Introduction to Quantum Mechanics

Quantum mechanics as Cambridge teaches it — from the wavefunction to the hydrogen atom.

About this course

An outrageous departure from the familiar world: baffling, disturbing, and undeniably right — the greatest achievement in the history of science, as David Tong puts it. He works the mathematics long before the rules arrive, so that when they do, you can see why they had to.

What you will be able to do

  • Work with wave functions and the Schrödinger equation
  • Solve bound-state and scattering problems
  • Use operators, eigenstates and measurement postulates
  • Solve the hydrogen atom

What this course covers

  • Introduction & Motivation for Quantum Mechanics
  • Spectral Lines of Atoms as Course Punchline
  • Course Structure & Teaching Approach
  • Classical State vs. Quantum State
  • The Wave Function and the Born Rule
  • Inherent Quantum Probability vs. Classical Ignorance & Reconciling Particle Tracks
  • Introduction & Recap
  • Normalization of the Wave Function
  • Normalizable vs. Non-Normalizable Wave Functions
  • Global Phase Invariance of the Wave Function
  • Principle of Superposition
  • Gaussian Wave Function & Superposition Example
  • Introduction to the Schrödinger Equation
  • Structure of the Schrödinger Equation and Its Terms
  • Planck's Constant ℏ: Units and Notation
  • The Hamiltonian Operator
  • Q&A: Origins, Nature, and Examples of the Hamiltonian
  • Classical Analogy: Hamiltonian and Energy Conservation
  • Fundamental vs. Emergent Forces and the Quantum-Classical Divide
  • Numerical Simulation of the Schrödinger Equation
  • Motivation: Conservation of Probability
  • Setting Up the Calculation
  • Continuity Equation and Probability Current
  • Collapse of the Wave Function
  • The Double-Slit Experiment: Classical Balls and Waves
  • The Double-Slit Experiment: Quantum Particles
  • Course Roadmap & Recap of Wave Function
  • Classical vs. Quantum Particle in 1D
  • Time-Dependent vs. Time-Independent Schrödinger Equation
  • Free Particle: Solutions and Non-Normalizability
  • De Broglie Relation and Momentum from the Wave Function
  • Particle on a Circle and Quantization of Momentum
  • Spectrum of the Hamiltonian and Wave Function Visualizations
  • Experimental Evidence: Davisson–Germer and GP Thomson
  • Motivation: Non-Normalizable Wave Functions
  • Linearity of the Schrödinger Equation and Superposition
  • Free Particle Solutions and Superposition via Integration over k
  • Choosing A(k): The Gaussian Profile and Completing the Square
  • Evaluating the Gaussian Integral and the Wave Packet Solution
  • Probability Density: Motion and Spreading of the Wave Packet
  • Simulation of the Gaussian Wave Packet
  • Q&A: Initial Position, Complex Phase Factor, and Dissociation of Modes
  • Introduction to the Quantum Harmonic Oscillator
  • Setting Up the Time-Independent Schrödinger Equation
  • Ground State Solution and the Gaussian Wave Function
  • Solving for All Energy Eigenstates via Power Series
  • Quantization of Energy and the Harmonic Oscillator Spectrum
  • Hermite Polynomials and the Wave Functions
  • Visualization and Classical Correspondence
  • Variable Substitution & Simplified Schrödinger Equation
  • Ansatz and Second Differentiation of Wave Function
  • Power Series Solution and Recursion Relation
  • Introduction to Bound States
  • General Framework: Potentials Tending to Zero
  • Scattering States vs. Bound States: Asymptotic Solutions
  • Physical Implications and Reference Energy
  • The Finite Square Well: Setup and Solutions
  • Matching Conditions and Transcendental Equations
  • Graphical Solution and Number of Bound States
  • Introduction to Scattering Theory
  • Classical vs. Quantum Scattering Setup
  • Gaussian Wave Packet Simulation
  • Time-Independent Approach and Beam Interpretation
  • Square Barrier Example: Setting Up the Solution
  • Solving the Schrödinger Equation Inside the Barrier
  • Reflection and Transmission Probabilities & Quantum Tunneling
  • Transmission Probability Simulation and Inverse Scattering
  • Problem Setup: Quantum Scattering
  • General Solution in Three Regions
  • Boundary Conditions: Four Matching Equations
  • Algebraic Solution for Coefficients
  • Transmission Probability and Final Result
  • Introduction & Motivation
  • Overview of the Formalism of Quantum Mechanics
  • States as Vectors
  • Mathematical Definition of Vectors and the Principle of Superposition
  • Qubits: Two-State Quantum Systems
  • Connection Between Qubits and Wave Functions
  • Inner Products and Hilbert Space
  • Introduction to Observables
  • Classical Observables
  • Quantum Observables as Operators
  • Operators in Finite-Dimensional Hilbert Spaces (Matrices)
  • Operators on Wave Functions: Position and Momentum
  • Angular Momentum and Hamiltonian Operators
  • Q&A: Clarifications on Operators and Observables
  • Introduction to Eigenvalues and Eigenstates
  • The Eigenvalue Equation
  • Matrix Eigenvalue Example
  • Physical Meaning: Measurement Outcomes Are Eigenvalues
  • Q&A: States, Operators, and Uncertainty Principle Preview
  • Eigenstates and Eigenvalues of the Momentum Operator
  • Eigenstates and Eigenvalues of the Position Operator
  • Q&A: Dirac Delta Function and Shared Eigenstates
  • Review and Lecture Overview
  • Definition of the Adjoint Operator
  • Hermitian Operators: Definition and Physical Motivation
  • Adjoint for Finite-Dimensional Matrices
  • Hermiticity of Position and Momentum Operators
  • Key Properties of Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates
  • Orthonormality, Degeneracy, and Spectral Terminology
  • Completeness of the Eigenbasis and Fourier Decomposition
  • Closing Questions and Summary
  • Recap of Quantum Mechanics Facets
  • Time Evolution and the Schrödinger Equation
  • Time Evolution in Energy Eigenstates
  • Time Evolution of a General Superposition
  • Comparison with Classical Mechanics
  • Simulation: Particle on a Circle and Quantum Revivals
  • Q&A: Randomness, Probability, and Classical Analogy
  • Introduction to Measurement and Probability
  • Eigenvalues as Possible Measurement Results
  • The Born Rule
  • Wave Function Collapse
  • Q&A: Measurement, Collapse, and Interpretation
  • Expectation Values
  • Introduction to Qubits
  • Qubit State Space and Vector Representation
  • Hermitian Matrices and the Pauli Matrices
  • Properties and Significance of Pauli Matrices
  • Eigenvalues and Eigenvectors of Pauli Matrices
  • Physical Interpretation: Electron Spin
  • Measuring Spin in the Z Direction and the Born Rule
  • Quantum vs. Classical Angular Momentum
  • Incompatible Observables and Measurement Disturbance
  • Q&A and Connection to Heisenberg Uncertainty
  • Recap: Incompatible Observables & Lecture Roadmap
  • Commutators and Canonical Commutation Relations
  • Quantum Uncertainty (Variance) Definition
  • Heisenberg Uncertainty Principle: Statement, Proof & Physical Intuition
  • Ehrenfest's Theorem: Deriving Classical Mechanics from Quantum Mechanics
  • Commuting Observables and Shared Eigenstates
  • Statement of the Commutator Eigenstates Theorem
  • Setup: Assumptions for the Converse Proof
  • Proof: Commuting Operators Share Eigenstates
  • Remark on the Degenerate Case
  • Course Overview and Goals
  • Quantum Particle in 3D and Central Potential
  • Strategy: Angular Momentum as the Key to the 3D Problem
  • Angular Momentum Operators: Definition and Components
  • Commutation Relations of Angular Momentum
  • Total Angular Momentum L² and Simultaneous Eigenfunctions
  • Lecture Overview & Goals
  • Spherical Polar Coordinates
  • Angular Momentum Operators in Polar Coordinates
  • Eigenfunctions of L₃ and Quantization of m
  • Eigenfunctions of L² and Spherical Harmonics Defined
  • Visualizing Spherical Harmonics
  • Completeness of Spherical Harmonics & Outlook
  • Course Roadmap & Angular Momentum Review
  • Lecture Plan for the Radial Problem
  • Commutation Relations: H, L², and Lᵢ
  • Simultaneous Eigenstates and Their Quantum Numbers
  • Laplacian in Spherical Coordinates and the Angular Momentum Term
  • Derivation of the Radial Schrödinger Equation
  • Degeneracy of Energy Eigenstates
  • Q&A: Degeneracy and Hydrogen Orbitals
  • Lecture Roadmap
  • The Hydrogen Atom and Coulomb Force
  • The Bohr Model: Early Attempt and Its Flaws
  • Quantum Numbers and the Radial Schrödinger Equation
  • Simplifying the Radial Equation: Rescaling Variables
  • Asymptotic Analysis of the Radial Wave Function
  • Recap: Radial Wave Function Structure
  • Differential Equation for the Polynomial f(r)
  • Power Series Solution and Recurrence Relation
  • Termination Condition and Quantization of Energy
  • Physical Interpretation: Bohr Radius and Fine Structure Constant
  • Hydrogen Spectrum, Rydberg Formula, and Degeneracy
  • Foundation of the Schrödinger Equation: Noether's Theorem
  • Spectral Lines and Photon Emission
  • Visualizing Hydrogen Wave Functions
  • Outlook: Beyond Hydrogen — Future Directions in Quantum Mechanics
  • Radial Schrödinger Equation for Hydrogen
  • Ansatz and First Differentiation
  • Second Differentiation and Substitution into the Radial Equation
  • Reduced Differential Equation for F
  • Recurrence Relation for Series Coefficients

Syllabus

  1. Course Lectures

    1. The Wavefunction
    2. Normalisation and Superposition
    3. The Schrödinger Equation
    4. Collapse and the Double Slit Experiment
    5. A Particle in 1D
    6. The Gaussian Wavepacket
    7. The Harmonic Oscillator
    8. The Harmonic Oscillator · Bonus
    9. Bound States
    10. Scattering
    11. Scattering · Bonus
    12. States
    13. Observables
    14. Eigenstates and Eigenvalues
    15. Hermitian Operators
    16. Time Evolution
    17. Measurement
    18. The Qubit
    19. Commutation Relations
    20. Commutation Relations · Bonus
    21. Angular Momentum
    22. Spherical Harmonics
    23. Solving the 3D Schrödinger Equation
    24. The Hydrogen Atom
    25. More Hydrogen Atom
    26. More Hydrogen Atom · Bonus