Quantum Mechanics
The Theoretical Minimum
A rigorous, interactive study platform covering five foundational lectures — from the qubit and spin measurements to the Schrödinger equation and Ehrenfest's theorem. Theory, derivations, and exercises with full solutions.
$i\hbar\,\dfrac{d}{dt}|\psi\rangle = H\,|\psi\rangle$
Ten Complete Lectures
From the qubit and Stern-Gerlach to entanglement, Bell's theorem, and continuous systems — a complete course.
Classical vs quantum states, the qubit, Stern-Gerlach experiments, wave function collapse, and complex vector spaces.
Invasive measurements, spin states in $\mathbb{C}^2$, the Bloch sphere, inner products, Born rule, and the failure of classical logic.
Linear operators, eigenvectors and eigenvalues, Hermitian operators, the four postulates, and derivation of the Pauli matrices.
Unitary operators, the Hamiltonian, derivation of Schrödinger's equation, expectation values, commutators, and Poisson brackets.
Negative energy states, the Dirac sea, wave functions, rigorous proof of the Heisenberg Uncertainty Principle, and classical emergence.
Compatible vs incompatible observables, stationary states, Larmor precession, Rabi oscillations, and the Robertson uncertainty relation.
Composite systems, tensor products, the singlet and triplet states, EPR correlations $\langle\sigma_A\sigma_B\rangle = -\hat{n}\cdot\hat{m}$, and the no-signaling theorem.
Pure vs mixed states, reduced density matrices, von Neumann entropy $S=-\text{Tr}(\rho\log\rho)$, decoherence, and the measurement chain.
Local hidden variables, CHSH inequality proof, quantum violation $S=2\sqrt{2}$, position eigenstates, the Dirac delta, and momentum eigenstates.
Fourier transforms, momentum operator derivation, free particle Schrödinger equation, group vs phase velocity, wave packet spreading, and Ehrenfest.
Core Concepts
Understand why quantum states live in $\mathbb{C}^N$ and how inner products connect algebra to probability.
Derive $\sigma_x, \sigma_y, \sigma_z$ from first principles and understand their commutation algebra.
Derive $i\hbar\dot{|\psi\rangle}=H|\psi\rangle$ from unitarity and solve it via energy eigenstates.
$P(\lambda_i) = |\langle i | a \rangle|^2$ — the fundamental link between amplitudes and measurable outcomes.
Prove $\Delta x \cdot \Delta p \geq \hbar/2$ rigorously from the Cauchy-Schwarz inequality.
Understand why $\langle x\rangle$ and $\langle p\rangle$ obey Newton's laws — the classical world emerges from quantum mechanics.