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One-particle Green's function of interacting two electrons using analytic solutions for a three-body problem:
Taichi Kosugi1, Yu-Ichiro Matsushita2
1Department of Physics, University of Tokyo, Tokyo 113-0033, Japan.
Journal of Physics. Condensed Matter : an Institute of Physics Journal
|September 20, 2018
Summary
Researchers analytically solved the Schrödinger equation for a three-electron system, obtaining exact solutions for energy and spin eigenstates. This work derives analytic expressions for the one-particle Green's function (GF) in two-electron systems, advancing the understanding of interacting systems.
Area of Science:
- Quantum mechanics
- Condensed matter physics
- Atomic physics
Background:
- Understanding electron-electron interactions is crucial in quantum systems.
- Analytical solutions for interacting many-body systems are often limited.
- The Kohn-Sham (KS) approach is widely used but relies on approximations.
Purpose of the Study:
- To analytically solve the Schrödinger equation for a three-electron system in a 1D harmonic trap.
- To derive exact one-particle Green's functions (GFs) for a two-electron system from the three-electron solution.
- To investigate the impact of electron interactions on GF properties and compare with KS approximations.
Main Methods:
- Analytical solution of the Schrödinger equation for a three-electron system.
- Construction of simultaneous energy and total spin eigenstates.
- Derivation of analytic expressions for the one-particle Green's function (GF).
- Frequency-domain and real-space calculations of GFs.
- Comparison of exact GFs with those obtained using the exact Kohn-Sham potential.
Main Results:
- Exact analytical solutions for the three-electron system, including energy and total spin eigenstates, were obtained for the first time.
- Analytic expressions for the exact one-particle Green's function (GF) of the corresponding two-electron system were derived.
- The behavior of the GF was systematically examined in the frequency domain, showing increased discrepancy in the energy gap with stronger interactions when compared to KS approximations.
- Numerical analysis revealed significant differences in the real-space behavior between exact and KS GFs.
Conclusions:
- The developed analytical model provides exact solutions for interacting electron systems in a 1D harmonic trap.
- The study highlights limitations of the Kohn-Sham approach for strongly interacting systems by comparing exact and KS Green's functions.
- This work offers insights into the generic characteristics of interacting Green's functions, valuable for theoretical and computational physics.