Hydrogen Like Atom Calculator - Radius, Energy, and Photon Wavelength

A hydrogen like atom calculator solves the radius, energy, velocity, and period of any electron orbit in a one-electron ion, then turns a chosen transition between two orbits into a photon wavelength, photon energy, and named spectral series. Hydrogen-like atoms are atoms or ions with exactly one electron bound to a nucleus, so the same Bohr model formulas apply to neutral hydrogen (H, Z=1), helium ion (He+, Z=2), lithium ion (Li2+, Z=3), and so on up through Ne9+.

• • Hydrogen homework: Compute the n=1, 2, 3 orbits and the Lyman or Balmer series photon wavelengths for a physical chemistry or modern physics problem set.
• • Hydrogen-like ions: Re-run the same formulas for He+, Li2+, Be3+, and other one-electron ions just by changing the atomic number Z while keeping the model assumptions.
• • Spectroscopy check: Match a measured spectral line to the named series (Lyman, Balmer, Paschen) using the n_i to n_f transition you identified in the lab.
• • Concept review: Show how radius scales with n^2/Z, energy with -Z^2/n^2, and velocity with Z/n to make the textbook algebra concrete for a class discussion.

The Bohr model treats the electron as a particle in a Coulomb potential that orbits a stationary nucleus, then quantizes its angular momentum. The model gives exact predictions for one-electron systems and matches the Rydberg series that real hydrogen-like ions produce.

For multi-electron atoms the model breaks down because electron-electron repulsion shifts the energy levels, but the hydrogen like atom calculator still gives the right answer for any one-electron species. It covers Z from 1 to 10 paired with principal quantum numbers from 1 to 10 without any re-derivation.

For the closely related Bohr-style problem on neutral hydrogen and the textbook Z=1 spectral series, Bohr Model Calculator pairs these one-electron formulas with extra worked examples.

The calculator uses the standard Bohr model expressions for one-electron ions, with the Bohr radius a_0, the Rydberg constant R_infinity, the Hartree energy E_h, and the fine-structure constant alpha all taken from the 2018 NIST CODATA review. Each output is a direct evaluation of the formula that matches the printed textbook values to four or more significant figures.

• Z: Atomic number of the hydrogen-like ion. Z=1 gives neutral hydrogen, Z=2 gives He+, and so on.
• n: Principal quantum number of the orbit whose radius, energy, velocity, and period you want to inspect.
• n_i: Upper level for the photon transition. Must be greater than n_f and at least 2.
• n_f: Lower level for the photon transition. Determines the named spectral series.

All four input fields feed a single pure function. Atomic number Z and orbit n drive the orbital properties, while the upper and lower transition levels feed the Rydberg formula for the photon wavelength and photon energy.

The orbital velocity uses v_n = alpha * c * Z / n, where alpha is the fine-structure constant. That keeps the speed of a Z=1, n=1 electron at about alpha * c, which is roughly 1/137 of the speed of light, a textbook number worth remembering.

According to NIST CODATA 2018 - Bohr radius, the Bohr radius is 5.29177210903e-11 m, with a relative standard uncertainty of 1.7e-10

According to NIST CODATA 2018 - Rydberg constant, the Rydberg constant R_infinity is 1.0973731568160e7 m^-1, which sets the scale of the spectral lines the calculator reproduces

When the homework problem moves from the Bohr-model energy levels to a different atomic-structure question like electron configuration or mass number, Atom Calculator keeps the same input style but covers a broader atom-level view.

Four ideas make the hydrogen-like atom predictable: a fixed unit of length, quantized angular momentum, hydrogen-like scaling with Z, and a transition rule that turns level gaps into photons.

These four ideas let the same formulas describe hydrogen, deuterium, and He+. The Z^2 factor in energy and the 1/Z factor in radius are the easiest clues for predicting an answer before computing it.

The Bohr model assumes one electron and a point nucleus, so real hydrogen's small reduced-mass correction and finite nuclear-size shift are not included here.

For the spectral lines these energy gaps produce, Rydberg Equation Calculator solves the same 1/n^2 level spacing directly from the upper and lower quantum numbers.

Enter the atomic number of your hydrogen-like ion, the orbit you want to inspect, and the two levels of the transition you want to study. The result panel updates as you type.

1. 1 Choose the atomic number: Set Z from 1 (hydrogen) up to 10 for any one-electron species the Bohr model covers, from H through Ne9+.
2. 2 Pick the orbit n: Use the principal quantum number of the orbit whose radius, energy, velocity, and period you want to see.
3. 3 Set the upper transition level: Enter n_i between 2 and 10. This is the orbit the electron drops from.
4. 4 Set the lower transition level: Enter n_f between 1 and n_i - 1. This is the orbit the electron lands in and it sets the named spectral series.
5. 5 Read the photon wavelength: The first result is the wavelength of the photon emitted for the n_i to n_f transition, in nanometers.
6. 6 Check the spectral series: Match the named series label to a textbook line list (Lyman, Balmer, Paschen, Brackett, Pfund, or Humphreys).

For the textbook Balmer-alpha line, leave Z=1, set n=2 to inspect the upper level, and use n_i=3 with n_f=2. The wavelength result should read about 656 nm and the spectral series label should show Balmer (visible). If you flip to n_f=1 the same calculation produces the Lyman-alpha line at 121.5 nm.

After the Bohr model predicts the photon energy for a transition, the population of each level at a given temperature needs the Boltzmann factor, which Boltzmann Factor Calculator computes with the same NIST CODATA constants.

The calculator is most useful when a chemistry or physics problem asks for a single numerical Bohr-model answer for a one-electron ion and you want it without rebuilding the formulas by hand.

• • Direct Bohr formulas: Uses the standard r_n, E_n, and Rydberg expressions with NIST CODATA constants, so the answer matches a printed textbook to four or more significant digits.
• • Covers every hydrogen-like ion: Z up to 10 means the same tool handles H, He+, Li2+, Be3+, and the rest of the one-electron series without re-deriving the algebra.
• • Two problems in one place: Computes both the static orbit properties (radius, energy, velocity, period) and the photon from a chosen transition.
• • Series identification: Labels the transition as Lyman, Balmer, Paschen, Brackett, Pfund, or Humphreys, so a measured wavelength can be matched to a line list quickly.
• • Worked-example friendly: Defaults are set to the hydrogen Lyman-alpha problem, so the result panel can be cross-checked against a sample solution in seconds.

For one-electron problems the Bohr numbers are the right ones to quote, and the calculator delivers them in seconds without setting up a spreadsheet.

For multi-electron atoms, transitions in solids, or fine-structure effects, the Bohr model stops being the right tool and the result should be cross-checked with a fuller quantum model.

For the nuclear side of the same atomic-structure unit, where you need the isotope mass that sets the reduced-mass correction behind the Bohr model, Atomic Mass Calculator handles neutral atoms and ions by isotope.

Three input factors change every result, and the spectral line depends on which lower level you choose. The caveats below describe where the model is reliable and where it stops being exact.

• • The Bohr model assumes one electron. For neutral helium, lithium, or any multi-electron atom the model overestimates the binding energy because it ignores electron-electron repulsion.
• • Spin, relativistic corrections, and fine structure are not included. The 2p and 2s levels are degenerate in the Bohr model, and real hydrogen shows a 4.5 GHz fine-structure splitting between them.
• • The nucleus is treated as a point mass at the center. The reduced-mass correction and the finite nuclear size shift the levels by parts per million, which matters in precision spectroscopy.

If you need precision beyond the simple Bohr model, the next step is the Dirac fine-structure formula or a full Schrodinger-solution comparison.

For the n_i to n_f transition the calculator assumes emission (an electron dropping down). The validator requires n_f to be strictly less than n_i, so any input where n_f is greater than or equal to n_i is rejected with an invalid-transition error and the wavelength result is left at zero.

According to NIST Atomic Spectra Database - Hydrogen energy levels, hydrogen energy levels are tabulated in the same -R_infinity / n^2 form the calculator uses, so the listed series limits and Lyman, Balmer, and Paschen wavelengths agree to seven significant figures