Syllabus

Textbook Used: 1) Introduction to Quantum Mechanics by David Griffiths and Darrell F. Schroeter 2) Modern Quantum Mechanics by J.J. Sakurai and Jim Napolitano 3) Quantum Field Theory for the Gifted Amateur by Tome Lancaster and Stephen Blundell.

Lecture 2 Time-Independent Perturbation Theory:  A bump in the Infinite Square Well
Since we learned how to find the 1st order correction in time-independent perturbation theory, let us apply to a simple example.

Lecture 3 Time-Independent Perturbation Theory: 2nd Order Correction

What if you are not satisfied with the 1st order correction and you want an approximate solution more closer to the true answer? We will find the formula for the second order correction.

Lecture 4 Matrix Representation Review and Degenerate Perturbation Theory

When the unperturbed part of the Hamiltonian is degenerate, the perturbed part of the Hamiltonian may lift that degeneracy. We will learn degenerate perturbation theory. But firrst, we will begin by reviewing how to represent quantum operators as matrices. You will discover that the secret to simplifying these problems lies in choosing the right basis—specifically, identifying the 'Good States' that diagonalize the perturbation and make an otherwise daunting calculation manageable. 

Lecture 5 Degenerate Perturbation Theory and Review of Angular momentum

We will wrap up degenerate perturbation theory and shift gears to a new topic. We will first review angular momentum. This review will serve as the essential foundation for our next major topic: the addition of angular momenta 

Lecture 6 Ladder Operators in Angular Momentum

Just as they were for the harmonic oscillator, ladder operators are incredibly powerful tools for solving angular momentum problems. 

Lecture 7 Addition of Angular Momenta

When dealing with two types of angular momentum, such as orbital (L) and spin (S), we are faced with two competing ways to describe the system. Can we simultaneously measure the individual components (Lz, Sz) and the total angular momentum (J^2, Jz)? We will explore this 'commutation conflict' and learn how to transition between these descriptions using a change of basis—moving from the uncoupled to the coupled representation. 

Lecture 8 Addition of Angular Momenta 2

When you express the uncoupled states in terms of a sum of coupled states or vice versa, you will need to know the coefficients for each sum. We will learn how to find these coefficicients.

Lecture 9 Addition of Angular Momenta 3 and Identical Particles

After we wrap up the addition of angular momenta, we’ll shift our focus to one of the most significant leaps in the course: transitioning from single-particle systems to multi-particle quantum mechanics. 

Lecture 10 Identical Particles 2

When transitioning to multi-particle systems, your first critical consideration is whether the particles are distinguishable or identical (indistinguishable). If they are identical, the laws of nature branch into two distinct paths: fermions or bosons.

Lecture 11 A taste of 2nd Quantization

This session introduces the fundamentals of Second Quantization. We will discuss first that the n! complexity of multi-particle states necessitates a more efficient way to describe quantum mechnanics. We will then develop the second quantization formalism, highlighting its conceptual parallels with the harmonic oscillator’s ladder operators to make the transition to graduate-level topic more accessible. 

Lecture 12 Pauli exclusion principle in 2nd quantization

In second quantization, the Pauli exclusion principle for fermions is not enforced by manually antisymmetrizing wavefunctions, but is instead "baked" into the fundamental algebra of the operators. 

Lecture 13 Change of Basis and Operators in 2nd Quantization

In second quantization we learnd so far, the key spirit is delegating the what the state vectors use to do to creation and annihilation operators. This transition raises important questions: how do we handle a change of basis in this new framework, and how do we represent physical observables that were already defined as operators in the first quantization? Lastly, I will introduce how physicists model Hamiltonians by introducing the Hubbard model.

Lecture 14 Operators in 2nd Quantization: Interaction Terms

In the lecture 13, we introduced one-body operators in 2nd quantization language.Today, we advance to two-body operators, allowing us to model the complex interactions between particles.

Lecture 15 BCS Hamiltonian

At this point, you might be wondering: 'What are the real-world applications of second quantization?' To answer that, I will introduce a compact version of the Bardeen-Cooper-Schrieffer (BCS) model. This theory explains superconductivity, and we will use our new toolkit to demonstrate exactly why there is an energy gap in superconductors. 

Lecture 16 Introduction to Symmetry in Quantum Mechanics

Having symmetry is not just about beauty. We will first define what do you mean by symmetric in quantum mechanics and why do we care about having a symmetry in systems. 

Lecture 17 Parity Operators and the meaning of conservation in quantum mechanics

We will introduce the parity operator to explore the physical implications of inversion symmetry. Furthermore, we will examine what it truly means for a quantity to be 'conserved' in a quantum system; since observables like momentum are represented by operators, we must formalize conservation as a commutation relation with the Hamiltonian.

Lecture 18 Parity Selection Rules, Pseudo vector and a true vector

Knowing an operator's behavior under parity symmetry is a powerful time-saver; it allows you to spot which matrix elements are zero without performing a single integral. We will categorize operators as vectors or pseudovectors to determine how they dictate the parity of the states they connect. 

Lecture 19 Rotational symmetry and Degeneracy

We will explore how rotational symmetry leads to the conservation of angular momentum. Beyond providing conservation laws and defining the generators of Hermitian operators, symmetry has a third profound implication: it is the fundamental origin of degenerate states. 

Building on lecture 18, we will conduct a more thorough investigation into the distinctions between true vectors, pseudovectors, and true scalars and pseudo scalars. We will see that these classifications are not arbitrary, but are rigorously defined by the parity and the rotational operators. 

Lecture 20 Rotation Selection Rules, Schrodinger Picture vs Heisenberg Picture

Much like the parity operator, rotation operators are indispensable for identifying vanishing matrix elements before you even begin a calculation. I will briefly summarize these results.

Up to this point, we have operated within the Schrödinger picture, where the time dependence is carried by the quantum state or wavefunction. However, we can make an alternative choice: the Heisenberg picture, which shifts the time dependence onto the operators.

Lecture 21 Hamiltonian that results in Lorentz force classically, Landau levels quantum mechanically

In classical mechanics, what is the specific Hamiltonian that recovers the Lorentz force law when a uniform magnetic field is applied to a moving charged particle? After finding that Hamiltonian, we will promote this Hamiltonian to a quantum mechanical operator and solve the time-independent Schrödinger equation. Surprisingly, the problem maps directly onto the harmonic oscillator formalism, leading us to the quantized energy states we call Landau levels. 

Lecture 22 Quantum Dynamics: Interaction Picture

We now turn our attention to Quantum Dynamics where the Hamiltonian itself carries an explicit time dependence. This is interesting because we can finally deal with transitions from one state to another upon a time-varying perturbation. To study such cases, I will first introduce you to the Interaction (Dirac) Picture. 

Lecture 23 Time-Dependent Perturbation Theory and Dyson Series

In the previous lecture, we learned that the time-varying Hamiltonian influences the expansion coefficients c(t) of the states. This lecture details the calculation of the expansion coefficients c(t) using time-dependent perturbation theory. We will develop the Dyson series as an iterative solution to the Schrödinger equation in the interaction picture, providing a pathway to determine first, second, and higher-order perturbative corrections. 

Lecture 24 Transition Probability, Sudden Approximation, and Adiabatic Approximation

We’ll begin by addressing the question of 'when' a transition is likely to occur in a driven system. Next, we will explore the two extremes of quantum dynamics: the sudden limit and the adiabatic limit.

Lecture 25 Adiabatic Approximation and the Berry Phase

When a Hamiltonian evolves at an adiabatic (extremely slow) rate, one might naively assume the system simply behaves as if the Hamiltonian were static. However, this slow evolution leads to a profound result: the states acquire a non-trivial phase factor beyond the usual dynamical phase. 

Lecture 26 Berry Phase, Relation to Topology, and Aharonov-Bohm Effect

We will more rigorously show that an anomalous velocity emerges because of a Berry curvature. We will then extend the Berry phase concept to topology, and briefly introduce how modern quantum materials scientists are viewing materials. Lastly, I will close this lecture series by introducing you the Aharonov-Bohm effect.

Course feedback from students