Statistical Physics: relation to quanta and thermodynamics

The mathematics used in the discovery of quantum physics, the foundations of thermodynamics, phase transitions.

4.60 (21 reviews)
Udemy
platform
English
language
Math
category
247
students
8 hours
content
Nov 2023
last update
$49.99
regular price

What you will learn

Planck's mathematical trick which led to the discovery of quantum physics

rigorous definition of entropy

history of the physics of the 1900's

Basics of Statistical Mechanics

Einstein's papers on thermodynamics

Brownian motion

Lioville theorem

Ideal gas law

photoelectric effect

canonical transformations

Hamilton equations

Black body problem

Phase transitions

Description

First part of the course:

The first part of the course showcases the beautiful mathematics that, in the late 19th century/ early 20th century, led to the discovery of a revolutionary branch in physics: Quantum Mechanics.

Planck postulated that the energy of oscillators in a black body is quantized. This postulate was introduced by Max Planck in his derivation of his law of black body radiation in 1900. This assumption allowed Planck to derive a formula for the entire spectrum of the radiation emitted by a black body (we will also derive this spectrum in this course). Planck was unable to justify this assumption based on classical physics; he considered quantization as being purely a mathematical trick, rather than (as is now known) a fundamental change in the understanding of the world.

In 1905, Albert Einstein adapted the Planck postulate to explain the photoelectric effect, but Einstein proposed that the energy of photons themselves was quantized (with photon energy given by the Planck–Einstein relation), and that quantization was not merely a "mathematical trick". Planck's postulate was further applied to understanding the Compton effect, and was applied by Niels Bohr to explain the emission spectrum of the hydrogen atom and derive the correct value of the Rydberg constant.

In addition to the very useful mathematical tools that will be presented and discussed thoroughly, the students have the opportunity to learn about the historical aspects of how Planck tackled the blackbody problem.

Calculus and multivariable Calculus are a prerequisite to the course; other important mathematical tools (such as: Fourier Series, Perseval's theorem, binomial coefficients, etc.) will be recalled, with emphasis being put on mathematical and physical insights rather than abstract rigor.

Second part of the course

By the end of June 1902, just after being accepted as Technical Assistant at the Federal Patent Office in Bern, Albert Einstein, 23, sent to the renowned journal Annalen der Physik a manuscript with the bold title “Kinetic Theory of Thermal Equilibrium and of the Second Law of Thermodynamics”. In the introduction, he explains that he wishes to fill a gap in the foundations of the general theory of heat, “for one has not yet succeeded in deriving the laws of thermal equilibrium and the second law of thermodynamics using only the equations of mechanics and the probability calculus”. He also announces “an extension of the second law that is of importance for the application of thermodynamics”. Finally, he will provide “the mathematical expression of the entropy from the standpoint of mechanics”.

In particular, in the second part of the course we will see the mathematics Einstein used in his paper from 1902.

Besides, other concepts from Classical mechanics are explained, such as Liouville's theorem (this theorem is used by Einstein in his article), as well as Hamilton equations and more.

For the second part, the student should already be familiar with phase space and other concepts from classical physics (such as Lagrange equations).

Third part of the course

In the third part of the course some of the articles of Einstein's Annus Mirabilis are explained. In particular, the article on the photoelectric effect and that on the Brownian motion.

Fourth part of the course

In the last section of this course we focus on the derivation of phase transitons from the Ising model. All the previous sections will be useful in contextualizing this last part of the course.

Content

Introduction

Introduction
Introduction to the blackbody problem
Definition of blackbody

Analysis of blackbody cavities

Irrelevance of Shape of the Cavity
Wave equation for Electromagnetic Waves
Solution to the Wave Equation
Satisfaction of Boundary Conditions
Number of Modes per Frequency
Average Energy per Mode

Quantization: Systems of Particles

Distribution of the Average Energy, Planck's idea
Systems of Particles, Binomial Coefficient
Number of Arrangements of the Particles into the Energy Levels
Sterling's Approximation
Preparing to maximize the Number of Arrangements

Derivation of the Average Energy per Mode

Maximizing the Number of Arrangements
Expression for the Energy per Mode
Planck's Mathematical Trick
Calculation of the Average Energy per Mode part 1
Calculation of the Average Energy per Mode part 2

Derivation of the Stefan-Boltzman Law

Classical vs Quantum
Energy per Volume per Wavelength and Energy per Volume
Planck's Integral
Brief Summary of Fourier Analysis
Parseval's Theorem
Ultraviolet Catastrophe and Energy per Unit Surface
Calculation of the Series 1/n^4
Putting Results Together
Deriving the Stefan-Boltzmann Law part1
Deriving the Stefan-Boltzmann Law part2

Screenshots

Statistical Physics: relation to quanta and thermodynamics - Screenshot_01Statistical Physics: relation to quanta and thermodynamics - Screenshot_02Statistical Physics: relation to quanta and thermodynamics - Screenshot_03Statistical Physics: relation to quanta and thermodynamics - Screenshot_04

Reviews

Christopher
August 9, 2021
this course was very interesting & gave a great mathematical excursion. You notice the enthusiasm of the instructor especially.
Ron
January 1, 2021
I like the math. Instructor gets to the point and makes it simple. I didn't expect statistical mechanics, but it makes sense. I shall compare with some of the texts I've collected on quantum physics.

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5/17/2020
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5/29/2020
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