Numerical Second Quantization of Interacting Quantum Systems

Sammanfattning

Systems where the wavenature of particles and phenomena are of the same order as system properties themselves are governed by non-classical phenomena. The most elementary of these systems are still described by the Schrödinger differential equa- tion, and the complexity grows exponentially with particle number and system com- plexity. This thesis builds upon the second quantization formalism, in which any many-body eigenstate is represented via creation and annihilation operators acting on a Fock space. An approximation is adopted which limits the single-particle basis to a finite set of harmonic oscillator orbitals, and the Coulomb interaction is regu- larised with a softening parameter. A program is constructed that applies operator al- gebra and the fermionic anti-commutation rules to numerically build and diagonalise the many-body Hamiltonian. Spin correlations are verified in the ground and excited states, and the framework is extended to the optical dipole response. The two ex- act symmetry rules, together with the dipole coupling to the interaction-independent centre-of-mass coordinate, collapse the spectrum of the cold system onto a single line; this single line is the central result that is recovered numerically.