Oğuzhan Kaya | Searcharxiv

arXiv2026

Ground-State Energy Solutions of the Lithium Atom: Zeroth-, First-, and Second-Order Perturbation Theory and the Variational Method

In this work, the ground-state energy of the lithium atom is systematically investigated using both time-independent perturbation theory and the variational method to provide a comprehensive pedagogical analysis of many-body atomic systems. The unperturbed Hamiltonian is initially constructed by neglecting electron-electron interactions, treating the system as three independent hydrogen-like electrons to yield a zeroth-order energy baseline of -275.51 eV. The antisymmetric fermionic nature of the exact wave function is rigorously enforced through the Slater determinant formalism. First-order perturbation theory is applied to evaluate static inter-electronic repulsion using exact Coulomb and exchange integrals, refining the energy state to -192.01 eV. To account for dynamical electronic correlation, second-order perturbation theory is computed numerically for virtual single-electron s-orbital transitions, leading to a total perturbative energy of -196.36 eV. A brief discussion of two-electron excitations is also included to encapsulate further physical realism within the framework. Furthermore, a non-orthogonal two-parameter variational approach is employed to model the shell-specific shielding effect. By optimizing the effective nuclear charges, the variational method establishes a superior upper bound energy of -201.187 eV. The results of both methods are comprehensively contrasted against each other and the reference baseline to provide critical insights into the nature of electron correlation and screening in multi-electron atoms.

quant-ph↗

arXiv2023

Quantization in fibering polarizations, Mabuchi rays and geometric Peter--Weyl theorem

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-Kähler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant Kähler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the Kähler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.

math.SG↗