Aspects of Operator Theory in Quantum Mechanics: And a Study of Magnetic Many-body Quantum Systems in the Hartree-Fock Approximation
Translated title
Aspekter af operator teori i kvantemekanik: Og en undersøgelse af magnetiske flerlegeme kvantesystemer i Hartree-Fock approksimationen
Author
Thorn, Mikkel Hviid
Term
4. term
Education
Publication year
2025
Submitted on
2025-05-28
Pages
100
Abstract
Dette speciale udvikler den matematiske værktøjskasse, der ligger bag kvantemekanik, og er organiseret i tre dele: (1) selvadjungerede ubundne operatorer, som modellerer kvantemekaniske observabler, (2) pseudodifferentialregning, der generaliserer differentialoperatorer og kvantisering, og (3) enkle kvantesystemer i Euklidisk rum med og uden magnetfelt. Først gennemgår og beviser vi centrale resultater om selvadjungerede ubundne operatorer på Hilbertrum: grundlæggende definitioner og kriterier for selvadjungering, Kato–Rellich-sætningen om perturbationer, samt variationalt formulerede operatorer (inklusive Friedrichs’ udvidelse). I spektalteorien beviser vi spektralsætningen for både begrænsede og ubundne selvadjungerede operatorer, samt Stones formel og Helffer–Sjöstrand-formlen. Dernæst behandler vi tempererede distributioner og pseudodifferentiale operatorer. Vi introducerer Schwartz-rummet og rummet af tempererede distributioner og beviser blandt andet disse rums refleksivitet, Schwartz’ kernesætning og struktursætningen. Herefter studerer vi generelle kvantiseringsskemaer for pseudodifferentiale operatorer, især for Hörmander-klasser af glatte symboler med aftag kontrolleret af en tempereret vægt. For disse etablerer vi en Calderón–Vaillancourt-sætning (begrænsethed), et Moyal-produkt (en kompositionsformel) og, for visse kvantiseringer, et kommutatorkriterium af Beals-typen. Et centralt redskab er en moduleret stram Gabor-ramme, som gør det muligt at beskrive funktioner og operatorer via deres koordinater eller matricer i denne ramme. Til sidst analyserer vi enkle kvantesystemer: først uden magnetfelt, hvor vi gennemgår klassiske resultater for den frie Schrödinger-operator og den harmoniske oscillator; dernæst frie magnetiske Schrödinger-operatorer, hvor vi udleder Landau-spektret (kvantiserede energiniveauer i et konstant magnetfelt). Afslutningsvis undersøger vi et mange-partikel-system i Hartree–Fock-approksimationen under et konstant magnetfelt. Her reduceres problemet til en én-partikel Schrödinger-operator med et ekstra potentiale, der repræsenterer partikel-skyen. Dette potentiale opfylder en fastpunktligning, som vi løser.
This thesis develops the mathematical tools that support quantum mechanics, arranged in three parts: (1) self-adjoint unbounded operators, which model quantum observables, (2) pseudodifferential calculus, which generalizes differential operators and quantization, and (3) simple quantum systems in Euclidean space with and without magnetic fields. First, we review and prove core results about self-adjoint unbounded operators on Hilbert spaces: basic definitions and criteria for self-adjointness, the Kato–Rellich theorem on perturbations, and variationally defined operators (including Friedrichs’ extension). In spectral theory we prove the spectral theorem for bounded and unbounded self-adjoint operators, as well as Stone’s formula and the Helffer–Sjöstrand formula. Second, we study tempered distributions and pseudodifferential operators. We introduce the Schwartz space and the space of tempered distributions and prove results such as reflexivity of these spaces, the Schwartz Kernel Theorem, and the Structure Theorem. We then consider general quantization schemes for pseudodifferential operators, focusing on Hörmander classes of smooth symbols with decay controlled by a tempered weight. For these, we establish a Calderón–Vaillancourt theorem (boundedness), a Moyal product (a composition formula), and, for certain quantizations, a commutator criterion of Beals type. A key tool is a modulated tight Gabor frame, which lets us describe function and operator spaces through their coordinates or matrices in this frame. Lastly, we analyze simple quantum systems: first without magnetic fields, where we cover classical results for the free Schrödinger operator and the harmonic oscillator; then free magnetic Schrödinger operators, where we derive the Landau spectrum (quantized energy levels in a constant magnetic field). Finally, we study a many-body system in the Hartree–Fock approximation under a constant magnetic field. This reduces the problem to a single-particle Schrödinger operator with an added potential representing the particle cloud. The potential satisfies a fixed-point equation, which we solve.
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