Quantum Energy Levels Calculator
Calculate atomic energy levels, transition energies, photon wavelengths, and spectral series for hydrogen-like atoms.
About this calculator
This calculator applies the Bohr model for hydrogen-LIKE (single-electron) systems: Eₙ = -13.6 eV × Z² / n². That formula is exact only for a nucleus of charge Z orbited by a single electron -- hydrogen itself (Z=1), or ions stripped down to one electron, like He+ (Z=2), Li2+ (Z=3), or Fe25+ (Z=26). Entering an atomic number for a neutral multi-electron atom (e.g., Z=26 for neutral iron) still produces a number, but it does NOT describe a real neutral iron atom -- electron-electron repulsion, which this single-electron formula ignores entirely, dominates the energy levels of any atom with more than one electron. Transition Energy and Photon Wavelength describe the photon absorbed or emitted when an electron jumps between Initial Level (n) and Target Level; Ionization Energy is simply the magnitude of the energy at the initial level n (the energy needed to remove that one electron to infinity, so it depends only on Atomic Number (Z) and Initial Level (n), never on Target Level).
Orbital Radius follows the Bohr radius scaling r_n = a0 × n² / Z, and Orbital Velocity follows v_n = αc × Z / n (α is the fine-structure constant), both describing the electron at Initial Level (n) rather than the transition itself. Degeneracy (2n²) counts the distinct quantum states sharing that energy level (2 spin states × n² orbital states) -- depending only on Initial Level (n), not on Atomic Number (Z) or Target Level -- before any fine-structure or spin-orbit splitting is considered. Spectral Series names the transition by comparing Target Level to Initial Level (n): Lyman/Balmer/Paschen/Brackett/Pfund+ when Target Level is BELOW n (photon emitted), "Absorption" when it is ABOVE n (photon absorbed). When Initial Level and Target Level are set equal, there is no transition at all -- Transition Energy and Photon Wavelength both report 0, which means "no photon," not a zero-length wave.
Inputs
Results
Energy at n
-13.6 eV
Transition Energy
10.2 eV
Photon Wavelength
121.57 nm
How to Use This Calculator
- Enter Atomic Number (Z): 1 for hydrogen, 2 for helium (hydrogen-like), up to heavy elements for single-electron ions.
- Enter Initial Level (n) — the principal quantum number of the starting energy level (n = 1 is the ground state).
- Enter Target Level for the transition you want to analyze — below Initial Level is emission (a photon is released), above Initial Level is absorption; moving Target Level further away from Initial Level in either direction increases the transition energy and shortens the photon wavelength.
- Read Energy at n (eV), Transition Energy (eV), and Photon Wavelength (nm) to identify the spectral line.
- Check the Spectral Series output rather than the wavelength alone — real series overlap in wavelength (some Balmer lines fall below 400 nm), so the calculator names the series from Target Level directly: 1 = Lyman, 2 = Balmer, 3 = Paschen, 4 = Brackett, 5+ = Pfund and beyond.
What each input means
- Atomic Number (Z)
- Atomic number (1 = hydrogen)
- Initial Level (n)
- Initial principal quantum number
- Target Level
- Target quantum number for transitions
How this is calculated
Formula
Eₙ = -13.6 eV × Z² / n²Worked example, using the default values
- Identify Input Parameters3 parametersAtomic Number (Z) = 1, Initial Level (n) = 1, Target Level = 2 = 3 input(s) provided
- Calculate Energy at nEnergy at n-13.6 = -13.6
- Calculate Transition EnergyTransition Energy10.2 = 10.2
- Calculate Photon WavelengthPhoton Wavelength121.57 = 121.57
- Calculate Ionization EnergyIonization Energy13.6 = 13.6
- Calculate Orbital RadiusOrbital Radius0.529 = 0.529
Engine last updated . Checked against 5 independently-derived tests — how we verify calculators. Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why do I get a number for Iron (Z=26), a multi-electron atom, when this is a single-electron model?
Because the Bohr formula only takes a nuclear charge Z and an electron count of exactly one -- it has no way to represent 25 additional electrons or the repulsion between them. Entering Z=26 computes what a hydrogen-like Fe25+ ion (iron stripped to a single remaining electron) would look like, not neutral iron. Real neutral iron's electron energy levels are set by electron-electron repulsion and shielding that this formula ignores entirely, so treat any result for Z greater than 1 as a single-electron ion, not the everyday neutral element.
What determines whether a transition falls in the Lyman, Balmer, or Paschen series?
The series name is set by Target Level when it is BELOW Initial Level (n) -- meaning the electron is emitting a photon as it drops to a lower orbit. Target Level = 1 names a Lyman-series (ultraviolet) line, Target Level = 2 a Balmer-series (visible) line, Target Level = 3 a Paschen-series (infrared) line, and so on. When Target Level is instead ABOVE Initial Level (n), the calculator labels the transition "Absorption" rather than naming a series, since the electron is climbing to a higher level rather than emitting.
Why is Ionization Energy the same size as Energy at n?
Ionization Energy is defined as the energy required to remove the electron from level n all the way to infinity, where a free electron's energy is zero by convention. Since Energy at n is already measured relative to that same zero point, the ionization energy is exactly the absolute value of Energy at n -- it isn't a separate calculation, just the same bound-state energy reported as a positive "energy to escape" rather than a negative "energy below free."
Does Degeneracy (2n²) account for real fine-structure splitting?
No. Degeneracy (2n²) is the simple non-relativistic count of distinct quantum states sharing one principal energy level -- two spin orientations for each of the n² orbital shapes available at that n. Real atoms split this degeneracy further through spin-orbit coupling (fine structure), the Lamb shift, and external fields (Zeeman splitting), none of which this calculator models; it reports the textbook Bohr-model degeneracy only.
What happens if Initial Level and Target Level are set to the same number?
There's no transition to describe -- Transition Energy is exactly 0 and Photon Wavelength reports 0 as well, meaning "no photon is emitted or absorbed," not a zero-length wave. Spectral Series reports "N/A" in this case since neither an emission series nor "Absorption" applies. Set Target Level to a different value than Initial Level to see a real transition.
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