Quantum 101

Table of Contents Introduction Classical vs Quantum Behavior Step Potential Definition Regions and Wavefunction Forms Solving the Schrödinger Equation Case 1: ( E > V_0 ) — Partial Transmission Case 2:...

Table of Contents Introduction Classical vs Quantum View of Barriers What Is Quantum Tunneling? Mathematical Description Tunneling Through a Rectangular Potential Barrier Transmission and Reflection Coefficients Probability Current and Flux Conservation Tunneling...

Table of Contents Introduction Physical Setup of the Finite Square Well Mathematical Formulation Schrödinger Equation in Different Regions Bound State Conditions Even and Odd Solutions Transcendental Equations for Energy Levels Number of...

Table of Contents Introduction Concept of the Infinite Square Well Mathematical Formulation Schrödinger Equation Inside the Well Boundary Conditions and Wavefunction Form Quantized Energy Levels Normalized Wavefunctions Probability Distributions and Nodes Expectation Values...

Table of Contents Introduction What Is Quantum Dynamics? Schrödinger Equation and Time Evolution Unitary Evolution and the Role of the Hamiltonian Schrödinger, Heisenberg, and Interaction Pictures Evolution of Wavefunctions Evolution of...

Table of Contents Introduction Motivation for the Interaction Picture Schrödinger vs Heisenberg vs Interaction Picture Definition of the Interaction Picture Time Evolution in the Interaction Picture State Vector Evolution Operator Evolution The...

Table of Contents Introduction Schrödinger vs Heisenberg Picture Core Idea of the Heisenberg Picture Time Evolution of Operators Heisenberg Equation of Motion Constant State Vectors Time Evolution of Expectation Values Comparison with...

Table of Contents Introduction Quantum Time Evolution Overview Schrödinger vs Heisenberg Pictures The Time-Dependent Schrödinger Equation (TDSE) Hamiltonian and the Generator of Time Evolution Time Evolution Operator Properties of the Evolution...

Table of Contents Introduction Origins of the Uncertainty Principle Heisenberg's Thought Experiment Mathematical Formulation Generalized Uncertainty Principle Position-Momentum Uncertainty Energy-Time Uncertainty Angular Momentum and Angular Position Spin Components and Uncertainty Interpretation and Physical Meaning Uncertainty...

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...