The Bohr Model — Energy Levels, The Complete Guide
In 1913, Niels Bohr proposed a radical model of the hydrogen atom that, for the first time, correctly predicted the wavelengths of light hydrogen emits and absorbs — a puzzle that had stumped physicists for decades. Bohr's key insight was that electrons don't orbit the nucleus at just any distance, as classical physics would allow, but only at specific, quantised energy levels. An electron can jump between these levels by absorbing or emitting a photon of exactly the right energy, but it cannot exist anywhere in between.
Though later superseded by the full quantum mechanical treatment of the atom (which describes electrons as probability clouds rather than orbiting particles), the Bohr model remains an exceptionally useful — and for hydrogen, exactly correct — tool for calculating atomic energy levels and predicting the precise colours of light atoms emit and absorb.
What are Bohr Energy Levels?
In the Bohr model, a hydrogen atom's electron can only occupy specific allowed orbits, labelled by a positive integer called the principal quantum number, n (n = 1, 2, 3, …). Each orbit corresponds to a specific, fixed energy — lower n means a smaller orbit closer to the nucleus and more negative (more tightly bound) energy; higher n means a larger orbit and energy closer to zero (more loosely bound). The lowest energy state, n = 1, is called the ground state; higher levels are called excited states.
Energies are negative by convention, representing a bound electron: zero energy corresponds to a free electron that has just barely escaped the atom's pull (ionisation), and every bound level has energy below this. As n increases toward infinity, the energy approaches zero — an infinite number of increasingly closely spaced levels converge just below the ionisation threshold.
The Formula Explained
Eₙ is the energy of level n, in electron-volts. −13.6 eV is the ground-state energy of hydrogen (n=1) — this specific number emerges from combining the electron's mass, charge, and Planck's constant, and represents the ionisation energy of hydrogen: exactly the energy needed to remove the electron completely from its lowest orbit. When an electron transitions between two levels, the photon energy involved equals the difference: ΔE = |E_final − E_initial|, related to frequency by ΔE = hf and to wavelength by λ = c/f = hc/ΔE.
If the electron falls from a higher level to a lower one, a photon of exactly that energy is emitted. If the electron absorbs an incoming photon of exactly that energy, it jumps from a lower level to a higher one — this is absorption. Only photons with precisely the right energy can be absorbed or are produced by emission; this is why atomic emission and absorption spectra consist of sharp, discrete lines rather than a continuous range of colours.
How to Use This Calculator
Use "Single level" to find the energy of any individual level n. Use "Transition" to find the photon energy, frequency, and wavelength involved when an electron moves between two levels — enter the initial level n₁ and final level n₂; the calculator automatically determines whether this represents emission (electron falling to a lower level) or absorption (electron rising to a higher level).
Worked Example 1 — Ground State Energy
E₁ = −13.6/1² = −13.6 eV — the hydrogen atom's ground state, and the energy required to fully ionise a hydrogen atom from rest.
Worked Example 2 — The Lyman-Alpha Line
Problem: Find the photon wavelength emitted when an electron falls from n=2 to n=1.
E₂ = −13.6/4 = −3.4 eV, E₁ = −13.6 eV
ΔE = |−3.4 − (−13.6)| = 10.2 eV
λ = hc/ΔE = 121.5 nm — the famous Lyman-alpha line, in the ultraviolet
Worked Example 3 — The Balmer-Alpha Line (Visible Light)
Problem: Find the wavelength emitted when an electron falls from n=3 to n=2 — one of the few hydrogen transitions visible to the naked eye.
E₃ = −1.511 eV, E₂ = −3.4 eV
ΔE = 1.889 eV
λ = 656.3 nm — the distinctive red line of hydrogen's Balmer series, visible in stellar spectra and neon-style discharge tubes
Spectral Series and Astronomical Applications
Transitions ending at n=1 form the Lyman series (ultraviolet), transitions ending at n=2 form the Balmer series (mostly visible light — the only series easily observed with ordinary optical telescopes), and transitions ending at n=3 form the Paschen series (infrared). Each series was discovered independently, decades before Bohr's model explained why they follow such a precise mathematical pattern.
Astronomers rely heavily on hydrogen's spectral lines to identify the composition, temperature, and even the motion (via Doppler shift) of distant stars and galaxies — the characteristic pattern of Balmer lines is one of the most recognisable signatures in all of astronomical spectroscopy, visible in the light from stars billions of light-years away exactly as it appears in a hydrogen discharge tube in a school laboratory. The 21 cm hydrogen line (a much lower-energy transition involving electron spin rather than principal quantum number) is likewise a cornerstone tool of radio astronomy, used to map the distribution of hydrogen gas throughout our own galaxy and beyond.
Limitations of the Bohr Model
The Bohr model works with remarkable precision for hydrogen (and hydrogen-like ions with a single electron), but fails for any atom with more than one electron, since it doesn't account for electron-electron repulsion or the true quantum mechanical wave nature of electrons. The full quantum mechanical model, developed shortly after by Schrödinger and Heisenberg, replaces Bohr's neat circular orbits with probability distributions (orbitals) and correctly handles multi-electron atoms — but for hydrogen specifically, the energy levels it predicts are identical to Bohr's simpler formula, making the Bohr model a genuinely useful and historically important stepping stone rather than merely an approximation.
Historical Context — Solving the Rydberg Mystery
Decades before Bohr, Swiss mathematician Johann Balmer had noticed in 1885 that the wavelengths of hydrogen's visible spectral lines fit a simple numerical pattern, later generalised by Johannes Rydberg into what became known as the Rydberg formula. Nobody, however, could explain why hydrogen's spectrum should follow this particular mathematical pattern — it was an empirical curiosity with no theoretical foundation. Bohr's genius was recognising that quantising the electron's angular momentum (allowing only specific, discrete orbital states) reproduced the Rydberg formula exactly, providing the first physical explanation for a pattern that had puzzled physicists for nearly thirty years.
This success was remarkable given how much of Bohr's original reasoning has since been superseded — he pictured electrons as tiny planets orbiting the nucleus in fixed circular paths, an image modern quantum mechanics has completely replaced with probability clouds. Yet the energy level formula itself survived entirely intact, a testament to how a physically incomplete model can still capture an exactly correct piece of mathematics.
The Correspondence Principle
Bohr also introduced the correspondence principle: the idea that quantum predictions must smoothly match classical physics predictions in the limit of large quantum numbers. For very high values of n, the energy levels become so closely spaced that they form an almost continuous range, indistinguishable from the continuous energies classical physics would predict for an orbiting electron. This principle became an important guiding tool in the development of quantum mechanics more broadly — any new quantum theory needed to reduce to familiar classical behaviour in the appropriate limit, providing a valuable check on whether a proposed quantum theory made physical sense.
This same idea appears throughout physics whenever a more fundamental theory must "contain" an older, simpler theory as a special case — Einstein's relativity reduces to Newtonian mechanics at low speeds, and general relativity reduces to Newtonian gravity for weak fields, in exactly the same spirit as Bohr's correspondence principle for quantum mechanics reducing to classical orbits at high n.