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.

Compare classical barriers with quantum tunneling using a rectangular barrier, show energy conservation, and derive a quantum tunneling probability via the Schrödinger equation that depends on width L.

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.

Derive the third-zone solution to the Schrödinger equation with zero potential, compare positive and negative energies, and introduce constants that determine the transmission coefficient for barrier penetration.

Impose continuity of the wave function and its derivative at barrier boundaries, solve for the coefficients, and define the transmission coefficient as the magnitude squared of the transmitted-to-incident amplitude ratio.

Trace the shift from Rutherford’s planetary model to quantum mechanics, showing classical collapse, discrete emission spectra, and Bohr and Schrödinger’s roles in hydrogen-like energy levels.

We separate the Schrodinger equation into a time-dependent part and a space-dependent part, yielding a constant energy eigenvalue and the stationary equation; solutions take the form F(x) e^{-iEt/ħ}.

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

Separate the time-independent Schrödinger equation in spherical coordinates. Write the wavefunction as Y(θ, φ) times a radial function, yielding angular and radial equations related to the energy spectrum.

Explore the radial Schrödinger equation and how L^2 reveals angular momentum and spherical harmonics. Learn to derive discrete energy levels for bound states via variable changes and negative energy.

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.

Substitute the expression into the differential equation, neglect small terms at large era, and show lambda equals minus energy quantized by n, yielding the discrete energy spectrum from Schrödinger equation.

Revises classical field theory by formulating the action from the lagrangian density, depending on a field and its derivatives, and derives the field equations via boundary-vanishing variations.

Set natural units and derive the time evolution of the lagrangian and hamiltonian densities for a scalar field, introducing momentum density pi and energy conservation, guiding toward the Klein-Gordon equation.

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†.

This lecture rewrites the hamiltonian to obtain the spectrum and shows that the annihilation operator a_k lowers the energy by omega, while its creation counterpart raises it by omega.

Define the ground state or vacuum for a field, show annihilation operators lower energy and creation operators generate quanta, and construct one- and two-quanta states with associated energy and momentum.

Define the number operator for bosons with creation and annihilation operators, apply normal ordering to the hamiltonian and momentum, and count particles in states generated from the vacuum.

Apply the no operator to a two-particle state and show it yields two times the two-particle state, using the commutation relations and the delta terms.

The lecture shows space-like events cannot influence each other by requiring the field commutator to vanish under the Lorenz metric, using positive and negative frequency parts and delta propagators.

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.

Using the Dirac delta and its invariance, the lecture shows the commutator equals delta(x - y) and vanishes for space-like events, preserving causality in quantum theory.

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.

Explore Wick's theorem by expanding time-ordered and normal-ordered products, performing contractions, and deriving vacuum expectations in quantum fields, with links to Feynman diagrams.

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.

Derive a general expression for the commutator of fields in the interaction picture for LSZ formula part 2, linking vacuum expectation to a delta function and a mass integral.

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.

Explain lsz formula part 4 by linking in and out states with delta functions, exploring single and multi-particle alpha states, and introducing endpoint correlation functions for transition amplitudes.

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.

Derive the LSZ formula in the context of in and out Hilbert spaces, the S-matrix, and asymptotic states using creation and annihilation operators and time-ordered phi fields.

Derive the LSZ formula part 7 by analyzing limits to plus or minus infinity and time-ordered products, linking vacuum expectations to the S-matrix's connected part.