What formula does this use?

The Bohr model formula: E_n = −13.6 eV / n², where n is the principal quantum number (1, 2, 3, …). The −13.6 eV constant is the ground-state energy derived from fundamental constants.

Why is the energy negative?

Negative energy means the electron is bound to the nucleus. Zero energy corresponds to a free electron at rest at infinity. The more negative the value, the more tightly the electron is bound.

What is the ionization energy of hydrogen?

The ionization energy from the ground state (n=1) is 13.6 eV — the energy needed to remove the electron completely, bringing E from −13.6 eV to 0 eV.

How do I find the wavelength of a transition?

Use the transition section: enter an initial level n_i and final level n_f. The calculator finds ΔE = E_f − E_i, then uses E = hc/λ to get the photon wavelength. If n_f n_i the photon is absorbed.

Does this work for hydrogen-like ions?

This tool is for neutral hydrogen (Z=1). For hydrogen-like ions (He⁺, Li²⁺, etc.) the formula scales as −13.6 × Z² / n². Multiply the displayed energy by Z² to adapt it.

Hydrogen Energy Level Calculator

Calculate the energy of any hydrogen electron shell using the Bohr model — enter a quantum number n and get energy in eV and joules.

Single Level

Principal quantum number n

Enter any positive integer (n ≥ 1).

Transition Photon

Initial level n_i

Final level n_f

Energy Level Result

n = 1

Energy (eV)

−13.6000

electron-volts

Energy (J)

−2.18×10⁻¹⁸

joules

Ionization energy

13.6000 eV

Orbit radius (a₀)

1.000 a₀

Velocity / c

1/137

Energy Level Diagram

Summary

This calculator uses the Bohr model to find the energy of any hydrogen electron shell given its principal quantum number n. Results are shown in both electron-volts and joules, with an energy-level diagram and a transition tool that converts ΔE to photon wavelength for common spectral series.

How it works

Enter a principal quantum number n (positive integer, 1 or greater).
The calculator applies the Bohr model formula: E_n = −13.6 eV / n².
Energy in electron-volts (eV) and joules (J) is shown instantly.
The energy diagram updates to show the selected level relative to n=1 through n=6.
Use the transition tool to find the photon energy emitted or absorbed between two levels.

Use cases

Find the ground-state energy of hydrogen (n=1, −13.6 eV).
Calculate the ionization energy for any shell.
Determine the photon energy and wavelength for spectral line transitions.
Study the Lyman, Balmer, and Paschen series transitions.
Verify textbook problems on atomic structure.
Explore how energy approaches zero as n increases toward ionization.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu