Quantum 101

Table of Contents Introduction Operators and Observables Definition of Commutator Mathematical Properties of Commutators Canonical Commutation Relations Position-Momentum Commutation Angular Momentum Commutation Relations Spin and Pauli Matrices Consequences for Measurement Commutators and Uncertainty Principle Lie...

Table of Contents Introduction What Are Operators in Quantum Mechanics? Definition of Hermitian Operators Mathematical Properties Physical Significance Hermitian vs Non-Hermitian Operators Spectral Theorem and Diagonalization Examples of Hermitian Operators Eigenvalues and Eigenfunctions Inner...

Table of Contents Introduction The Role of Observables in Quantum Theory Mathematical Representation of Observables Hermitian Operators and Their Properties Measurement Postulate Eigenvalues as Measurement Outcomes Projection Postulate and State Collapse Expectation...

Table of Contents Introduction Historical Background What Is the Born Rule? Mathematical Statement Connection to Probability Amplitudes Normalization and Completeness Interpretation of the Wavefunction Experimental Validation Role in the Measurement Process Born Rule in...

Table of Contents Introduction What Are Probability Amplitudes? Mathematical Definition Born Rule and Measurement Superposition Principle Inner Products and Transition Amplitudes Amplitudes in Different Representations Probability Densities and Continuum States Normalization and Orthogonality Interference...

Table of Contents Introduction The Schrödinger Equation Framework Time-Dependent vs Time-Independent Equations Derivation from Separation of Variables Mathematical Form of the Time-Independent Schrödinger Equation Physical Interpretation Boundary Conditions and Normalization One-Dimensional Potential...

Table of Contents Introduction Quantum States and Hilbert Space Bra-Ket Notation Properties of State Vectors Orthogonality and Completeness Observables as Operators Hermitian Operators and Measurement Eigenvalues and Eigenvectors Expectation Values and Variance Operator Algebra Commutators...

Table of Contents Introduction Why Postulates? Postulate 1: State Vector and Hilbert Space Postulate 2: Observables and Operators Postulate 3: Measurement and Expectation Values Postulate 4: Time Evolution and Schrödinger...

Table of Contents Introduction Historical Context and the Failure of Classical Models Basic Postulates of Bohr’s Model Quantization of Angular Momentum Energy Levels of the Hydrogen Atom Spectral Lines and...

Table of Contents Introduction Historical Background Classical Wave Behavior Young’s Original Experiment Single-Photon and Single-Electron Double-Slit Experiments Interference Patterns and Quantum Superposition Quantum Mechanical Description Probability Amplitudes and the Born Rule Which-Path Information...

Table of Contents Introduction Historical Background What Is Compton Scattering? Classical Wave Prediction Compton’s Experiment Derivation of the Compton Shift Key Equations and Interpretation Conservation of Energy and Momentum The Compton Wavelength Experimental Validation Implications...

Table of Contents Introduction Historical Context Light as a Wave Light as a Particle The Double-Slit Experiment de Broglie’s Hypothesis Matter Waves and Electron Diffraction Wave-Particle Duality of Photons Wave-Particle Duality of Electrons Interference...