Category: Elementary Physics

Electron Probability Distribution Function Etc. (c) James Pate Williams, Jr. December 2023

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More than Four Dimensions Why Worry a Blog Entry by James Pate Williams, Jr. December 27, 2023

Some modern physical models of our universe require more than Einstein’s four dimensions: three spatial dimensions and one time dimension. Why do people worry about introducing more dimensions into our understanding of chemistry and physics? When Erwin Schrödinger introduced his famous quantum mechanical two-body solution of the time independent hydrogen-like atom wave equation he went four dimensions to three spatial dimensions. Later, Wolfgang Pauli espoused his famous Pauli Exclusion Principle that simply stated no two electrons (fermions) in an atomic orbital can have the same quantum spin number. Atoms live in a four-dimensional quantum number space augmented by three spatial dimensions and one time dimension.

Some Helium Coulomb Integrals over Six Dimensions by James Pate Williams, Jr. Source Code in C++ Development over December 15 – 16, 2023

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Revised Translated Source Code from May 15, 2015, by James Pate Williams, Jr.

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New and Corrected Ground State Energy Numerical Computation for the Helium Like Atom (Atomic Number 2) by James Pate Williams, Jr.

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A New Calculus of Variations Solution of the Schrödinger Equation for the Lithium Like Atom’s Ground State Energy

This computation took a lot longer time to reach a much better solution than my previously published result.

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Guitar String and Piano Key Frequencies by James Pate Williams, Jr.

// FrequencyKey.cpp : Defines the entry point for the console application.
// James Pate Willims, Jr. (c) All Applicable Rights Reserved

#include "stdafx.h"
#include <math.h>
#include <iomanip>
#include <iostream>
#include <string>
#include <vector>
using namespace std;

vector<string> pnote;
double a = pow(2.0, 1.0 / 12.0);
double f0 = 440.0, gStrF[6];
double e2, a2, d3, g3, b3, e4;
double pfreq[9 * 12];
int offset = 0;

double fn(int n)
{
	return f0 * pow(a, n);
}

void printFrequency(char note, int octave, double frequency)
{
	cout << note << "\t" << octave << "\t";
	cout << setw(6) << fixed << setprecision(2);
	cout << frequency << endl;
}

int main()
{
	for (int octave = 0; octave <= 8; octave++)
	{
		pnote.push_back("C");
		pnote.push_back("C#");
		pnote.push_back("D");
		pnote.push_back("D#");
		pnote.push_back("E");
		pnote.push_back("F");
		pnote.push_back("F#");
		pnote.push_back("G");
		pnote.push_back("G#");
		pnote.push_back("A");
		pnote.push_back("A#");
		pnote.push_back("B");
	}

	pfreq[0] = 16.35;
	pfreq[1] = 17.32;
	pfreq[2] = 18.35;
	pfreq[3] = 19.45;
	pfreq[4] = 20.6;
	pfreq[5] = 21.83;
	pfreq[6] = 23.12;
	pfreq[7] = 24.5;
	pfreq[8] = 25.96;
	pfreq[9] = 27.5;
	pfreq[10] = 29.14;
	pfreq[11] = 30.87;
	
	for (int octave = 1; octave <= 8; octave++)
	{
		for (int i = 0; i < 12; i++)
		{
			pfreq[octave * 12 + i] = 2.0 * pfreq[(octave - 1) * 12 + i];
		}
	}

	gStrF[0] = e2 = fn(offset - 29);
	gStrF[1] = a2 = fn(offset - 24);
	gStrF[2] = d3 = fn(offset - 19);
	gStrF[3] = g3 = fn(offset - 14);
	gStrF[4] = b3 = fn(offset - 10);
	gStrF[5] = e4 = fn(offset - 5);

	cout << "Guitar\tOctave\tFrequency (Hz)" << endl;
	
	printFrequency('E', 2, e2);
	printFrequency('A', 2, a2);
	printFrequency('D', 3, d3);
	printFrequency('G', 3, g3);
	printFrequency('B', 3, b3);
	printFrequency('E', 4, e4);
	
	cout << endl;
	cout << "Piano Keys" << endl << endl;

	for (int octave = 0; octave <= 8; octave++)
	{
		for (int i = 0; i < 2; i++)
		{
			cout << octave << '\t';

			for (int j = 0; j < 6; j++)
			{
				{
					cout << pnote[(12 * octave + 6 * i + j) % 12] << '\t';
					cout << pfreq[(12 * octave + 6 * i + j)] << '\t';
				}
			}

			cout << endl;
		}
	}

	return 0;
}

https://en.wikipedia.org/wiki/Piano_key_frequencies#:~:text=%20Every%20octave%20is%20made%20of%20twelve%20steps,Hz%20and%20the%20sixth%20A%20is%20880%20Hz%29.