Personal Research Notes on Mathematics

Schrödinger Equation in the Free State (3 Dimensions)

In this article, we derive the solution to the three-dimensional Schrödinger equation when we consider a Gaussian wave packet model (white noise model) in the same way as in the one-dimensional case.

2024.08.25

Physics

Motion and Energy in the Theory of Relativity

Consider the hypothesis that the space-time of the theory of relativity is a conical coordinate system. We will summarize the discussion of what the equations of motion and conserved quantities are under this hypothesis, and show the problems.

2024.08.18

Physics

Kepler Equation

The orbits of planets and other objects moving around stars are elliptical, but the angular velocity is not constant. We consider a method of finding a solution to an algebraic equation called the Kepler equation by using Bessel functions.

2024.08.11

AnalysisPhysics

Elliptical Motion in Classical Mechanics

This article consider elliptical motion in three-dimensional space. Typically, it would be the motion of a planet orbiting a large star such as the Sun.

2024.08.06

Physics

Bessel Functions

Here is a brief summary of the properties of Bessel functions. They will be necessary when considering Kepler's equations (equations of elliptical motion) and the motion of electrons bound to atomic nuclei.

2024.08.02

Analysis

Uniform Circular Motion in the Theory of Relativity

This article explain the theory of uniform circular motion in classical mechanics and extend it to the case of the theory of relativity.

2024.07.29

Physics

Free Fall in the Theory of Relativity

The falling motion caused only by gravity, ignoring the effects of air resistance and friction, is called free fall. Find the equation of motion for free fall in the theory of relativity.

2024.07.25

Physics

The Kelly Criterion in Gambling: Implementation

Some readers may want to try using the Kelly Criterion, so I have included the source code in VBA and python. This will probably be the last in "The Kelly Criterion in Gambling" series.

2024.07.20

Probability

The Probability Distribution of Composites of Normal Random Numbers

For two random numbers that follow a normal distribution, the probability distribution that their sum, product, ratio, etc. follow is shown.

2024.07.18

Probability

Extension of the Kelly Criterion in Gambling: Mean-Variance Utility Function

As another variant of the Kelly criterion, we define the utility function in terms of the mean and variance of the assets and consider the problem of maximizing it.

2024.07.12

Probability