Tag: Core Quantum Mechanics

Table of Contents Introduction What Is Fine Structure? Sources of Fine Structure Spin-Orbit Coupling: The Physical Picture Mathematical Formulation of Spin-Orbit Interaction Derivation from Magnetic Interaction Total Angular Momentum ( vec{J}...

Table of Contents Introduction Motivation for Combining Angular Momenta Coupled and Uncoupled Bases What Are Clebsch–Gordan Coefficients? Mathematical Definition Orthogonality and Normalization How to Read and Use CG Coefficients Clebsch–Gordan Series Explicit Examples CG...

Table of Contents Introduction Why Add Angular Momenta? Individual Angular Momenta Total Angular Momentum Operator Commutation Relations and Conservation Allowed Values of Total Angular Momentum ( J ) Clebsch–Gordan Coefficients Coupled vs...

Table of Contents Introduction Why Spherical Harmonics Matter Definition and Notation The Angular Part of the Schrödinger Equation Associated Legendre Polynomials Mathematical Expression of Spherical Harmonics Properties and Symmetries Orthogonality and Normalization Quantum...

Table of Contents Introduction What Are Pauli Matrices? Matrix Definitions Algebraic Properties Commutation and Anticommutation Relations Eigenvalues and Eigenvectors Geometrical Interpretation Pauli Matrices and Spin Operators Rotations in Spin Space Pauli Matrices in Quantum...

Table of Contents Introduction What Is Spin? Spin vs Orbital Angular Momentum Mathematical Description of Spin Spin Quantum Numbers Spin Operators and Pauli Matrices Spin Eigenstates and Measurements Commutation Relations and Algebra Spin...

Table of Contents Introduction Classical vs Quantum Angular Momentum Angular Momentum in Quantum Mechanics Commutation Relations of Angular Momentum Operators Ladder Operators ( L_+ ) and ( L_- ) Eigenstates...

Table of Contents Introduction Central Potentials and Radial Schrödinger Equation Separation of Variables in Spherical Coordinates Angular Momentum and the Centrifugal Term Redefining the Radial Equation Effective Potential and Physical...

Table of Contents Introduction Importance of the Hydrogen Atom Classical vs Quantum Models The Coulomb Potential and Central Force Problem Schrödinger Equation in 3D Spherical Coordinates Separation of Variables Radial and...

Table of Contents Introduction The Classical vs Quantum View of Free Motion The Concept of a Free Particle in Quantum Mechanics Schrödinger Equation for a Free Particle General Plane...

Table of Contents Introduction What Are Coherent States? The Harmonic Oscillator Framework Definition via Displacement Operator Definition as Eigenstates of the Annihilation Operator Properties of Coherent States Uncertainty Minimization and Gaussian...

Table of Contents Introduction Classical vs Quantum Oscillator Potential and Schrödinger Equation Dimensionless Variables and Rescaling Series Solution and Hermite Polynomials Energy Quantization Wavefunctions and Normalization Probability Densities and Node Structure Ladder Operators:...