Trimester 06 · Course detail

Continuum Systems & Quantum Mechanics

Partial differential equations and Sobolev spaces, quantum mechanics from operator foundations to the hydrogen atom, complex analysis and geometric function theory, and a computational quantum and continuum lab

Credits: 15
Sequence: 6 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Evidence of mastery

PDE solver + quantum tunnelling study

Subject 01

Partial Differential Equations & Sobolev Spaces

Official syllabus code: MIMUW 1000-215bRR2

The modern theory of PDEs: classification of second-order equations, weak derivatives and Sobolev spaces, variational formulations, and the regularity theory that underlies continuum physics.

Classification & Classical Boundary Problems

Second-Order PDE Classification (Elliptic, Parabolic, Hyperbolic)

Classify second-order linear PDEs into elliptic, parabolic, and hyperbolic types, and connect each type to its characteristic physics — equilibrium, diffusion, and propagation.

Classical Boundary Problems: Laplace/Poisson, Heat Kernel & Wave Equation

Solve the canonical boundary value problems: Laplace and Poisson equations on bounded domains, the heat equation via the heat kernel, and the Cauchy problem for the wave equation.

Weak Derivatives & Variational Theory

Weak Derivatives, Distributions & Sobolev Spaces Hᵏ(Ω), Wᵏᵖ(Ω)

Define weak derivatives through duality with test functions, build the Sobolev scale Hᵏ(Ω) and Wᵏᵖ(Ω), and establish embedding and trace theorems.

Variational Formulations, Lax–Milgram Theorem & Poincaré Inequality

Rewrite boundary value problems in weak (variational) form, prove the Lax–Milgram theorem for coercive bilinear forms, and quantify the Poincaré inequality that makes coercivity possible.

Regularity Theory, Maximum Principles & Fredholm Alternative for PDEs

Establish interior and boundary regularity of weak solutions, master the weak and strong maximum principles, and apply the Fredholm alternative to characterize solvability.

Subject 02

Quantum Mechanics I: Operator Foundations & Systems

Official syllabus code: FUW 1100-3INZ13

Quantum mechanics built on operator theory: postulates and spectral measures, canonical commutation, one-dimensional and central-potential systems, and the Fock-space harmonic oscillator.

Postulates & Canonical Structure

Postulates of Quantum Mechanics, Dirac Bra-Ket & Spectral Measures

State the postulates in Dirac notation, formulate observables as self-adjoint operators with spectral measures, and derive the measurement and evolution postulates.

Generalized Uncertainty Relations & Canonical Commutation ([x̂, p̂] = iħ)

Derive the Robertson uncertainty relation from commutators and establish the canonical commutation relation as the algebraic heart of quantum kinematics.

Canonical Quantum Systems

1D Potentials: Tunneling, Transfer Matrices, Bound States & S-Matrix

Solve one-dimensional scattering and bound-state problems, use transfer matrices for layered potentials, and analyze quantum tunneling through the scattering matrix.

Harmonic Oscillator: Fock States, Creation/Annihilation & Coherent States

Solve the oscillator algebraically with ladder operators, construct Fock states, and introduce coherent states as the closest quantum analogs of classical motion.

Central Potentials: SO(3), Orbital Angular Momentum, Spin-1/2 & Hydrogen

Exploit rotational symmetry, build the theory of orbital angular momentum and spin-1/2 representations, and solve the hydrogen atom exactly.

Subject 03

Complex Analysis & Geometric Function Theory

Official syllabus code: MIMUW 1000-135AM2

One-variable complex analysis as a geometric and computational instrument: from conformal mappings and Cauchy theory to the residue calculus and the Gamma/Zeta infrastructure.

Holomorphic Functions & Cauchy Theory

Holomorphic Functions, Cauchy–Riemann Equations & Conformal Mappings

Characterize holomorphy via the Cauchy–Riemann equations and study conformal mappings as angle-preserving geometric transformations.

Cauchy's Integral Theorem, Cauchy Integral Formula & Morera's Theorem

Prove the Cauchy integral theorem and formula, and use Morera's theorem as the converse route to holomorphy.

Meromorphic Functions, Laurent Series, Isolated Singularities & Branch Cuts

Expand meromorphic functions in Laurent series, classify isolated singularities, and manage multi-valued functions through branch cuts.

Residue Calculus & Analytic Continuation

The Residue Calculus, Jordan's Lemma & Principal Value Integrals

Compute real integrals by residues, apply Jordan's lemma to decaying exponentials, and handle singular integrals in the principal value sense.

Analytic Continuation, the Gamma/Zeta Infrastructure & Riemann Mapping Theorem

Continue analytic functions along paths, develop the Gamma and Zeta functions as central special-function infrastructure, and prove the Riemann mapping theorem.

Subject 04

In-Silico Computational Quantum & Continuum Lab

Official syllabus code: COMP-301 / Finite Element

The trimester's computational laboratory: finite elements for continuum PDEs and spectral/time-split solvers for the Schrödinger equation, closing with the wave-packet tunneling capstone.

Numerical PDE & Quantum Solvers

Finite Element Method (FEM) for Elliptic & Parabolic PDEs in 2D/3D

Build finite element discretizations of elliptic and parabolic problems in two and three dimensions, from weak form and meshing to assembly and solvers.

Time-Dependent Schrödinger Numerical Solvers: Split-Step Fourier & Crank–Nicolson

Integrate the time-dependent Schrödinger equation with split-step Fourier and Crank–Nicolson schemes, analyzing their stability and unitarity preservation.

Numerical Contour Integration & Dispersion Discretization

Implement contour integration on the complex plane for numerical evaluation and discretize dispersive relations for spectral solvers.

Panel 6 Capstone Deliverable: High-Fidelity Wave-Packet Tunneling & PDE Solver

Assemble the capstone: a high-fidelity wave-packet tunneling simulation coupled with the trimester's finite element PDE solver, submitted as the panel's evidence artifact.