Hydrogen Atom Energy Levels Calculator

Calculate electron energy levels, photon wavelengths, and spectral series for hydrogen.

Higher energy level (e.g. 3,4,5...)
Lower level — Lyman(1), Balmer(2), Paschen(3)
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Hydrogen Energy Levels Are Quantized

In the hydrogen atom, the electron can occupy only discrete bound-state energies rather than any continuous negative energy. For the ideal hydrogen atom, En=−13.6eV/n2. The negative sign indicates a bound state relative to a free electron and proton at infinite separation, which is defined as zero energy. Larger n values lie closer to zero and represent less tightly bound states.

A photon is emitted or absorbed only when its energy matches the difference between two allowed levels: hf=|ΔE|=hc/λ. Transitions ending at n=1 form the Lyman series, ending at n=2 the Balmer series, and ending at n=3 the Paschen series. The simple formula neglects fine structure and other small corrections.

En=−13.6eV/n2,   |ΔE|=hf=hc/λ
SymbolMeaningWhy it appears / units
nPrincipal quantum numberPositive integer 1,2,3,… for bound hydrogen states.
EnLevel energyeV; approaches 0 from below as n increases.
λPhoton wavelengthm or nm; set by the energy difference between levels.

A downward transition releases a photon; an upward transition requires absorption. The ionization energy from the ground state is 13.6 eV because the electron must be raised from −13.6 eV to the zero-energy continuum.

Hydrogen levels should become less negative as n increases. Because En∝−1/n2, the spacing between adjacent high-n levels shrinks and the series approaches zero from below. A transition photon energy must equal the positive magnitude of the difference between the two levels.

Worked Examples

Example 1: Balmer Hα: n=3→2
ΔE=13.6(1/4-1/9)=1.89eV
Result: λ=656 nm — red line
Most prominent visible hydrogen line
Example 2: Lyman α: n=2→1
ΔE=13.6(1-1/4)=10.2eV
Result: λ=121.6 nm — UV
Most important line in astronomy
Example 3: n=3 to n=2
ΔE=13.6(1/22−1/32)eV
Result: 1.889eV, λ≈656.3nm
This is the red Hα line of the Balmer series.
Example 4: Ionizing n=2
Energy needed=0−(−13.6/4)eV
Result: 3.40eV
Excited hydrogen requires less energy to ionize than ground-state hydrogen.

Common Mistakes

⚠️
Dropping the negative sign without understanding it

Bound-state energies are negative relative to a separated proton and electron. Photon energy uses the positive magnitude of the level difference.

⚠️
Subtracting levels in the wrong order and reporting negative photon energy

Emission and absorption photon energies are positive; use |ΔE| and determine separately whether the transition is upward or downward.

⚠️
Applying the formula unchanged to multi-electron atoms

The −13.6/n² spectrum is for hydrogen and hydrogen-like one-electron ions with the appropriate nuclear charge scaling.

Frequently Asked Questions

Why only specific wavelengths?
Bohr model: electrons occupy discrete energy levels E_n=-13.6/n² eV. Photon emitted only when electron transitions between levels. Energy of photon = ΔE between levels → specific wavelength.
Hydrogen spectrum in astronomy?
Hydrogen is the most abundant element (75% of universe by mass). Its spectral lines appear in virtually all stars. Redshift of these lines reveals galaxy recession velocities — evidence for expanding universe.
Why are hydrogen energies negative?
Zero energy is chosen for a free electron and proton infinitely far apart. A bound atom has lower energy than that separated state, so its energy is negative.
What happens as n approaches infinity?
The level energy approaches zero from below and the level spacing becomes very small. Reaching the continuum corresponds to ionization.
Why are Balmer lines visible?
Balmer transitions end at n=2 and produce wavelengths that include several strong visible lines, such as Hα near 656 nm and Hβ near 486 nm.
Can hydrogen absorb any photon energy?
A bound electron absorbs discrete photon energies that match allowed upward transitions. Photons above an ionization threshold can also eject the electron, with excess energy appearing as kinetic energy.
Why are hydrogen bound-state energies negative?
The zero of energy is conventionally chosen for a free electron and proton infinitely far apart. A bound electron has lower total energy, so En is negative. Supplying energy equal to |En| can ionize the atom from that level and bring the total energy to zero.

Formula Explorer connections

Interpretation: This relationship connects motion, force, momentum, work or energy in a mechanical system. Assumption: Choose a consistent reference direction and unit system. The model may assume constant acceleration, rigid bodies, negligible losses or an isolated system.

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