Degeneracy in Separable Systems

Degeneracy means that more than one linearly independent state has the same energy. In a separable system, this happens often because the energy is built from several quantum labels, and different label combinations can produce the same number.

If

$$\hat H\lvert E,\alpha\rangle = E\lvert E,\alpha\rangle,$$

where $\alpha$ distinguishes independent states with the same energy, then the degeneracy of $E$ is the dimension of the corresponding energy eigenspace:

$$g_E = \dim\ker(\hat H-EI).$$

The important lesson is that “same energy” does not always mean “same state” or “same physical reason.” Degeneracies can come from exact symmetries, from additive separable spectra, from number-theoretic coincidences, or from hidden structure of a special Hamiltonian.

Why Separability Creates Repeated Energies

In many separable problems, the Hamiltonian splits into commuting pieces:

$$\hat H = \hat H_1+\hat H_2+\hat H_3, \qquad [\hat H_i,\hat H_j]=0.$$

Product eigenstates then have additive energies:

$$\psi_{abc} = \psi_a^{(1)}\psi_b^{(2)}\psi_c^{(3)}, \qquad E_{abc} = E_a^{(1)}+E_b^{(2)}+E_c^{(3)}.$$

Degeneracy occurs when two or more triples give the same sum:

$$E_{abc}=E_{a'b'c'}.$$

Sometimes the equality is enforced by a symmetry. Sometimes it is a coincidence caused by the arithmetic form of the spectrum. The distinction matters because symmetry-enforced degeneracies are stable under symmetry-preserving perturbations, while accidental coincidences are usually lifted by small generic changes.

Energy Labels Are Not Complete State Labels

If an energy is nondegenerate, the energy label may identify the energy eigenstate up to an overall phase. If an energy is degenerate, energy alone labels an eigenspace, not a unique vector.

Inside a degenerate eigenspace, any orthonormal basis is allowed unless additional commuting observables or boundary conditions select a preferred basis. A complete set of commuting observables supplies enough compatible labels to distinguish states within the model. For a box these labels may be $(n_x,n_y,n_z)$. For a central potential they may be $(E,\ell,m)$ or $(n,\ell,m)$ when the spectrum is discrete.

This is why degeneracy is not merely a counting issue. It changes how measurements, perturbations, and state expansions are described.

Cubic Box Degeneracy

For a rectangular hard-wall box, the energies are

$$E_{n_xn_yn_z} = \frac{\pi^2\hbar^2}{2m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right),$$

with

$$n_x,n_y,n_z=1,2,3,\ldots .$$

If the side lengths are all different, most energy levels are nondegenerate. In the cubic case,

$$L_x=L_y=L_z=L,$$

the energy depends only on the sum

$$n_x^2+n_y^2+n_z^2.$$

The triples

$$(1,1,2), \qquad (1,2,1), \qquad (2,1,1)$$

therefore have the same energy. This threefold degeneracy follows from permuting the equal axes of the cube.

Higher box levels can also show arithmetic coincidences. For example, two different unordered triples can sometimes have the same sum of three squares. Such coincidences are less robust than the degeneracy forced by exact geometric symmetry: if the box is slightly deformed away from a cube, permutation degeneracies generally split.

Harmonic-Oscillator Degeneracy

A three-dimensional isotropic oscillator has Hamiltonian

$$\hat H = \sum_{i=x,y,z} \left( \frac{\hat p_i^2}{2m} + \frac12m\omega^2\hat x_i^2 \right).$$

Because all three frequencies are equal, the energy is

$$E_{n_xn_yn_z} = \hbar\omega \left( n_x+n_y+n_z+\frac32 \right),$$

with

$$n_x,n_y,n_z=0,1,2,\ldots .$$

For a fixed shell

$$N=n_x+n_y+n_z,$$

the degeneracy is the number of nonnegative integer triples with that sum:

$$g_N = \frac{(N+1)(N+2)}{2}.$$

For example, $N=2$ has six states:

$$(2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), (0,1,1).$$

This degeneracy is larger than a simple axis-permutation count because all partitions of the same total excitation number have the same energy. If the frequencies become anisotropic,

$$E_{n_xn_yn_z} = \hbar\omega_x\left(n_x+\frac12\right) + \hbar\omega_y\left(n_y+\frac12\right) + \hbar\omega_z\left(n_z+\frac12\right),$$

the degeneracy usually disappears unless the frequencies have exact rational relations or residual symmetries.

Central-Potential Degeneracy

For a central potential,

$$V(\mathbf r)=V(r),$$

the wavefunction separates as

$$\psi(r,\theta,\phi) = R_{E\ell}(r)Y_\ell^m(\theta,\phi).$$

The radial equation depends on $\ell$ but not on $m$. Therefore, for any rotationally invariant central potential, the $2\ell+1$ states

$$m=-\ell,-\ell+1,\ldots,\ell$$

are degenerate within a fixed $\ell$ sector. This is symmetry-enforced rotational degeneracy: no axis is preferred, so the Hamiltonian cannot distinguish the different orientations.

Degeneracy between different $\ell$ sectors is a separate question. A generic central potential does not have it, because changing $\ell$ changes the centrifugal term in the effective radial potential:

$$\frac{\hbar^2\ell(\ell+1)}{2mr^2}.$$

The nonrelativistic Coulomb potential is special: its bound-state energy depends only on the principal quantum number $n$, producing the familiar $n^2$ spatial degeneracy. That stronger result is explained in Degeneracy of the Hydrogen Atom.

Symmetry, Accident, and Hidden Symmetry

It is useful to classify degeneracies by their source.

Source	Example	What Usually Lifts It
Exact visible symmetry	$m$ degeneracy in a central potential	external fields or symmetry-breaking boundaries
Equal separable parameters	cubic box or isotropic oscillator	unequal side lengths or unequal frequencies
Arithmetic coincidence	distinct box triples with the same sum of squares	small generic deformation
Hidden symmetry	Coulomb $\ell$ degeneracy	relativistic, radiative, or non-Coulomb corrections
Continuous label degeneracy	free-particle directions with fixed $\lvert\mathbf k\rvert$	boundaries, fields, or interactions selecting directions

The word “accidental” is often used for degeneracies not explained by the most obvious symmetry. It should not mean “unimportant” or “numerically approximate.” Some accidental degeneracies reveal a hidden symmetry, as in hydrogen; others are simple arithmetic coincidences.

Lifting Degeneracy

A perturbation lifts a degeneracy when it acts differently on states that previously had the same energy. The classic qualitative example is an external field selecting an axis.

For a central potential, a weak interaction proportional to $\hat L_z$ gives shifts of the form

$$\Delta E_m = \gamma\hbar m,$$

where $\gamma$ is a field-dependent constant in the simplified model. The different $m$ values, previously degenerate under full rotational symmetry, now separate because the field has chosen a preferred $z$ direction.

This is the basic idea behind magnetic-field splitting. Detailed Zeeman physics involves spin, magnetic moments, perturbation theory, and atomic structure, so it belongs in later volumes. The lesson here is more general: once the symmetry or exact equality responsible for a degeneracy is broken, energy levels usually split.

Common Mistakes

• Treating degeneracy as if it always has the same explanation.
• Assuming that energy alone labels an eigenstate in a degenerate spectrum.
• Calling every repeated energy “accidental” without checking for symmetry.
• Assuming cubic-box permutation degeneracy survives in a rectangular box.
• Treating hydrogen’s $\ell$ degeneracy as generic for all central potentials.
• Forgetting that a perturbation must be diagonalized inside a degenerate eigenspace, not just evaluated on one arbitrary basis vector.

Where This Is Used

• Three-Dimensional Box gives the first concrete separable example with cubic-box degeneracies.
• Density of States: First Encounter explains the large-box limit where dense levels are counted by energy intervals instead of exact degeneracy.
• Central Potentials gives the generic rotational setting behind $m$ degeneracy.
• Quantum Harmonic Oscillator supplies the one-dimensional oscillator; the isotropic three-dimensional oscillator uses three copies.
• Angular and Radial Separation explains why central-potential states carry $\ell$ and $m$ labels.
• Degeneracy of the Hydrogen Atom treats the special Coulomb degeneracy in detail.
• Complete Sets of Commuting Observables gives the operator language for resolving degenerate eigenspaces.
• Landau Levels gives a later magnetic-field example where degeneracy is tied to guiding-center labels.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.

Exercises

1. In a cubic hard-wall box, find the degeneracy of the shell with $n_x^2+n_y^2+n_z^2=14$.

Solution

The positive integer triple

$$(1,2,3)$$

has

$$1^2+2^2+3^2=14.$$

All three entries are distinct, so there are

$$3!=6$$

permutations. Thus this shell has at least sixfold degeneracy from permutations. If no other positive integer triple gives the same sum, the degeneracy is exactly $6$.

2. Count the degeneracy of the $N=3$ shell of a three-dimensional isotropic oscillator.

Solution

Use

$$g_N = \frac{(N+1)(N+2)}{2}.$$

For $N=3$,

$$g_3 = \frac{4\cdot5}{2} =10.$$

Equivalently, count the nonnegative integer triples satisfying

$$n_x+n_y+n_z=3.$$

3. Explain why a perturbation proportional to $\hat L_z$ splits an $m$ multiplet.

Solution

The states $Y_\ell^m$ obey

$$\hat L_zY_\ell^m = \hbar mY_\ell^m.$$

If the perturbation is $\gamma\hat L_z$, then the first-order shift is

$$\Delta E_m = \gamma\hbar m.$$

Different $m$ values generally receive different shifts. Physically, the perturbation has selected a $z$ axis, so the full rotational symmetry that protected the $m$ degeneracy is broken.