Topics#

Key Concepts#

• Fermat’s principle (stationary optical path)

• Optical path length \(L = \int n\,\mathrm{d}s\)

• Huygens’ principle (secondary wavelets, wavefront)

• Phase \(\Phi = k_0 L\) and why refractive index enters

• De Broglie relations: \(\lambda = h/p\), \(E = \hbar\omega\)

• The wavefunction \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\)

• Path integral: sum over all paths of \(\mathrm{e}^{\mathrm{i}S/\hbar}\)

Learning Objectives#

• Derive the laws of reflection and refraction from Fermat’s principle

• Re-derive these laws using the Huygens wavefront construction

• Explain why frequency is shared across media but wavelength is not

• State the identification Action = Phase and write \(\psi = A\,\mathrm{e}^{\mathrm{i}S/\hbar}\)

• Explain how the classical limit emerges from the path integral via stationary phase

Project: Path Integral Monte Carlo for Quantum Many-Body Systems#

Objective: Implement Path Integral Monte Carlo (PIMC) to compute finite-temperature properties of quantum systems and explore the connection between imaginary-time path integrals and classical statistical mechanics of “quantum worldlines.”

Background: The path integral formulation reveals a profound connection: finite-temperature quantum mechanics \(\hat{\rho} = \mathrm{e}^{-\beta H}\) is equivalent to sampling classical polymer chains (worldlines) in imaginary time. This insight is the foundation of ab initio computational methods for quantum materials, including electronic structure of real solids at finite temperature. PIMC is particularly powerful for bosons (no sign problem) and low-density systems, making it a workhorse method in quantum chemistry and materials science.

Suggested Approach:

• Literature Survey: Study the path integral formulation and PIMC methods. Read David Tong’s lecture notes on path integrals, or “Quantum Monte Carlo” (Ceperley & Alder, Rev. Mod. Phys. 1980). Understand the connection between imaginary-time path integrals and statistical mechanics of classical polymers.

• Start with a simple model: the harmonic oscillator or a chain of coupled oscillators at finite temperature.

• Implement a primitive PIMC algorithm: discretize imaginary time into \(M\) slices, represent each configuration as a path \((x_0, x_1, \ldots, x_M)\) on the imaginary-time lattice, and use Metropolis sampling to generate paths weighted by \(\mathrm{e}^{-S_{eff}}\).

• Verify against analytical results: compute ground state energy, thermal expectation values \(\langle x^2 \rangle\), and specific heat as functions of temperature.

• Extend to a two-body interacting system (e.g., a Lennard-Jones pair potential or a quartic potential) and study how interactions modify thermodynamic properties.

• Quantify convergence: investigate how results depend on the number of slices \(M\) (finer imaginary-time grid). Extrapolate to the continuum limit.

• Interpretation: explain how a classical random walk (polymer chain) in imaginary time computes quantum thermal averages.

Expected Deliverable: A research report (5–10 pages) with documented code. Include: (i) algorithm pseudocode and implementation details, (ii) numerical results (ground state, thermal properties) compared to exact or high-accuracy benchmarks, (iii) convergence analysis in \(M\), (iv) physical interpretation of worldlines and the imaginary-time ensemble, (v) discussion of PIMC applications in real materials and current limitations.

Key References: D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995); R. P. Feynman & A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill); path integral Monte Carlo reviews in modern computational physics.