Notes

Selected topics

• Linear spectra of a two-level system coupled to a harmonic bath

  Derivation of the linear absorption lineshape for a single two-level system linearly coupled to a harmonic bath, including Drude–Lorentz, underdamped Brownian, and combined spectral densities.

• Polaron transform of the Frenkel exciton model

  Term-by-term polaron transformation of an N-site Frenkel exciton Hamiltonian with site-local harmonic baths, showing how the linear system–bath coupling is removed and how the excitonic couplings are dressed by collective bath momentum operators.

• Variational polaron transform

  Derivation of the variational polaron transformation for the Frenkel exciton model with site-local harmonic baths, including the dressed couplings and self-consistent conditions for the variational parameters.

• Schrieffer–Wolff transform

  Derivation of the dispersive Hamiltonian for a qubit coupled to two harmonic modes via Tavis–Cummings interaction, using the Schrieffer–Wolff transformation.

• Discrete Variable Representation (Fourier DVR) in 1D

  Construction of a Fourier-based DVR for a 1D particle in a periodic box, including the analytic kinetic-energy matrix, diagonal potential, and the resulting DVR Hamiltonian eigenvalue problem.

• Fast Fourier Transform

  Operator-based derivation of the (radix-2) FFT using an even–odd decomposition of the Hilbert space, twiddle-factor operators, and a Hadamard on the first qubit.

• Chebyshev polynomial propagator

  An introduction to chebyshev propagator in quantum dynamics. We discuss how the need for efficient time-evolution methods arises, and how the chebyshev propagator becomes a naturally efficient and stable choice for large Hamiltonians with an approximately known spectral radius.

• Basics of Electronic Structure

  Notes on the birth of electronic structure: Born–Oppenheimer approximation, antisymmetry and spin, Hartree products, Slater determinants, Hartree–Fock (SCF), configuration interaction, and Coulomb/exchange integrals.