last y-coordinate
System.out.printf("last-y:%.2f ", yy);
// display de Broglie wavelength of 1/4 orbit
System.out.printf("WN:%.6f\n", WN);
} r=r+1;
}}}