Comparing Hydrogen Atom Models: Bohr, Sommerfeld, and Quantum Wave MechanicsThe hydrogen atom — the simplest atom with a single electron bound to a single proton — played a central role in the development of atomic physics. Over the late 19th and early 20th centuries, physicists proposed successive models to explain observed spectral lines, atomic stability, and the behavior of electrons. Each model contributed important concepts and helped reveal limitations that pointed toward the next advance. This article compares three landmark approaches: the Bohr model, Sommerfeld’s extension, and the modern quantum wave-mechanical (Schrödinger) treatment. We describe each model’s assumptions, successes, limitations, and how they connect to one another.
Historical context and experimental clues
By the end of the 19th century, experiments had produced precise measurements of hydrogen’s emission and absorption spectra — distinct lines at well-defined wavelengths. The Rydberg formula empirically captured the wavelengths of spectral lines, but classical physics couldn’t explain why atoms were stable or why discrete spectral lines appeared. The early 20th century provided new ideas: quantization of energy and momenta, discrete electronic states, and eventually a wave description of matter. The Bohr model (1913) offered the first successful theory giving energy levels that matched the Rydberg formula. Sommerfeld (1916–1917) refined Bohr’s model to include elliptical orbits and special-relativistic corrections, improving agreement for fine-structure splitting. Later, Schrödinger’s wave mechanics (1926) provided a deeper, more general framework that subsumed the earlier models and added predictive power for multi-electron atoms and chemical behavior.
The Bohr Model
Core assumptions
- The electron moves in circular orbits around a fixed proton under the Coulomb force (classical mechanics for motion).
- Only orbits with quantized angular momentum are allowed: L = nħ, where n = 1, 2, 3, …
- Electrons in these allowed orbits do not radiate energy (stationary states).
- Radiation occurs only when an electron jumps between allowed orbits; the emitted photon energy equals the difference between orbital energies: ΔE = hν.
Key results
- Quantized radii: rn = a0 n^2, where a0 = 0.529 Å (Bohr radius).
- Energy levels: En = −13.6 eV / n^2. This reproduces the Rydberg formula and correctly predicts spectral line series (Lyman, Balmer, Paschen, etc.).
- The model gives accurate wavelengths for hydrogenic atoms (single-electron ions like He+, Li2+) when replacing the proton charge by Ze.
Successes
- First theory to derive the Rydberg formula from physical principles.
- Simple, intuitive picture linking discrete spectra to quantized energy levels.
- Correct scaling of energy with nuclear charge Z for hydrogen-like ions.
Limitations
- Treats the proton as a fixed center (ignores reduced mass correction unless added).
- Assumes circular orbits — no explanation for orbital shapes or angular distributions.
- Cannot explain fine structure (small splittings of spectral lines), hyperfine structure, or Zeeman/Stark effects quantitatively.
- Does not account for electron spin or the Pauli exclusion principle.
- Fundamentally semi-classical: uses classical trajectories together with a single quantum rule (quantized angular momentum).
Sommerfeld Extension (Old Quantum Theory refinement)
Motivations and modifications
Arnold Sommerfeld extended Bohr’s ideas to explain observed fine structure and to generalize allowed orbits beyond perfect circles. He introduced:
- Elliptical orbits characterized by two quantum numbers: the principal quantum number n and the azimuthal (or angular) quantum number k (related to orbital eccentricity).
- Quantization of action variables (integrals of momentum over a cycle) using the Bohr–Sommerfeld quantization condition: ∮ p_i dq_i = n_i h for each independent coordinate.
- Inclusion of special-relativistic corrections to the electron’s kinetic energy for high orbital velocities near the nucleus.
Key outcomes
- Predicts energy level splitting (fine structure) due to relativistic corrections and dependence on orbital eccentricity.
- Introduces the quantum number ℓ (orbital angular momentum) informally, enabling a richer set of allowed states than Bohr’s simple circular orbits.
- Produces more accurate spectral line positions for hydrogen by accounting for small shifts (fine structure).
Successes
- Improved agreement with high-resolution spectral measurements (fine structure) compared with Bohr.
- Maintained intuitive orbital picture while introducing multi-quantum-number classification.
Limitations
- Still semi-classical: relies on quantized classical orbits and action integrals, lacking a full wave interpretation.
- Unable to account for phenomena emerging from full quantum mechanics: electron spin, intrinsic magnetic moment, the correct multiplicity of states, and precise selection rules derived from wavefunctions.
- Broke down for systems where action-angle variables are not separable; lacked general applicability.
- Could not explain intensities of spectral lines or all selection rules rigorously.
Quantum Wave Mechanics (Schrödinger model)
Core framework
- The electron is described by a wavefunction ψ(r, t) whose dynamics follow the Schrödinger equation (time-dependent or time-independent for stationary states):
- Time-independent: Hψ = Eψ, where H is the Hamiltonian operator (kinetic + potential).
- For hydrogen (stationary proton), the potential is the Coulomb potential V® = −(Ze^2)/(4πε0 r) with Z = 1.
- Solutions yield quantized energy eigenvalues and spatial wavefunctions (orbitals) ψ_{nℓm}(r, θ, φ) labeled by quantum numbers n, ℓ, m.
Quantum numbers and their meanings
- n (principal): determines energy level; En = −13.6 eV / n^2 for hydrogen.
- ℓ (orbital angular momentum): integer with 0 ≤ ℓ ≤ n−1; determines orbital shape (s, p, d…).
- m (magnetic): integer with −ℓ ≤ m ≤ ℓ; determines orientation-related properties.
- Spin s (intrinsic; added later via Pauli spinors): electron spin ⁄2 gives two spin states, introducing additional multiplicity.
Wavefunctions and probability interpretation
- |ψ|^2 gives the probability density of finding the electron at position r — a fundamentally probabilistic interpretation that replaces definite orbits.
- Radial and angular parts separate: ψ{nℓm}(r, θ, φ) = R{nℓ}® Y{ℓm}(θ, φ), where Y{ℓm} are spherical harmonics describing angular dependence.
- Orbitals exhibit nodes and characteristic shapes (spherical s orbitals, dumbbell-shaped p orbitals, cloverleaf d orbitals).
Explanatory power and successes
- Exactly reproduces Bohr energy levels for hydrogen (En = −13.6 eV/n^2) while providing the physical meaning of quantum numbers ℓ and m.
- Explains fine structure (when relativistic corrections and spin–orbit coupling are included via Dirac equation or perturbation theory) and hyperfine structure with additional physics.
- Provides selection rules for transitions (Δℓ = ±1, etc.) from matrix elements of the dipole operator, predicting which spectral lines are allowed or forbidden.
- Extends naturally to multi-electron atoms (with approximations) and forms the basis of modern chemistry and atomic physics.
- Predicts electron probability distributions, chemical bonding behavior, and angular momentum quantization in a consistent, general framework.
Limitations and the Dirac refinement
- Non-relativistic Schrödinger equation neglects electron spin and relativistic effects. The Dirac equation (relativistic quantum mechanics) accounts for spin naturally and predicts fine-structure corrections and the existence of antiparticles.
- Many-body problems (multi-electron atoms) require approximation methods (Hartree–Fock, DFT, CI) because exact solutions are intractable.
Direct comparison: Bohr vs. Sommerfeld vs. Schrödinger
| Feature | Bohr | Sommerfeld | Schrödinger (Wave Mechanics) |
|---|---|---|---|
| Nature | Semi-classical, circular orbits | Semi-classical, elliptical orbits + action quantization | Fully quantum, wavefunctions |
| Quantum rules | L = nħ (angular momentum quantization) | Action integrals quantized: ∮ p dq = n h | Operators and eigenvalue equations; multiple quantum numbers arise naturally |
| Energy levels (hydrogen) | En = −13.6 eV / n^2 (correct) | Same principal energies; fine corrections included | Same principal energies; full structure from solutions |
| Predicts fine structure? | No | Partially (via relativity) | Yes (with relativistic extensions like Dirac) |
| Predicts orbital shapes | Circular only | Elliptical allowed | Probability distributions (s, p, d shapes) |
| Spin included? | No | No | No (in Schrödinger) — included in Dirac or via spinors |
| Applicability | Hydrogenic atoms qualitatively | Improved hydrogenic spectra | General, foundation for modern atomic and molecular physics |
How earlier models connect to full quantum theory
- Bohr and Sommerfeld captured essential quantization rules that hinted at a discrete structure of atomic states. Mathematically, the Bohr quantization condition can be seen as a special-case quantization of action variables; in the semiclassical (WKB) limit, Bohr–Sommerfeld quantization emerges as an approximation to the full quantum eigenvalue problem.
- The Bohr radius a0 and the energy scale −13.6 eV appear naturally in Schrödinger’s hydrogen solutions; thus the new theory preserved quantitatively successful predictions while offering a radically different interpretation (probability waves vs. definite orbits).
- Sommerfeld’s relativistic corrections anticipated results that later follow from the Dirac equation when one treats relativity correctly and includes spin.
Physical interpretation: orbits versus orbitals
The Bohr and Sommerfeld pictures imagine electrons traveling along definite trajectories (circles or ellipses). Quantum wave mechanics replaces trajectories with orbitals: spatial distributions describing probabilities. This shift is more than aesthetic — it changes how we predict and understand experiments. Interference, tunneling, and chemical bonding are natural in a wave description but incompatible with a strict particle-on-a-path picture. The classical-like orbits survive as semiclassical approximations: for large quantum numbers (n ≫ 1), the probability distribution concentrates near classical trajectories (correspondence principle).
Practical implications and modern relevance
- For quick back-of-the-envelope calculations of hydrogenic energy levels and spectral line wavelengths, Bohr’s formulas remain useful.
- Sommerfeld’s extensions are historically important and useful pedagogically to show how relativity modifies quantized systems and to introduce action-angle quantization.
- Schrödinger wave mechanics (and its relativistic generalizations like Dirac) underpin all accurate modern calculations in atomic, molecular, and optical physics, as well as quantum chemistry.
Summary
- Bohr introduced quantized energy levels and explained hydrogen spectra with a simple, semi-classical circular-orbit model.
- Sommerfeld generalized Bohr by allowing elliptical orbits and by adding relativistic corrections, accounting for fine-structure splitting more accurately.
- Schrödinger (wave mechanics) replaced classical orbits with wavefunctions, providing a consistent, general framework that reproduces earlier energy results while explaining orbital shapes, selection rules, and a wide range of quantum phenomena.
Each model marks a conceptual step: from discrete rules imposed on classical motion to a full wave-based theory where quantization is intrinsic. Together they trace the path of physics from the intuitive old quantum theory to the deeper, predictive structure of modern quantum mechanics.