-13.6
eV
-3.4
eV
10.2
eV
984.15
kJ/mol
121.55
nm
2.4663e+15
Hz
1
1
-13.6
eV
-3.4
eV
10.2
eV
984.15
kJ/mol
121.55
nm
2.4663e+15
Hz
1
1
The hydrogen atom is the simplest atomic system and the only one for which the Schrödinger equation can be solved exactly. Its energy levels follow a beautifully simple pattern: En = −13.6/n² eV, where n is the principal quantum number. The Hydrogen Atom Energy Levels Calculator computes the energies of any two levels and determines the transition energy, photon wavelength, and frequency for the corresponding spectral line. These transitions produce the characteristic spectral series of hydrogen: Lyman (UV), Balmer (visible), Paschen (infrared), and higher series. The hydrogen spectrum provided crucial evidence for quantum theory and led to the development of the Bohr model and ultimately quantum mechanics. Understanding hydrogen energy levels is essential for atomic spectroscopy, astrophysics (hydrogen is the most abundant element), plasma physics, and the precise determination of fundamental constants. The 13.6 eV ground state energy equals the ionization energy of hydrogen.
The energy levels of the hydrogen atom are given by:
$$E_n = -\frac{13.6 \text{ eV}}{n^2} = -\frac{m_e e^4}{8\epsilon_0^2 h^2 n^2}$$
where n = 1, 2, 3, ... is the principal quantum number. The negative sign indicates bound states (below the ionization threshold E = 0).
The energy of a photon emitted or absorbed in a transition between levels n₁ and n₂ (n₂ > n₁) is:
$$\Delta E = 13.6 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV}$$
The corresponding wavelength and frequency:
$$\frac{1}{\lambda} = R_\infty \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$
$$\nu = \frac{\Delta E}{h}$$
The spectral series are named by the lower level: Lyman (n₁=1, UV), Balmer (n₁=2, visible/UV), Paschen (n₁=3, near-IR), Brackett (n₁=4, IR), and Pfund (n₁=5, far-IR).
The ground state (n=1) at −13.6 eV is the most tightly bound. As n increases, levels converge toward 0 eV (ionization limit). The Lyman series transitions to n=1 produce high-energy UV photons (10.2–13.6 eV). The Balmer series (n₁=2) produces visible lines: Hα (656.3 nm, red), Hβ (486.1 nm, blue-green), Hγ (434.0 nm, violet), and Hδ (410.2 nm, violet). These are the lines originally observed by Balmer in 1885. The ionization energy is 13.6 eV—the energy needed to completely remove the electron from the ground state. For transitions to very high n, the photon energy approaches 13.6/n₁² eV (the series limit). These results apply strictly to hydrogen (Z=1); for hydrogen-like ions, multiply by Z².
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The Lyman-alpha line at 121.6 nm is the strongest hydrogen emission line, crucial in astrophysics for studying distant galaxies and the intergalactic medium.
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The Hα line at 656.5 nm (red) is the most prominent visible hydrogen line, responsible for the red color of emission nebulae in astronomy.
The negative sign indicates a bound state—the electron has less energy than a free electron at rest (E = 0). The magnitude |E_n| represents the energy needed to ionize the atom from level n. More negative = more tightly bound.
The ionization energy is the energy to remove the electron from the ground state: |E₁| = 13.6 eV = 1312 kJ/mol. This corresponds to photon wavelength of 91.2 nm—the Lyman series limit.
Lyman (n₁=1): UV, 91.2–121.6 nm. Balmer (n₁=2): visible/UV, 364.6–656.3 nm. Paschen (n₁=3): near-IR, 820.4–1875 nm. Brackett (n₁=4): IR. Pfund (n₁=5): far-IR.
13.6 eV = m_e·e⁴/(8ε₀²h²) is determined entirely by fundamental constants: electron mass, electron charge, permittivity of free space, and Planck's constant. It is also equal to one Rydberg (Ry) of energy.
Only for hydrogen-like ions (one electron): He⁺, Li²⁺, Be³⁺, etc. For these, E_n = −13.6Z²/n² eV. Multi-electron atoms require approximate methods (Hartree-Fock, DFT) due to electron-electron repulsion.
Johann Balmer (1885) empirically found: 1/λ = R(1/4 − 1/n²) for n = 3, 4, 5, ..., predicting visible hydrogen lines. This was later generalized by Rydberg to all series and explained by Bohr's quantum theory in 1913.
Since E_n = −13.6/n², the spacing ΔE between adjacent levels decreases as n increases: ΔE ≈ 27.2/n³ eV for large n. At the ionization limit (n→∞), levels merge into a continuum.
Four quantum numbers: n (principal, 1,2,3...), l (angular momentum, 0 to n−1), m_l (magnetic, −l to +l), and m_s (spin, ±1/2). Energy depends only on n in the non-relativistic treatment; fine structure lifts the l-degeneracy.
Light from distant quasars passes through intervening hydrogen clouds, each absorbing at the Lyman-alpha wavelength (121.6 nm) but redshifted by cosmic expansion. This produces a 'forest' of absorption lines revealing the distribution of matter in the universe.
The hydrogen spectrum was a key puzzle that classical physics could not explain. Bohr's quantized orbits (1913) predicted the observed spectral lines perfectly, and Schrödinger's wave mechanics (1926) provided the full theoretical framework, including quantum numbers and selection rules.
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