What is degeneracy?

Short Answer

Degeneracy refers to the condition in which two or more atomic orbitals have the same energy level. Orbitals that are equal in energy are called degenerate orbitals. For example, the three p orbitals (px, py, pz) in an atom are degenerate because they all have identical energy.

Degeneracy is an important concept in quantum mechanics because it explains how electrons distribute themselves in subshells. Hund’s rule, electron configuration, and spectral behaviour all depend on the idea of degeneracy in orbitals.

Detailed Explanation :

Degeneracy

Degeneracy is an important concept in quantum mechanics and atomic structure. It refers to the situation where multiple quantum states or orbitals have the same energy. When orbitals or states are said to be degenerate, it means electrons occupying those orbitals possess equal energy even though the orbitals may differ in orientation or shape. Degeneracy arises naturally from the mathematical solutions of Schrödinger’s wave equation for atoms, especially in hydrogen-like systems.

In simpler terms, degeneracy explains why certain orbitals come in groups. For example, p orbitals always come in sets of three and all have the same energy. Similarly, d orbitals come in sets of five, and f orbitals come in sets of seven. Even though each orbital may point in a different direction in space, their energies are exactly the same unless an external effect breaks this equality.

Understanding degeneracy is essential for learning how electrons fill orbitals, how atoms behave in magnetic and electric fields, and why spectral lines split under certain conditions.

Meaning of degeneracy in orbitals

Orbitals belonging to the same subshell have the same azimuthal quantum number (l). These orbitals also have different magnetic quantum number (mₗ) values but still share the same energy. This gives rise to degeneracy.

Examples:

  • The three p orbitals (px, py, pz) are degenerate.
  • The five d orbitals are degenerate.
  • The seven f orbitals are degenerate.

Degeneracy depends on how many values mₗ can take. The formula for the number of degenerate orbitals is:

Number of degenerate orbitals = 2l + 1

This matches the number of orbitals in each subshell.

Degeneracy in the hydrogen atom

In the hydrogen atom, degeneracy is very strong. All orbitals with the same principal quantum number (n) have exactly the same energy. This means:

  • 2s and 2p are degenerate
  • 3s, 3p, and 3d are degenerate
  • and so on

This happens because hydrogen has only one electron, so electron–electron repulsion does not exist. The energy depends only on the principal quantum number.

As soon as we move to multi-electron atoms, this degeneracy is removed, and orbitals with the same n value have different energies.

Degeneracy in multi-electron atoms

In atoms with more than one electron, degeneracy is partially removed. Orbitals with the same n but different l values no longer have equal energy because:

  • electrons repel each other,
  • shielding and penetration effects arise,
  • orbitals with different shapes experience different nuclear attractions.

For example, in multi-electron atoms:

  • 2s is lower in energy than 2p
  • 3s < 3p < 3d
  • 4s < 3d (in many cases)

However, orbitals within the same subshell (same n and same l) remain degenerate. This means that:

  • all three 2p orbitals are degenerate
  • all five 3d orbitals are degenerate

even though the energy of 2p is higher than 2s.

Importance of degeneracy

Degeneracy plays a major role in several areas of atomic physics and chemistry.

  1. Electron configuration
    When electrons fill a subshell, they first occupy degenerate orbitals singly. Hund’s rule explains this behaviour. Degeneracy determines how many orbitals need to be filled.
  2. Spectral lines
    Transitions between degenerate states can produce characteristic spectral lines. The splitting of these lines under external fields is also explained using degeneracy.
  3. Magnetic properties
    The number of unpaired electrons in degenerate orbitals determines whether an atom is paramagnetic or diamagnetic.
  4. Chemical bonding
    Degenerate p orbitals form directional covalent bonds. Degeneracy is essential for understanding molecular shapes and hybridization.
  5. External fields and degeneracy breaking
    Degeneracy can be broken by external forces:
  • A magnetic field causes Zeeman splitting.
  • An electric field causes Stark splitting.

These effects reveal fine details of atomic structure.

Degeneracy and quantum numbers

Degeneracy arises when all quantum numbers except mₗ are the same. The set of degenerate orbitals share the same:

  • principal quantum number (n),
  • azimuthal quantum number (l),
  • energy.

They differ only in their magnetic quantum number (mₗ), which represents orientation.

Mathematical view of degeneracy

When Schrödinger’s wave equation is solved for the hydrogen atom, the resulting energy depends only on n. This leads to:

Eₙ = −13.6 eV / n²

Since l and mₗ do not appear in this energy expression, orbitals with the same n but different l or mₗ values have identical energy, creating degeneracy.

Conclusion

Degeneracy refers to orbitals or quantum states that share the same energy level. Orbitals within the same subshell are degenerate because they have equal energy even though they differ in orientation. Degeneracy explains electron distribution, spectral behaviour, chemical bonding, and magnetic properties. It plays a central role in quantum mechanics, atomic physics, and the modern understanding of atomic structure.