The key difference between accidental degeneracy and normal degeneracy is that accidental degeneracy is energy degeneracy that occurs coincidentally, without any protection by symmetry, whereas normal degeneracy occurs with the protection by symmetry.
The term degeneracy is discussed mainly under quantum mechanics. It states that an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. The number of different states that corresponds to a particular energy level is named as the degree of degeneracy. Degeneracy can occur in two forms as accidental degeneracy and normal degeneracy. Accidental degeneracy refers to the energy degeneracy that occurs without any protection by symmetry while normal degeneracy refers to the energy degeneracy that occurs with the protection by symmetry.
CONTENTS
1. Overview and Key Difference
2. What is Accidental Degeneracy
3. What is Normal Degeneracy
4. Side by Side Comparison – Accidental Degeneracy vs Normal Degeneracy in Tabular Form
5. Summary
What is Accidental Degeneracy?
Accidental degeneracy refers to the energy degeneracy that occurs without any protection by symmetry. This type of degeneracy is known to be coincidental. This type of degeneracy results in some special features of the system or the functional form of the potential that we are considering. It is also possibly related to a hidden dynamical symmetry in the system. Furthermore, accidental degeneracy results in conserved quantities that are often not easy to identify.
Figure 01: Degenerate Energy Levels in a Quantum System
Generally, an accidental degeneracy occurs due to accidental symmetries that can lead to additional degeneracies in the discrete energy spectrum. As an example of accidental degeneracy, we can consider a particle in a constant magnetic field. Moreover, a particle moving under the influence of a constant magnetic field that is undergoing a cyclotron motion on a circular orbit has an accidental symmetry.
What is Normal Degeneracy?
Normal degeneracy refers to the energy degeneracy that occurs with protection by symmetry. In other words, a normal degeneracy occurs in a system having a symmetry. Moreover, the representation that is obtained by a normal degeneracy is irreducible, and the corresponding eigenfunctions form a basis for this representation.
What is the Difference Between Accidental Degeneracy and Normal Degeneracy?
Degeneracy can occur in two forms as accidental degeneracy and normal degeneracy. The key difference between accidental degeneracy and normal degeneracy is that accidental degeneracy is energy degeneracy that occurs coincidentally, without any protection by symmetry, whereas normal degeneracy occurs with the protection by symmetry. In other words, if the energy level of the system of consideration contains all symmetric transformations of the system, we call it normal degeneracy. In contrast, the accidental degeneracy is related to the existence of some undiscovered transformations of the system of consideration. Furthermore, normal degeneracy is irreducible while accidental degeneracy is reducible.
The following infographic summarizes the difference between accidental degeneracy and normal degeneracy.
Summary – Accidental Degeneracy vs Normal Degeneracy
The term degeneracy refers to the fact that an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Moreover, degeneracy can occur in two forms as accidental degeneracy and normal degeneracy. The key difference between accidental degeneracy and normal degeneracy is that accidental degeneracy is energy degeneracy that occurs coincidentally, without any protection by symmetry, whereas normal degeneracy occurs with the protection by symmetry.
Reference:
1. “Degenerate Energy Levels.” Wikipedia, Wikimedia Foundation, 22 June 2020, Available here.
2. “ Normal Degeneracy and Accidental Degeneracy .” Group Theory for Physicists, by Zhongqi Ma, World Scientific, 2019.
Image Courtesy:
1. “Degenerate energy levels” By Anypodetos – Own work, based on File: Degenerate energy levels.png (CC BY-SA 4.0) via Commons Wikimedia