Foundations Of Quantum Mechanics: From Basics To Application

Lectures for a semester-long flipped course on quantum mechanics

What you'll learn

Introduction to fundamental principles and the need for quantum mechanics

Domain of quantum mechanics and its applicability to microscopic systems

Key concepts such as wave-particle duality, quantization, and superposition

Complex numbers and their role in quantum mechanics

Probability theory, probability distributions, and variance in quantum mechanics

Concept and physical meaning of the wavefunction

Normalization of the wavefunction and its importance

Extracting position, velocity, and momentum from the wavefunction

Heisenberg's uncertainty principle and its implications

Derivation and interpretation of Schrodinger's equation

Separation of variables and stationary states in quantum systems

Superposition of stationary states and their physical consequences

Potential functions and their impact on quantum behavior

Infinite square well (particle in a box) and its quantized energy states

Orthogonality and completeness of quantum states

Fourier series applications in quantum mechanics

Quantum harmonic oscillator using ladder operators and power series solutions

Free particles and their corresponding wavefunctions

Example computations and simulations for quantum systems

Requirements

Basic Mathematics – Algebra, trigonometry, and exponentials/logarithms

Complex Numbers – Understanding 𝑖 = − 1 i= −1 ​ and simple operations

Probability – Basic ideas of probability and distributions

Basic Physics – Newton's laws, energy, and momentum

Description

Course OverviewThis course offers a clear and approachable introduction to quantum mechanics, designed to guide students, professionals, and enthusiasts through the essential concepts, mathematical tools, and real-world applications of this exciting field. Whether you're a student looking to understand the foundations of quantum physics or a curious mind eager to dive into one of science's most fascinating areas, this course will provide you with a solid grounding in quantum mechanics.What You Will LearnBy the end of this course, you will have a deep understanding of the following core topics:Fundamental Concepts:Understand the key principles of quantum mechanics, including wave-particle duality, quantization, and the superposition principle.The Wavefunction & Its Properties:Learn how the wavefunction represents quantum systems and how to calculate probabilities, positions, momentum, and more.Schrödinger Equation:Dive into the Schrödinger equation and its role in describing quantum systems, with an emphasis on separation of variables and solving for stationary states.Uncertainty Principle:Grasp the implications of Heisenberg's uncertainty principle and how it defines the limits of measurement in quantum mechanics.Advanced Quantum Systems:Explore the infinite square well (particle in a box), quantum harmonic oscillators, free particles, and scattering phenomena.Mathematical Tools:Learn the mathematical foundations, including complex numbers, linear algebra (eigenvalues, eigenvectors), and the use of Fourier series in quantum mechanics.Applications:Apply your knowledge to real-world systems, such as the hydrogen atom and its wavefunctions, angular momentum in quantum mechanics, and spin 1/2 particles.Quantum Mechanics in Practice:Develop practical skills in quantum simulations and calculations, gaining insight into how quantum mechanics influences modern technology and research.Who This Course is ForPhysics and Engineering Students: Those studying quantum mechanics as part of their curriculumSTEM Enthusiasts: Anyone curious about the fundamental principles of quantum physicsMathematics Students: Individuals interested in the mathematical foundations of quantum mechanicsSelf-Learners: Anyone with a basic background in math and physics who wants to explore quantum mechanicsResearchers and Professionals: Scientists or engineers looking for a refresher on quantum principles and their applicationsNo previous experience with quantum mechanics is required—just a basic understanding of math and physics will help you succeed!Why Take This Course?Quantum mechanics is the foundation of modern physics, providing the framework for understanding the behavior of particles at the microscopic scale. This course will empower you with the knowledge to understand the quantum world, a realm where classical physics breaks down and strange phenomena like wave-particle duality, superposition, and entanglement reign. By learning quantum mechanics, you'll gain insight into a variety of real-world applications, such as semiconductors, quantum computing, and molecular chemistry.Whether you're preparing for advanced studies, looking to apply quantum mechanics in your research, or simply curious about the quantum nature of the universe, this course will give you the tools to begin that journey with confidence.Enroll NowDive into the quantum realm and gain a deeper understanding of the universe. Let's explore the world of quantum mechanics together!

Overview

Section 1: Introduction to Quantum Mechanics & Key Concepts

Lecture 1 Introduction to quantum mechanics

Lecture 2 The domain of quantum mechanics

Lecture 3 Key concepts in quantum mechanics

Section 2: Mathematical Foundations

Lecture 4 A review of complex numbers

Lecture 5 Complex numbers examples

Lecture 6 Probability in quantum mechanics

Lecture 7 Probability distributions and their properties

Lecture 8 Variance of probability distributions

Lecture 9 Normalization of the wavefunction

Lecture 10 Position, velocity, and momentum from the wavefunction

Section 3: The Uncertainty Principle

Lecture 11 Introduction to the uncertainty principle

Lecture 12 Key concepts of QM, revisited

Section 4: The Schrödinger Equation

Lecture 13 Separation of variables and the Schrödinger equation

Lecture 14 Stationary solutions to the Schrödinger equation

Lecture 15 Superposition of stationary states

Lecture 16 Potential functions in the Schrödinger equation

Lecture 17 Infinite square well (particle in a box)

Lecture 18 Infinite square well states, orthogonality and completeness (Fourier series)

Lecture 19 Infinite square well example computations and simulation

Physics and Engineering Students – Those studying quantum mechanics as part of their curriculum,STEM Enthusiasts – Anyone curious about the fundamental principles of quantum physics,Mathematics Students – Those interested in the mathematical foundations of quantum theory,Self-Learners – Individuals with a basic background in math and physics who want to explore quantum mechanics,Researchers and Professionals – Scientists or engineers looking for a refresher on quantum principles