Degeneracy in the Hydrogen Atom

Introduction

The phrase “an energy level” can hide several distinct wavefunctions. In the elementary hydrogen Hamiltonian, every spatial orbital with the same principal quantum number n has the same energy even if its l and mₗ values differ. This shared energy is degeneracy. Counting the states is straightforward once the allowed quantum-number ranges are known, but interpreting the count requires care about electron spin and about effects that can split levels.

Core explanation

For a fixed n, l can be any integer from 0 to n − 1. For each l, mₗ takes 2l + 1 values from −l through +l. Therefore the number of spatial hydrogen states in shell n is Σₗ₌₀ⁿ⁻¹(2l + 1) = n². For n = 3, the subshells have one 3s, three 3p and five 3d spatial states: 1 + 3 + 5 = 9. At the gross nonrelativistic hydrogen level, their shared energy is Eₙ = −hcR H/n².

This n² is a count of orbital wavefunctions, not the maximum number of electrons in a single hydrogen atom. A hydrogen atom has one electron. If electron spin is included, each spatial orbital can be paired with m s = +1/2 or −1/2, giving 2n² one-electron spin-orbitals. In a many-electron filling calculation, the Pauli principle permits at most one electron per spin-orbital, but applying that capacity to hydrogen itself would confuse possible states with simultaneous occupants.

There are two aspects to the degeneracy. The multiple mₗ values at fixed n and l reflect rotational symmetry when no direction is physically preferred. The equality across different l values at fixed n is a special property of the ideal Coulomb potential. Change the physical Hamiltonian and these equalities may fail. In a magnetic field, states with different angular-momentum projections can have different energies, producing Zeeman splitting. In a many-electron atom, shielding and electron–electron repulsion make s, p and d subshell energies differ even for the same n. Smaller corrections within hydrogen itself include fine and hyperfine structure. “Hydrogen degeneracy” therefore always means degeneracy under a specified approximation.

Degeneracy affects spectroscopy because energy differences may coincide even when several state pairs are involved. A single gross spectral line can represent multiple transitions that become distinguishable when a field or high-resolution measurement resolves their slightly different energies. Counting states helps establish how many possibilities exist, but selection rules and populations determine which transitions are actually observed.

Step-by-step reasoning

Choose n, list l = 0 through n − 1, and count 2l + 1 values of mₗ for each. Add them to get n², then double only if the question explicitly includes electron spin. State the Hamiltonian assumption before declaring them equal in energy. If a magnetic field or electron–electron interaction is present, identify which symmetry or interaction has changed and avoid asserting the ideal hydrogen equality without qualification.

Visual explanation

Draw one horizontal energy line labelled n = 3. Along it place nine small circles grouped as one s, three p and five d states. Beneath, show a field-free magnetic quantum-number group on one line; above, draw separated horizontal lines after a magnetic field is applied. The separation represents energy splitting, not a change in the number of states.

Real-world analogy

Several different songs can have exactly the same duration. Their shared numerical duration does not make them the same song. Likewise, different wavefunctions can share an energy. If a change in conditions slows some recordings differently, their equal durations disappear. The analogy is about equal values for distinct objects, not about how atomic states physically behave.

Real-world example

In a laboratory magnetic field, hydrogen emission can show closely separated components rather than one ideal line. This is evidence that different magnetic states respond differently to the field. The simple n-only energy formula remains a useful baseline, but it cannot predict the full splitting without adding the relevant interaction to the Hamiltonian.

Why?

An energy eigenvalue can correspond to more than one independent eigenfunction. In ideal hydrogen, the Coulomb potential and its symmetries lead to energies depending only on n. Once an external field chooses a direction or other electrons modify the potential, the mathematical operator changes. Its eigenvalues can then distinguish states that were previously tied.

Common misconception

The n² rule does not say that hydrogen contains n² electrons, nor that there are n² different energies. It says there are n² spatial states carrying one ideal gross energy. Another mistake is to treat 2n² as the same count; that includes spin. Finally, equal energies in an approximate model need not be exactly equal in every experiment.

Worked example

For n = 4, l = 0, 1, 2 and 3. These contribute 1, 3, 5 and 7 mₗ values, so the spatial count is 16 = 4². With two spin projections the one-electron spin-orbital count is 32. If a question asks how many distinct gross energies those states have in nonrelativistic field-free hydrogen, the answer is one: E₄. If it instead introduces a magnetic field, it is no longer valid to assume all mₗ choices have the same energy.

Quick check

1. How many spatial states with n = 2 share the ideal hydrogen gross energy? Answer: There is one 2s state with l = 0 and three 2p states with l = 1, for four spatial states. This equals n² = 4; including spin would give eight spin-orbitals.

Exam focus

Write the sum Σ(2l + 1) rather than memorising n² without its origin. Define whether you count spatial orbitals or spin-orbitals. Name the approximation when using n-only energies. A field generally splits energies; it does not delete the previously counted quantum states.

Advanced insight

Degenerate perturbation theory treats a collection of equal-energy basis states together, because a weak interaction may mix them before producing new energy eigenstates. Simply calculating a separate first-order shift for each arbitrary basis function can miss that mixing. This is especially relevant when an added electric or magnetic interaction selects a preferred combination within a degenerate manifold.

Summary

Shell n contains n² spatial hydrogen states because each l = 0, …, n − 1 contributes 2l + 1 mₗ values. Including spin doubles the count. In the basic Coulomb model these states share one gross energy; fields, electron–electron interactions and finer corrections can split the equality. Degeneracy describes energy equality, not electron population.

Practice questions

1. Derive the number of spatial states for n = 5 by subshell counting. Answer: l = 0, 1, 2, 3, 4 contributes 1 + 3 + 5 + 7 + 9 = 25 states, equal to 5². There are 50 one-electron spin-orbitals if spin is included. 2. Why are 3s and 3p equal in the elementary hydrogen model but generally unequal in a many-electron atom? Answer: Ideal one-electron Coulomb hydrogen has an n-only energy. Other electrons introduce repulsion and shielding, so the effective potential and penetration differ by l, breaking that special equality. 3. A magnetic field splits a set of mₗ states. Has the number of spatial orbitals changed? Answer: No. The available state space still has the same dimension. The field changes the energies and may change which combinations are convenient eigenstates; it does not remove the states.