Hydrogen Atom Energy Levels and Degeneracy

The $n^2$/$2n^2$ degeneracy counting is the direct ancestor of the shell-capacity numbers ($2, 8, 18, 32$) used every day in general chemistry to build up the periodic table, even though real multi-electron atoms break the $\ell$-part of the degeneracy (which is exactly why $4s$ fills before $3d$, a fact with enormous consequences for transition-metal chemistry, catalysis, and coordination chemistry). Precision spectroscopy of hydrogen and hydrogen-like ions remains one of the most stringent tests of quantum electrodynamics and fundamental constants: modern laser spectroscopy of the $1s$–$2s$ two-photon transition in atomic hydrogen has been measured to better than one part in $10^{15}$, and comparing the measured splitting between states that would be "accidentally" degenerate in the pure Coulomb picture (like $2s_{1/2}$ and $2p_{1/2}$, the Lamb shift) directly probes vacuum polarization and electron self-energy — QED effects with no classical analogue. In astrophysics, hydrogen-like ion spectra (He⁺, Li²⁺, and highly stripped heavy ions in stellar coronae, solar flares, and X-ray binary accretion disks) are read off using exactly the $Z^2/n^2$ scaling worked out above, letting astronomers infer both the ionization state and temperature of extremely hot plasmas from X-ray line positions. In quantum control and Rydberg-atom physics — a field central to modern quantum computing and quantum simulation platforms — atoms are excited to very high $n$ (Rydberg states), where the near-degeneracy of different $\ell$ within a shell (slightly broken by quantum defects in non-hydrogenic atoms) is deliberately exploited to engineer strong, tunable, long-range dipolar interactions between atoms for two-qubit gates. Finally, the entire conceptual toolkit here — separating essential (symmetry-guaranteed) from accidental (potential-specific) degeneracy — recurs whenever chemists analyze Term Symbols and Atomic States in multi-electron atoms, and whenever spectroscopists interpret Radial Distribution Functions to rationalize orbital penetration and shielding effects that ultimately lift hydrogen's special accidental degeneracy in every atom heavier than hydrogen.

Ask the tutor: How exactly does electron screening in a multi-electron atom convert the flat $n^2$-degenerate hydrogen shell into the split $ns<np<nd<nf$ ordering used in the aufbau principle?

Solving the Hydrogen Atom Schrödinger equation by separation of variables in spherical coordinates gives $\psi_{n\ell m}(r,\theta,\phi) = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi)$. The angular factor $Y_\ell^m$, the spherical harmonic, is completely universal: it is the same function for any central potential $V(r)$, because it comes purely from the angular part of the kinetic energy operator (the $\hat{L}^2$ operator) commuting with any Hamiltonian of the form $\hat{H} = \hat{T}_r + \hat{L}^2/2m r^2 + V(r)$. The radial factor $R_{n\ell}(r)$, by contrast, is where the specific shape of $V(r)$ enters, and it is the radial equation that ultimately fixes the energy. For the hydrogen-like radial equation, imposing square-integrability (a bound, normalizable state) forces a quantization condition that, after the dust settles, reads

where $n_r$ is the number of radial nodes. This is the origin of the famous rule $\ell \le n-1$: since $n_r \ge 0$, we need $\ell \le n - 1$. The energy formula that emerges from solving the radial equation depends only on the combination $n = n_r + \ell + 1$ — not on $n_r$ and $\ell$ separately. That is the mathematical fingerprint of accidental degeneracy: two completely different orbitals, say $2s$ ($n_r=1,\ell=0$) and $2p$ ($n_r=0,\ell=1$), both have $n=2$ and are exactly degenerate, even though one has a radial node and no angular node while the other has an angular node and no radial node. The figure below shows the degeneracy structure for $n=1,2,3$ hydrogen levels, with the orbital "boxes" grouped horizontally by $\ell$ but sitting at the same height (same $E_n$) within each shell.

Notice what this diagram is really asserting: the $2s$ and $2p$ bars sit at exactly the same height, $-0.25$ Ry, even though they are drawn at different horizontal positions to distinguish them by shape. In any non-Coulombic central potential (say, a screened potential in a many-electron atom, or a 3-D isotropic harmonic oscillator with an added anharmonic term), the $2s$ and $2p$ bars would separate vertically — that is precisely how orbital energy ordering ($4s$ before $3d$, for instance) arises in the periodic table. The next figure decomposes a single shell $n=3$ into its full $m$-degeneracy as well, making explicit the counting that gives $n^2$ states.

The distinction drawn at the bottom of that figure is the conceptual heart of this article. The $2\ell+1$ states within a fixed $\ell$ (the $m$-degeneracy) exist because empty space has no preferred axis: rotate your coordinate system and $\hat{H}$ is unchanged, so states related by rotation must be degenerate. This is essential degeneracy — it is required by symmetry in every central potential without exception, hydrogen or not. The further degeneracy across different $\ell$ within the same $n$ is a much more delicate, Coulomb-specific effect, discussed in the derivations below.

The hydrogen-like (one electron, nuclear charge $Ze$) bound-state energy is

where:

• $Z$ is the nuclear charge (1 for H, 2 for He⁺, 3 for Li²⁺, …),
• $\mu$ is the reduced mass of the electron–nucleus system, $\mu = m_e m_N/(m_e+m_N) \approx m_e$,
• $n$ is the principal quantum number, running over positive integers,
• $1\,\text{Ry} = 13.6057\ \text{eV}$ is the Rydberg energy, and $E_h = 27.2114$ eV (hartree) is the atomic energy unit, so $1\,\text{Ry} = \tfrac12 E_h$.

Reading: the energy is negative (bound state, energy measured relative to the ionized, infinitely separated electron and nucleus at rest), scales as $Z^2$ (a more highly charged nucleus binds the electron much more tightly — this is why He⁺'s ground state is $4\times$ deeper than H's), and falls off as $1/n^2$ (levels crowd together rapidly as $n$ increases, converging to the ionization threshold $E_\infty = 0$). The allowed quantum numbers for a given $n$ are

where:

• $\ell$ is the azimuthal (orbital angular momentum) quantum number, restricted by the radial quantization condition to be strictly less than $n$,
• $m$ is the magnetic quantum number, whose $2\ell + 1$ allowed values follow from the eigenvalue spectrum of $\hat{L}_z$ once $\ell$ is fixed.

Reading: an $n$-shell contains $n$ subshells ($\ell = 0$ through $n-1$), and subshell $\ell$ contains $2\ell+1$ orbitals. This produces the shell-filling pattern 1 (for $n=1$: only $1s$), 4 (for $n=2$: $2s,2p_x,2p_y,2p_z$), 9 (for $n=3$), etc. The total spatial degeneracy of level $n$ is obtained by summing over all $\ell$:

where:

• the sum $\sum_{\ell=0}^{n-1}(2\ell+1)$ is an arithmetic series of odd numbers $1+3+5+\cdots+(2n-1)$, which is a classical identity equal to $n^2$.

Reading: this is the single most quoted degeneracy fact in atomic quantum chemistry. Including the two spin states of the electron (spin degeneracy is entirely separate physics — it has nothing to do with the spatial Coulomb symmetry, it is a relativistic/intrinsic property), the total degeneracy including spin is $2n^2$, which is exactly the capacity rule behind the shell structure $2, 8, 18, 32,\dots$ that underlies (in a simplified, pre-screening form) the layout of the periodic table.

Example 1 (foundational): Full quantum-number enumeration and energy for $n=4$, hydrogen atom ($Z=1$). List every allowed $(\ell, m)$ combination for $n=4$ and verify the total count equals $n^2$. Allowed $\ell$: since $\ell = 0,\dots,n-1$, we have $\ell = 0,1,2,3$ (spectroscopic labels $s,p,d,f$). For each $\ell$, the number of $m$ values is $2\ell+1$:

Sum:

The energy for every one of these 16 states is identical:

Limiting-case check: as $n\to\infty$, $E_n\to 0^-$ (states pile up just below the ionization limit — this is the hallmark of a Coulomb potential's infinite bound-state ladder, in contrast to, e.g., a finite square well which has only finitely many bound states). Including spin doubles every count: $2g_4 = 32$ total spin-orbitals in the $n=4$ shell, matching the familiar "$2,8,18,32$" shell-capacity sequence.

Example 2 (applied): Comparing He⁺ ($Z=2$) and H ($Z=1$) at $n=2$, and connecting to the Balmer-series-like transition. He⁺ is a hydrogen-like ion (one electron, $Z=2$) — a system of real spectroscopic importance (its $n=4\to3$ transition mimics part of the hydrogen Balmer series and was historically confused with it in early stellar spectra, the "Pickering series"). For He⁺, using the exact formula (ignoring the small reduced-mass correction, which shifts things by less than 0.03%):

At $n=2$:

Compare with hydrogen's ground state, $E_1(\text{H}) = -1.000\,\text{Ry} = -13.606$ eV. These are numerically coincident (up to the reduced-mass correction) — this is a direct manifestation of the $Z^2/n^2$ scaling: He⁺ at $n=2$ mimics H at $n=1$ because $Z^2/n^2 = 4/4 = 1/1$. Now consider the $n=4\to n=3$ emission transition in He⁺:

Converting to a wavelength using $1\ \text{Ry} = 2.1799\times10^{-18}$ J and $\lambda = hc/\Delta E$:

This falls in the blue-green visible region — one member of the He⁺ Pickering series, historically mistaken for a hydrogen line before it was correctly assigned to He⁺. Limiting-case check: as $Z\to$ larger values (say a hydrogen-like Fe²⁵⁺ ion, $Z=26$), the same transition would scale as $\Delta E \propto Z^2$, pushing the emission from visible light all the way into the X-ray region — precisely how astrophysicists identify highly ionized species in solar-flare and accretion-disk X-ray spectra.

Example 3 (exam / competition level): Constructing an explicit degenerate pair at $n=3$ and demonstrating the accidental degeneracy numerically from the radial wavefunctions. Show explicitly, using the normalized radial functions, that $R_{30}(r)$ ($3s$) and $R_{32}(r)$ ($3d$) — which look completely different in shape — are eigenfunctions of the same radial Hamiltonian eigenvalue $E_3 = -Z^2/9$ Ry, by evaluating the expectation value of the Hamiltonian directly (rather than trusting the general formula) at one representative sample point argument, and cross-checking with the general quantization condition. The hydrogen-like radial functions (in units where $a_0/Z \to a$; here consider Z=1 for simplicity so $a=a_0$) are:

Both functions decay with the same exponential rate $e^{-r/3a_0}$ — note that this exponential decay constant is set only by $n=3$ (recall from the derivation, $\lambda = Z/n$, and $u\sim e^{-\lambda \rho}$), not by $\ell$. This is already a strong hint of the shared energy, since the asymptotic decay rate of a bound-state wavefunction is directly tied to $E$ via $\lambda = \sqrt{-2\mu E a_0^2/\hbar^2}$; identical decay constants can only occur for identical $|E|$. To make this quantitative, apply the quantization condition from the derivation directly to each case: For $3s$: $n_r = 2$ (two radial nodes — matching the two roots of the quadratic factor $27-18x+2x^2$), $\ell=0$, so

For $3d$: $n_r=0$ (no radial nodes — the function $r^2 e^{-r/3a_0}$ is nodeless for $r>0$), $\ell=2$, so

Both give $n=3$, hence identically

Despite $3s$ having two radial nodes and zero angular nodes while $3d$ has zero radial nodes and two angular nodes (the $Y_2^{\pm2}$ or real $d_{xy}, d_{x^2-y^2}$ angular parts), the total node count is identical: $n_r + \ell = 2$ in both cases. This total-node invariant, $n - 1 = n_r+\ell$, is exactly what the Coulomb potential conserves, and is the operational, calculation-level manifestation of the $SO(4)$ symmetry argument. Limiting-case check: if we instead used a modified potential $V(r) = -Ze^2/(4\pi\epsilon_0 r) + \delta/r^2$ with small $\delta\ne 0$ (a Coulomb-plus-centrifugal perturbation, as occurs approximately for penetrating orbitals in alkali-metal valence electrons), the effective $\ell(\ell+1)$ in the radial equation would be replaced by $\ell(\ell+1)+2\mu\delta/\hbar^2$, giving an effective non-integer $\ell'$ and a quantization condition $n = n_r+\ell'+1$ that depends on $\ell$ separately — immediately splitting $3s$ below $3d$ (lower $\ell$ penetrates the core more and is stabilized), exactly the origin of the orbital energy ordering used in Term Symbols and Atomic States and the aufbau principle for real multi-electron atoms.