Course Structure
All Lectures
All ten lectures from Susskind's Theoretical Minimum — from the qubit to Bell's theorem and continuous systems.
LECTURE 01
The Language of Quantum Mechanics
Classical vs quantum states
The qubit and spin
Stern-Gerlach experiments
Wave function collapse
Complex vector spaces & $\mathbb{C}^2$
Born rule introduction
LECTURE 02
Measurement & Quantum Logic
Invasive measurements
Spin-½ particle in $\mathbb{C}^2$
All 6 spin eigenstates
Complex inner product
Orthogonality and distinguishability
Non-commutative quantum logic
LECTURE 03
Operators & the Four Postulates
Dirac bra-ket notation
Completeness relation
Linear operators as matrices
Hermitian operators
The four quantum postulates
Derivation of Pauli matrices
LECTURE 04
Time Evolution & the Hamiltonian
Unitary time evolution operator
Conservation of information
The Hamiltonian from unitarity
Schrödinger equation derivation
Expectation values $\langle L\rangle = \langle\psi|L|\psi\rangle$
Commutators & Poisson brackets
LECTURE 10
Dirac Sea, Uncertainty & Ehrenfest
Negative energy and the Dirac sea
Position & momentum wave functions
Fourier transform duality
Proof of Heisenberg $\Delta x \Delta p \geq \hbar/2$
Schrödinger equation for particles
Ehrenfest's theorem
LECTURE 05
Uncertainty, Energy Eigenstates & Spin Precession
Compatible vs incompatible observables
Stationary states and beat frequencies
Spin Hamiltonian $H=\frac{\omega}{2}\sigma_z$
Larmor precession and Rabi oscillations
Full Pauli algebra: $\sigma_i\sigma_j=\delta_{ij}\mathbf{1}+i\varepsilon_{ijk}\sigma_k$
Robertson uncertainty relation
LECTURE 06
Entanglement, Tensor Products & EPR Correlations
Tensor product $\mathcal{H}_A\otimes\mathcal{H}_B$
Entangled vs product states (det test)
Singlet and triplet states
EPR anti-correlations $\langle\sigma_A\sigma_B\rangle=-\hat{n}\cdot\hat{m}$
No-signaling theorem
Wave function collapse in entangled systems
LECTURE 07
Density Matrices & Entanglement Entropy
Pure vs mixed states: $\rho=|\psi\rangle\langle\psi|$
Properties: $\text{Tr}(\rho)=1$, $\text{Tr}(\rho^2)\leq 1$
Partial trace and reduced density matrix
Von Neumann entropy $S=-\text{Tr}(\rho\log\rho)$
Decoherence and measurement chain
$S_A=S_B$ for pure bipartite states
LECTURE 08
Bell's Theorem, Locality & Continuous States
Local hidden variable hypothesis
CHSH inequality proof $|S|\leq 2$
Quantum violation $S_{QM}=2\sqrt{2}$
Continuum states and Dirac delta
Position eigenstates $\langle x|x'\rangle=\delta(x-x')$
Momentum eigenstates as plane waves
LECTURE 09
Fourier Analysis, Particle Mechanics & Wave Packets
Fourier transform duality position/momentum
Momentum operator $\hat{p}=-i\hbar\,d/dx$
Free particle Schrödinger equation
Dispersion: $\omega=\hbar k^2/2m$
Group vs phase velocity
Wave packet spreading and Ehrenfest
Quick Reference
Essential Equations
Born Rule
$$P(\lambda_i) = |\langle i | \psi \rangle|^2$$
Schrödinger Equation
$$i\hbar\,\frac{d}{dt}|\psi\rangle = H|\psi\rangle$$
Expectation Value
$$\langle L \rangle = \langle\psi|L|\psi\rangle$$
Uncertainty Principle
$$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$
Completeness Relation
$$\sum_i |i\rangle\langle i| = \mathbf{1}$$
Canonical Commutator
$$[\hat{x}, \hat{p}] = i\hbar$$