Symmetry and Degeneracy in Quantum Systems

Hello! Welcome back to our course on quantum mechanics.

In our last lesson, we investigated the discrete symmetries of parity and time reversal. We saw a powerful consequence of time-reversal symmetry in systems with half-integer spin: Kramers' theorem, which guarantees that every energy level must be at least doubly degenerate. This was our first concrete example of a symmetry forcing a degeneracy to exist.

Today, we will generalize this profound connection between symmetry and degeneracy. This lesson addresses the learning outcome: Relate the symmetry group of a Hamiltonian to the degeneracy of its energy levels. We will see that the degeneracies observed in a quantum system are not random; they are rigorously determined by the mathematical structure of the system's symmetry group. The key insight is that degenerate energy levels form multiplets that correspond to the irreducible representations of the symmetry group, with the degree of degeneracy being the dimension of the representation. This is one of the most powerful predictive tools in quantum physics, essential for classifying states in atoms, molecules, and crystalline solids.

1. From Symmetry to Degeneracy

Let's begin by formalizing the argument. A symmetry of a quantum system is described by a set of transformations that leave the Hamiltonian $\hat{H}$ unchanged. These transformations form a mathematical group $G$. For each group element $g\in G$, there is a corresponding unitary operator $\hat{U}(g)$ that acts on the system's Hilbert space. The condition that $G$ is a symmetry group of the Hamiltonian is expressed as:

This commutation relation is the direct cause of degeneracy. If $|\psi ⟩$ is an eigenstate of $\hat{H}$ with energy $E$, then any state transformed by a symmetry operation, $\hat{U}(g)|\psi ⟩$, is also an eigenstate with the exact same energy:

If $\hat{U}(g)|\psi ⟩$ is linearly independent from $|\psi ⟩$, then the energy level $E$ is degenerate.

The crucial insight, which we will build upon, is that it is often the presence of non-commuting symmetries that guarantees degeneracy. To develop a strong physical intuition for this, let's watch a lecture by Leonard Susskind where he masterfully explains how the algebraic structure of symmetry operators dictates the structure of the energy spectrum.

2. The Language of Group Theory: Irreducible Representations

As Susskind's lecture demonstrated, the collection of degenerate eigenstates for a particular energy level forms a vector space that is closed under the action of the symmetry group. In the language of group theory, this subspace of degenerate states "furnishes a representation of the group".

The next step is to introduce a crucial concept: irreducible representations (irreps). An irrep is a representation that cannot be broken down into smaller, independent representations. They are the fundamental "building blocks" of a group's symmetry.

The central principle of this lesson connects these irreps directly to the energy spectrum of a quantum system.

To summarize the key idea from the reading:

If a Hamiltonian $\hat{H}$ has a symmetry group $G$, its eigenstates can be grouped into multiplets. Each multiplet transforms according to an irreducible representation $\Gamma$ of $G$, and all states within that multiplet are degenerate. The degree of degeneracy is precisely the dimension, $d_{\Gamma }$, of the irrep $\Gamma$.

This allows us to label energy eigenstates not just with energy, but with the irrep they belong to: $|E,\Gamma ,\mu ⟩$, where $\mu$ is an index from 1 to $d_{\Gamma }$ that labels the different degenerate states within the multiplet.

A familiar example is the rotational symmetry group SO(3). Its irreps are labeled by the angular momentum quantum number $l=0,1,2,\dots$, and the dimension of the irrep $l$ is $d_l=2l+1$. For a system with full rotational symmetry (like the hydrogen atom, ignoring spin for a moment), its energy levels must have degeneracies of $1,3,5,\dots$, corresponding to s, p, d, ... orbitals. The index $\mu$ here is just the magnetic quantum number $m_l$.

3. Non-Abelian Symmetries and Accidental Degeneracies

The connection between irrep dimension and degeneracy leads to a vital distinction:

• Abelian groups (where all elements commute) have only one-dimensional irreducible representations. Therefore, symmetry under an Abelian group does not guarantee degeneracy.
• Non-Abelian groups (which have at least one pair of non-commuting elements) possess higher-dimensional irreps. Symmetry under a non-Abelian group guarantees the existence of degenerate energy levels.

This provides a powerful shortcut for identifying guaranteed degeneracies. We can formalize the argument Susskind alluded to.

The argument is as follows: Suppose a Hamiltonian $\hat{H}$ has two symmetries represented by operators $\hat{A}$ and $\hat{B}$ such that $[\hat{H},\hat{A}]=0$ and $[\hat{H},\hat{B}]=0$, but $[\hat{A},\hat{B}]\ne 0$. If the energy spectrum of $\hat{H}$ were entirely non-degenerate, then its eigenstates would be unique. Since they must also be eigenstates of $\hat{A}$ and $\hat{B}$ (or could be chosen to be), in this basis $\hat{H}$, $\hat{A}$, and $\hat{B}$ would all be diagonal matrices. However, all diagonal matrices commute, which contradicts the fact that $[\hat{A},\hat{B}]\ne 0$. Therefore, the initial assumption must be false: the spectrum of $\hat{H}$ must contain degenerate energy levels.

Accidental Degeneracy

What happens if we find a degeneracy that is larger than the dimension of any single irrep of the system's symmetry group? This is known as an accidental degeneracy.

As the notes explain, there are two primary reasons for accidental degeneracies:

1. Hidden Symmetry: The system may possess a larger symmetry than initially apparent. The famous example is the non-relativistic hydrogen atom. From the perspective of just SO(3) rotational symmetry, the fact that the 2s ($l=0$, degeneracy 1) and 2p ($l=1$, degeneracy 3) states are degenerate is accidental. This degeneracy is, in fact, explained by a "hidden" SO(4) symmetry related to the conservation of the Runge-Lenz vector.

2. Parameter Fine-Tuning: It's possible for energy levels belonging to different irreducible representations to have the same energy if the parameters of the Hamiltonian (like hopping amplitudes in a tight-binding model) are tuned to specific values. This type of degeneracy is generally not robust; a small perturbation that preserves the symmetry will typically lift the degeneracy by causing the levels to repel (an "avoided crossing"). This is in contrast to the degeneracies within a single irrep, which are protected by the symmetry.

Your background in condensed matter physics makes the C60 "buckyball" example from the reading particularly illustrative. The Ih symmetry group has irreps with dimensions up to 5, yet a 9-fold degenerate level appears in the simple tight-binding model. This is an accidental degeneracy caused by the crossing of a 4-dimensional and a 5-dimensional irrep, which occurs only when the two distinct types of C-C bonds are assumed to have the same hopping integral. Making the model more realistic by distinguishing the bonds lifts this accidental degeneracy.

Conclusion

This lesson has established the fundamental link between symmetry and degeneracy in quantum mechanics, moving from specific examples to a general, powerful framework provided by group theory.

Key Takeaways:

• If a Hamiltonian commutes with the operators of a symmetry group $G$, its energy eigenstates can be organized into degenerate multiplets.
• Each multiplet corresponds to an irreducible representation (irrep) of the group $G$.
• The degeneracy of a multiplet is equal to the dimension of its corresponding irrep.
• Non-Abelian symmetry groups have irreps with dimensions greater than one, thus guaranteeing the presence of degenerate energy levels.
• Degeneracies that are not explained by the dimensions of the irreps of the known symmetry group are called accidental. They may signal a hidden, larger symmetry or result from fine-tuning Hamiltonian parameters.

Preview of the next lesson:
We have now established that the symmetry group determines degeneracy. The next logical step is to see how we can use this in practice. In the next lesson, we will learn how to apply group representation theory to explicitly label the degenerate eigenstates and, even more powerfully, to block-diagonalize a Hamiltonian matrix, which can dramatically simplify the problem of finding its energy spectrum.