Lectures
Lecture notes
PART I: Introduction and Background
Introduction to the course.
A primer on classical wave phenomen
An intr
PART II: The Schrodinger Equation, the Wavefunction, and their Interpretation
Heruistic
Observables in quantum physics, mean values of observables.
Applications of
PART III: Quantum Mechanics in the Dirac Notation
A math primer for quantum physics, vector
spaces and Hilbert spaces, operators, inner and exterior products,
adjoint operators, Hermitian operators, eigenvalues and
eigenvectors, basis sets, orthonormal basis sets, representations,
commutation relations, Hilbert space of square integrable functions,
position basis and plane wave basis.
Quantum physics in Dirac notation, bras and kets,
Schrodinger equation in Dirac notation, observables, operators, and
quantum states in the Dirac notation
Time evolution in quantum physics, stationary states, measurement
and state collapse.
Commutation relations, measurements, disturbances, and Heisenberg
uncertainty relations in quantum mechanics, various forms of the
Heisenberg uncertainty relation, simultaneous measurements of
commuting and non-commuting observables.
The formal postulates of quantum physics.
The finite potential well, bound st
Simple harmonic oscillator (SHO) in classical and quantum physics, electromagnetic modes in an optical cavity, superconducting LC ciruits, quantization of a SHO, creation and destruction operators, operator algebra, energy eigenstates and eigenvalues, number states, quantization of an optical cavity mode and photons, quantization of a superconducting LC ciruit, introduction to vacuum fluctuations.
Completely commuting set of observables (CSCOs), time-energy uncertainty relation in quantum physics, quasi-bound states and their lifetimes.
PART IV: Two-Level Systems (TLS), Qubits, Entanglement, and Decoherence
Spin in quantum physics, spin operators, quantum larmor precession, Zeeman splitting, the bit and the qubit, spin 1/2 qubit as a quantum two-level system (TLS), spin operators and Pauli matrices, spin qubit in a time-dependent magnetic field, spin Rabi oscillations, introduction to single-qubit quantum gates.
Composite systems and quantum entanglement, joint Hilbert spaces and tensor products, entangled and unentangled states of bipartite systems, entangled states and the EPR paradox, absence of local realism in quantum physics.
Quantum decoherence, entanglement, and the conscious observer, the double-split experiment, decoherence via entanglement, quantum which-path measurements by conscious and unconscious observers, the Schrodinger's cat paradox.
Electron-photon interaction Hamiltonian, classical and quantum descriptions, electron in a finite potential well interacting with light as a two-level system (TLS), mapping to a spin Hamiltonian, Rabi Oscillations.
PART V: Perturbation Approaches in Quantum Physics
Time-dependent perturbation theory and Fermi's Golden Rule, optical transitions in energy bands, stimulated emission and absorption, spontaneous emission.
Time-independent perturbation theory, first order and second order changes in energy eigenstates and energy eigenvalues.
Time-independent perturbation approach using a finite basis expansion and sub-matrix diagonalization.
PART VI: Quantum Information Processing, Quantum Gates, Quantum Circuits, and Quantum Computation
Interactions, entanglement, and two-qubit quantum gates. Coupled spins, coupled superconducting LC circuits, implementation of control-X, control-Y, and control-Z gates.
Quantum bits and classical bits, single-qubit quantum gates (X,Y,Z,H,S, and T gates), two-qubit quantum gates (control-X, control-Y, control-Z, etc), linearity, unitarity, and reversability of quantum gates, quantum circuits, Bell circuit, swap circuit. Quantum information processing, quantum memory search and Grover’s algorithm, quantum no-cloning theorem, quantum teleportation, quantum superdense coding, quantum parallelism in computations, the Deutsch algorithm, the Bernstein-Vazirani algorithm.
PART VII: Orbital and Spin Angular Momentum, Spin and Statistics, Fermions and Bosons, Identical Paticles and Many-Particle States, Hydrogen Atom, Energy Bands in Solids
Orbital angular momentum, commutation relations, eigenvalues and eigenstates, spherical harmonics, spin angular momentum and its commutation relations, eigenvalues and eigenstates, Fermions and Bosons, spin-statistics theorem and Pauli's exclusion principle, spinor states and wavefunctions.
Many-particle wavefunctions, spin-statistics theorem, Pauli's exclusion principle, identical particles.
Hydrogen atom problem in relative and center-of-mass coordinates, angular momentum and spherical harmonics, CSCO, eigenvalues and eigenstates, atomic orbitals.
Electrons in periodic potentials, Bragg scattering, breakdown of perturbation theory, finite basis expansion, opening of energy gaps, introduction to energy bands in solids.
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