AkCalculators

Quantum energy levels — Bohr model, transitions, Balmer series and how to compute wavelengths

The historical Bohr model introduced quantized electron orbits for hydrogen and hydrogen-like ions. Although supplanted by modern quantum mechanics, the Bohr expressions for energy levels remain an accurate and extremely useful closed-form description for single-electron systems. For a hydrogen-like atom (a nucleus with charge +Ze and a single bound electron) the energy of level n is:

E_n = −13.6 eV · (Z² / n²)

Here −13.6 eV is the ground-state energy for hydrogen (Z=1, n=1). The negative sign indicates bound (negative) energy; a photon is emitted when the electron moves to a lower (more negative) energy and absorbed when excited to a higher (less negative) energy. The magnitude of energy differences determines the photon wavelength via the Planck relation.

Transition energies and photon wavelength

For a transition between an initial level n_i and final level n_f, the energy change (photon energy) is

ΔE = E_f − E_i

Because energies are negative, for an emission (n_i > n_f) ΔE is negative; the photon energy is the absolute value |ΔE|. Convert ΔE to joules (1 eV = 1.602176634×10⁻¹⁹ J) and then compute wavelength

λ = h c / |ΔE|

where h = 6.62607015×10⁻³⁴ J·s and c = 299,792,458 m/s. Express λ in metres and convert to nanometres (1 nm = 1×10⁻⁹ m) for visible spectral lines.

Worked examples — Balmer series (visible hydrogen lines)

The Balmer series corresponds to transitions that end at n_f=2. The most famous lines include:

• Hα (n=3 → n=2): Using E_n = −13.6/n² eV, E₃ = −13.6/9 = −1.511... eV; E₂ = −13.6/4 = −3.4 eV. ΔE = E₂ − E₃ = −3.4 − (−1.511...) = −1.8889 eV so photon energy ≈ 1.8889 eV. Convert to joules: 1.8889 × 1.602176634e−19 ≈ 3.027×10⁻¹⁹ J. Then λ = hc/ΔE ≈ (6.62607e−34 × 2.99792458e8) / 3.027e−19 ≈ 6.563×10⁻⁷ m = 656.3 nm (red)."
• Hβ (n=4 → n=2): ΔE ≈ 2.551 eV → λ ≈ 486.1 nm (blue-green).
• Hγ (n=5 → n=2): λ ≈ 434.0 nm (violet).

These computed wavelengths match measured values to high precision for hydrogen, illustrating that the Bohr energy expression accurately captures spectral lines for one-electron systems.

Hydrogen-like ions

For ions with Z>1 (e.g., He⁺ with Z=2), energies scale as Z². For example, the ground-state energy of He⁺ is −13.6×4 = −54.4 eV. Transitions produce correspondingly higher-energy (shorter-wavelength) photons. The calculator supports arbitrary integer Z for hydrogen-like atoms.

Units, constants and careful conversions

Key constants used here (exact SI definitions where possible):

• Planck constant h = 6.62607015 × 10⁻³⁴ J·s
• Speed of light c = 299,792,458 m/s
• 1 eV = 1.602176634 × 10⁻¹⁹ J

Compute energies in eV using the Bohr expression and convert to joules for λ calculation. Many mistakes come from unit mismatches (forgetting to convert eV→J or nm→m). The calculator performs these conversions automatically when you choose output units.

Limitations and quantum-mechanical context

The Bohr model is semiclassical and applies exactly only to single-electron systems; multi-electron atoms require quantum mechanics with electron-electron interactions, spin-orbit coupling, and screening effects. Modern quantum mechanics (Schrödinger equation, Dirac equation) explains energy levels more fully (fine structure, Lamb shift). Nevertheless, the Bohr energies are excellent for hydrogenic ions and good pedagogical approximations.

Selection rules and line strengths

Not every pair of levels yields a strong spectral line. Quantum selection rules (dipole transitions) require Δl = ±1 (change in orbital angular momentum quantum number) for electric dipole transitions. The Bohr formula ignores angular momentum quantum number l (it averages over degeneracy), so for detailed spectral intensities use quantum electrodynamics or spectroscopic data.

Practical applications

Calculating transition energies and wavelengths is used in astrophysics (identifying elements in stars), plasma diagnostics, laboratory spectroscopy and teaching. For example, observing the Balmer series in a star’s spectrum confirms the presence of hydrogen and provides redshift measurements from λ shifts.

Using the calculator

1. To compute Eₙ: enter Z and n, choose unit (eV or J), and click Compute.
2. To compute a transition: enter Z, n_i and n_f (n_i > n_f for emission), select output units (eV/J for energy, nm/m/Å for wavelength), and click Compute. Use the step-by-step checkbox to see conversions and intermediate values.

The examples above (Hα, Hβ, Hγ) are widely used benchmarks — you can reproduce those numbers in the Transition tab by entering Z=1 and the appropriate n_i,n_f pair.