Elementary Physics – Page 7 – Numerical Explorations

Blog Entry © Saturday, November 30, 2024, by James Pate Williams, Jr. Partial Solution of the Schrödinger Equation for Hydrogen in Parabolic Coordinates

Solution of the Schrödinger Equation for Hydrogen in Parabolic Coordinates Download

Blog Entry © Friday, November 29, 2024, by James Pate Williams, Jr. Christoffel Symbols of the Second Kind in Parabolic Coordinates

Christoffel Symbols of the Second Kind in Parabolic Coordinates Download

Blog Entry (c) Thursday, November 21, 2024, by James Pate Williams, Jr. Comparison of Homegrown Fifth Order Runge-Kutta Method Versus a Limited Number of Predictor-Corrector Algorithm

The first table is based on Conte-de Boor Fourth Runge-Kutta formulas that I converted to Fifth Order Runge-Kutta. Initial values: V = 2600 feet per second angle of elevation 30 degrees diameter 16 inches coefficient of form 0.61 density ratio 1.00 are from LCDR Ernest Edward Herrmann’s Exterior ballistics, 1935 My results are given first then LCDR Herrmann’s results:

x	deg	min	sec	time	v	vx	vy	y
0	30	0	0	0.00	2600	2252	1300	0
563	29	50	45	0.25	2578	2236	1283	325
1122	29	41	24	0.50	2556	2221	1266	646
1677	29	31	57	0.75	2535	2206	1250	962
2229	29	22	25	1.00	2515	2191	1233	1275
2776	29	12	47	1.25	2494	2177	1217	1583
3321	29	3	4	1.50	2474	2163	1201	1887
3861	28	53	15	1.75	2455	2149	1186	2188

x	deg	min	sec	time	v	vx	vy	y
0	30	0	0	0.00	2600	2252	1300	0
561	29	50	7	0.25	2582	2259	1285	323
1120	29	41	4	0.50	2564	2227	1270	642
1675	29	32	2	0.75	2546	2216	1255	958
2228	29	22	5	1.00	2529	2204	1241	1270

It is amazing how accurate Herrmann’s results were based on only a couple iterations using the Mayevski seven zone velocity retardation formulas.

Blog Entry © Sunday, November 17, 2024, by James Pate Williams, Jr. Three Methods of Solving the Exterior Ballistics Problem for the Fast Battleship USS Iowa (BB-61) 16-Inch/50 Caliber Rifles

Blog Entry Sunday 11 17 2024 Download

Blog Entry © Thursday, November 14, 2024, by James Pate Williams, Jr.

Blog Entry Thursday November 14 2024 Download

Blog Entry © Wednesday, November 6, 2024, by James Pate Williams, Jr. Particle in a Finite Spherical Three-Dimensional Potential Well

Spherical Finite Potential Well Download

Blog Entry (c) Wednesday, November 6, 2024, by James Pate Williams, Jr. Small Angular Momentum Quantum Numbers Gaunt Coefficients

// GauntCoefficients.cpp (c) Monday, November 4, 2024
// by James Pate Williams, Jr., BA, BS, MSWE, PhD
// Computes the Gaunt angular momentum coefficients
// Also the Wigner-3j symbols are calculated 
// https://en.wikipedia.org/wiki/3-j_symbol
// https://doc.sagemath.org/html/en/reference/functions/sage/functions/wigner.html#
// https://www.geeksforgeeks.org/factorial-large-number/
#include <iostream>
using namespace std;
typedef long double real;
real pi;
// iterative n-factorial function
real Factorial(int n)
{
    real factorial = 1;

    for (int i = 2; i <= n; i++)
        factorial *= i;
    if (n < 0)
        factorial = 0;
    return factorial;
}
real Delta(int lt, int rt)
{
    return lt == rt ? 1.0 : 0.0;
}
real Wigner3j(
    int j1, int j2, int j3,
    int m1, int m2, int m3)
{
    real delta = Delta(m1 + m2 + m3, 0) * 
        powl(-1.0, j1 - j2 - m3);
    real fact1 = Factorial(j1 + j2 - j3);
    real fact2 = Factorial(j1 - j2 + j3);
    real fact3 = Factorial(-j1 + j2 + j3);
    real denom = Factorial(j1 + j2 + j3 + 1);
    real numer = delta * sqrt(
        fact1 * fact2 * fact3 / denom);
    real fact4 = Factorial(j1 - m1);
    real fact5 = Factorial(j1 + m1);
    real fact6 = Factorial(j2 - m2);
    real fact7 = Factorial(j2 + m2);
    real fact8 = Factorial(j3 - m3);
    real fact9 = Factorial(j3 + m3);
    real sqrt1 = sqrtl(
        fact4 * fact5 * fact6 * fact7 * fact8 * fact9);
    real sumK = 0;
    int K = (int)fmaxl(0, fmaxl((real)j2 - j3 - m1,
        (real)j1 - j3 + m2));
    int N = (int)fminl((real)j1 + j2 - j3, 
        fminl((real)j1 - m1, (real)j2 + m2));
    for (int k = K; k <= N; k++)
    {
        real f0 = Factorial(k);
        real f1 = Factorial(j1 + j2 - j3 - k);
        real f2 = Factorial(j1 - m1 - k);
        real f3 = Factorial(j2 + m2 - k);
        real f4 = Factorial(j3 - j2 + m1 + k);
        real f5 = Factorial(j3 - j1 - m2 + k);
        sumK += powl(-1.0, k) / (f0 * f1 * f2 * f3 * f4 * f5);
    }
    return numer * sqrt1 * sumK;
}
real GauntCoefficient(
    int l1, int l2, int l3, int m1, int m2, int m3)
{
    real factor = sqrtl(
        (2.0 * l1 + 1.0) *
        (2.0 * l2 + 1.0) *
        (2.0 * l3 + 1.0) /
        (4.0 * pi));
    real wigner1 = Wigner3j(l1, l2, l3, 0, 0, 0);
    real wigner2 = Wigner3j(l1, l2, l3, m1, m2, m3);
    return factor * wigner1 * wigner2;
}
int main()
{
    pi = 4.0 * atanl(1.0);
    cout << "Gaunt(1, 0, 1, 1, 0, 0)  = ";
    cout << GauntCoefficient(1, 0, 1, 1, 0, 0);
    cout << endl;
    cout << "Gaunt(1, 0, 1, 1, 0, -1) = ";
    cout << GauntCoefficient(1, 0, 1, 1, 0, -1);
    cout << endl;
    real number = -1.0 / 2.0 / sqrtl(pi);
    cout << "-1.0 / 2.0 / sqrt(pi)    = ";
    cout << number << endl;
    return 0;
}