Quantum 101

Table of Contents Introduction What Is Quantum Entanglement? Historical Background and EPR Paradox Entangled States: Definition and Examples Tensor Product and Composite Systems Bell States and Maximally Entangled Qubits Nonlocal Correlations...

Table of Contents Introduction Conceptual Overview The Wavefunction and Pure States Statistical Mixtures and Mixed States Density Matrix Representation Mathematical Criteria Visualization: Bloch Sphere Physical Interpretation Examples of Pure States Examples of Mixed States Quantum...

Table of Contents Introduction Motivation for the Density Matrix Pure and Mixed States Definition of the Density Operator Properties of the Density Matrix Expectation Values and Observables Evolution of the Density...

Table of Contents Introduction The Measurement Problem and Superpositions What Is Decoherence? The Environment and Open Quantum Systems Formal Definition and Mathematical Framework Reduced Density Matrix and Tracing Out the...

Table of Contents Introduction What Is Measurement in Quantum Mechanics? Observables and Hermitian Operators The Role of the Wavefunction The Born Rule and Probability Collapse Postulate Mathematical Representation of Collapse Measurement of...

Table of Contents Introduction Motivation and Physical Context Basic Idea of the WKB Method Mathematical Derivation The WKB Ansatz Validity and Conditions Classical Turning Points Matching at Turning Points Connection Formulas Bohr-Sommerfeld Quantization Example: Particle...

Table of Contents Introduction Motivation and Importance The Variational Principle Statement of the Theorem Constructing a Trial Wavefunction Applying the Variational Method Example: Ground State of the Hydrogen Atom Example: Helium Atom...

Table of Contents Introduction Historical Background and Significance Motivation for the Rule Derivation from Time-Dependent Perturbation Theory Mathematical Expression Transition Rate and Density of States Physical Interpretation Validity Conditions Example: Spontaneous Emission of...

Table of Contents Introduction Motivation and Relevance General Setup of Time-Dependent Perturbation The Interaction Picture First-Order Time-Dependent Perturbation Theory Transition Probabilities Fermi’s Golden Rule Periodic Perturbations and Resonance Example: Two-Level Atom in an...

Table of Contents Introduction Why Degeneracy Requires Special Treatment The Failure of Non-Degenerate Formulas Basic Setup of Degenerate Perturbation Theory Perturbation Within the Degenerate Subspace Constructing the Effective Hamiltonian Diagonalizing the...

Table of Contents Introduction Motivation for Perturbation Theory The General Setup Unperturbed and Perturbed Hamiltonians Expansion of Energy and Wavefunctions First-Order Corrections to Energy First-Order Corrections to Wavefunctions Second-Order Corrections to Energy Degenerate...

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}...