# Consequences of Reducing Symmetry in Quantum Systems

## Document Type

Oral Presentation

## Event Website

https://source2022.sched.com/

## Start Date

18-5-2022

## End Date

18-5-2022

## Keywords

Symmetry, Triangular Antiferromagnet, Quantum Mechanical Systems

## Abstract

Symmetry is a fundamental characteristic of any physical system and it plays a clear role in biology, chemistry, classical physics, mathematics, and other disciplines. The role of symmetry in quantum mechanics is more abstract, and this research project constituted an investigation of how the symmetry of a quantum system governs the possible energies of the system’s states. A triangular arrangement of three spin-1/2 particles was used as a test system. The eigenenergies and eigenstates of a triangular arrangement with high-symmetry (equal exchange interactions between the particles) were calculated and compared to the eigenenergies and eigenstates of a triangular arrangement with low-symmetry (one of the exchange interactions was different than the other two). The eigenstates of the Hamiltonian of each arrangement were calculated and expressed in terms of the eigenstates of the spin-squared and z-component of spin operators, and the eigenenergies were calculated by applying the Hamiltonian of each arrangement to those eigenstates. The high-symmetry arrangement had a significant number of different states with the same energy (significant degeneracy), while the low-symmetry arrangement had less degeneracy. Both the high-symmetry and low-symmetry arrangements contained time-invariance symmetry, which can be broken by applying a magnetic field. The consequences for the eigenenergies of the two arrangements when a magnetic field is applied were calculated and a reduction in degeneracy was observed for both arrangements. The results of these calculations show that, in general, reduction of symmetry in quantum systems leads to degeneracy in energy levels being lifted.

## Recommended Citation

Klein, Nicholas, "Consequences of Reducing Symmetry in Quantum Systems" (2022). *Symposium Of University Research and Creative Expression (SOURCE)*. 64.

https://digitalcommons.cwu.edu/source/2022/COTS/64

## Department/Program

Physics

## Additional Mentoring Department

Physics

Consequences of Reducing Symmetry in Quantum Systems

Symmetry is a fundamental characteristic of any physical system and it plays a clear role in biology, chemistry, classical physics, mathematics, and other disciplines. The role of symmetry in quantum mechanics is more abstract, and this research project constituted an investigation of how the symmetry of a quantum system governs the possible energies of the system’s states. A triangular arrangement of three spin-1/2 particles was used as a test system. The eigenenergies and eigenstates of a triangular arrangement with high-symmetry (equal exchange interactions between the particles) were calculated and compared to the eigenenergies and eigenstates of a triangular arrangement with low-symmetry (one of the exchange interactions was different than the other two). The eigenstates of the Hamiltonian of each arrangement were calculated and expressed in terms of the eigenstates of the spin-squared and z-component of spin operators, and the eigenenergies were calculated by applying the Hamiltonian of each arrangement to those eigenstates. The high-symmetry arrangement had a significant number of different states with the same energy (significant degeneracy), while the low-symmetry arrangement had less degeneracy. Both the high-symmetry and low-symmetry arrangements contained time-invariance symmetry, which can be broken by applying a magnetic field. The consequences for the eigenenergies of the two arrangements when a magnetic field is applied were calculated and a reduction in degeneracy was observed for both arrangements. The results of these calculations show that, in general, reduction of symmetry in quantum systems leads to degeneracy in energy levels being lifted.

https://digitalcommons.cwu.edu/source/2022/COTS/64

## Faculty Mentor(s)

Benjamin White