Venue: TBA
Class Timings: TBA
First Meeting: TBA
Course Description:
1. Mathematical Structure of Quantum Mechanics
Hilbert spaces, linear operators, eigenvalues and eigenvectors. Commutators, uncertainty relations. Projection operators and measurement postulates. Density matrices and mixed states. Unitary evolution and time evolution operators.
2. Approximate Methods
Time-independent perturbation theory (non-degenerate and degenerate). Time-dependent perturbation theory. Fermi’s golden rule. The variational principle. The WKB approximation.
3. Path Integral Formulation
Feynman's path integral for the propagator. Path integral for the free particle and harmonic oscillator. Connection with classical action and stationary phase. Semiclassical methods.
4. Scattering Theory (Nonrelativistic)
Scattering amplitude and cross section. Born approximation. Partial wave analysis. Phase shifts. Lippmann-Schwinger equation.
5. Quantum Computing and Information Theory
Qubits, quantum gates, and quantum circuits. Entanglement and Bell states. Basic quantum algorithms (Grover’s algorithm and Shor's algorithm). Quantum teleportation. Basics of quantum error correction. Von Neumann and Relative entropy. Entropy inequalities.
6. Quantum Many-Body Physics (Foundations)
Tensor product structure and entanglement in many-body systems. Spin chains: Heisenberg, Ising, and XY models in 1+1 dimensions. Ground states, excitations, and symmetries.
Course Outcome:
1. Demonstrate a deep understanding of the mathematical structure of quantum mechanics, including Hilbert spaces, operators, and the formalism of states and observables.
2. Apply approximate methods such as time-independent and time-dependent perturbation theory, and the WKB method to solve complex quantum systems.
3. Formulate and interpret quantum mechanics using the path integral formalism, and apply it to systems such as the free particle and harmonic oscillator and to perturbation theory of anharmonic oscillators.
4. Understand nonrelativistic scattering theory, including concepts like cross sections and phase shifts. Understand how scattering data and bound state properties are related by analytic continuation.
5. Explain the foundational principles of quantum computing and quantum information theory, including basic quantum algorithms.
6. Describe key concepts in quantum many-body physics, with a focus on simple spin chain models in 1+1 dimensions.
- Teacher: Raghu Mahajan