Math Intuition for Quantum Mechanics & Quantum Field Theory

Derive the quantum analog of classical motion by formulating the Schrodinger equation, defining the energy operator as p̂^2/2m, incorporating potential energy, and describing state evolution for one or many particles.

Derive the Schrödinger equation solution for a one-particle system using a complex exponential with momentum and energy, and show how eigenstates form a normalized superposition over discrete or continuous spectra.

Derive the uncertainty principle from the operator framework by defining delta A and delta B, linking variances to mean values, and showing sigma_A sigma_B is bounded by the commutator [A,B].

Derives the Heisenberg uncertainty principle for position and momentum, showing Delta X Delta B >= hbar/2 via the X–B commutator.

Explore how unitary operators drive time evolution in quantum systems, derive the Schrödinger equation from U(t,t0)=exp(-i H (t-t0)/ħ), and confirm unitarity.

Solve the one-dimensional Schrödinger equation for a rectangular barrier using separation of variables, deriving time and space equations and applying boundary conditions across x<0, 0<x<L, and x>L.

Solve the time-dependent and time-independent Schrödinger equations by separating variables, obtaining exponential solutions and wave propagation for a particle near a barrier, including incident and reflected waves for positive energies.

Apply separation of variables to solve the Schrödinger equation for a barrier, deriving time and space components and examining energy conditions for passage through the barrier.

Separate the Schrödinger equation into time and space parts to derive the time-independent equation, energy eigenvalues, and use the Laplacian in spherical coordinates.

Explore solving the radial Schrödinger equation via separation of variables, deriving the radial and angular momentum eigenvalue equations, and analyzing large-radius behavior to infer the discrete energy spectrum.

Examine a complex scalar field with a phi and phi star lagrangian, its potential, and a global phase symmetry yielding the Noether current, plus a general transformation for action invariance.

Explain canonical quantization of a classical field, promoting fields to operators, deriving position–momentum commutators, and formulating the Klein-Gordon plane-wave expansion with creation and annihilation operators.

Quantize a classical field by deriving mode expansions for phi and pi, express a_k and a_k^\dagger, evaluate commutators, and relate to the hamiltonian.

Derive the commutator between a(k) and a†(k′) using phi and pi, obtaining 2ω (2π)^3 δ^3(k−k′). Then express the Klein–Gordon Hamiltonian in terms of a and a†.

Explore the causality of space-like events by analyzing commutators, delta of x minus y, and Lorentz-invariant momentum integrals to establish delta(x−y) invariance.

Explore vacuum expectation values, propagators, and time ordering in quantum fields. Derive Green's functions and the Feynman propagator from field operators and contour integration.

Explore the complex Klein-Gordon field: derive equations of motion, canonical momenta and commutation relations, expand in modes with A and B operators, and obtain the conserved charge from U(1) symmetry.

Explore how a complex scalar field interacts with an electromagnetic field, deriving a gauge-invariant lagrangian, introducing the covariant derivative, and illustrating invariance under local phase transformations.

Explore the covariant formulation of interacting classical field theory, deriving the Lagrangian, equations of motion, and how interaction terms modify Maxwell's equations and field dynamics.

Apply the interaction picture to decompose the full Hamiltonian into free and interacting parts, derive operator evolution, and study a real scalar field with phi^4 interactions.

Learn how normal-ordered and time-ordered products relate, derive the S-matrix in the interaction picture, and set up Wick's theorem for future expansion.

The lecture applies weeks theorem to define scattering cross sections for two-particle processes, linking incident and target densities to event counts, and introduces the transition amplitude as a matrix element.

In the lab frame, derive the two-particle scattering cross section as a ratio of transition rate to target density and flux, using the covariant Feynman amplitude.

The lecture analyzes the LSZ formula part 3, examining the no-interaction limit and how the constant Z relates to normalization and delta functions in quantum field theory.

Substitute 132 and 130 to rewrite the integrals and evaluate time limits from minus to plus infinity, connecting the field to Z to the minus one half.