Hydrogen Like Atom Energy Levels Calculator
Calculate energy levels, ionization energy, Bohr radius, orbital velocity, and transition wavelengths for hydrogen-like atoms using the Bohr model formula.
About This Calculator
The Hydrogen Like Atom Energy Levels Calculator computes the quantized energy levels, ionization energy, Bohr radius, orbital velocity, and spectral transition wavelengths for any hydrogen-like (hydrogenic) atom or ion. A hydrogen-like atom has exactly one electron orbiting a nucleus with atomic number Z. Examples include neutral hydrogen (Z=1, H), singly ionized helium (Z=2, He⁺), doubly ionized lithium (Z=3, Li²⁺), and triply ionized beryllium (Z=4, Be³⁺).
The calculator uses the Bohr model formula Eₙ = −13.606 × Z² / n² eV where n is the principal quantum number. The results include the electron energy in electronvolts and joules, the ionization energy required to remove the electron, the Bohr orbital radius (scales as n²a₀/Z), and the orbital velocity (scales as Zαc/n). When you specify a transition between two quantum levels n₁ and n₂, the calculator computes the transition energy, wavelength in nanometers, and identifies the spectral series (Lyman, Balmer, Paschen, Brackett, or Pfund).
This calculator is designed for students studying quantum mechanics, atomic physics, and spectroscopy at the high school and undergraduate level. It is also useful for researchers working with Rydberg atoms, exotic hydrogenic systems, or plasma diagnostics. The non-relativistic Bohr model provides excellent accuracy for low-Z elements (Z ≤ 20) and remains qualitatively correct for heavier hydrogenic ions.
Regional Notes
India (IN): The Bohr model and hydrogen energy levels are part of the CBSE and NCERT Class 11 and 12 physics curriculum. Students preparing for JEE and NEET exams often solve hydrogen atom problems using these formulas. The Rydberg constant and energy levels are fundamental topics in atomic structure.
United States (US): The hydrogen atom and Bohr model are covered in AP Physics 2 and AP Chemistry courses. Undergraduate physics and engineering students encounter hydrogenic atoms in quantum mechanics and modern physics courses. The Balmer series wavelengths (656 nm, 486 nm, 434 nm) are standard reference points in spectroscopy labs.
United Kingdom (UK): Hydrogen energy levels and the Bohr model appear in A-level Physics and Chemistry curricula (AQA, OCR, Edexcel). UK students study atomic spectra and the Rydberg formula as part of quantum physics modules at both secondary and university levels.
Frequently Asked Questions
What is a hydrogen-like atom?
A hydrogen-like atom or hydrogenic atom is any atom or ion that has exactly one electron orbiting the nucleus. Examples include hydrogen H, singly ionized helium He+, doubly ionized lithium Li2+, and triply ionized beryllium Be3+. These systems are described by the Bohr model where energy levels scale with the square of the atomic number Z.
How do you calculate hydrogen energy levels?
The energy of an electron in a hydrogen-like atom at principal quantum number n is given by En = -13.606 times Z squared divided by n squared electronvolts where Z is the atomic number and n is the energy level. For hydrogen Z=1 the ground state n=1 has energy -13.606 eV.
What is the ionization energy of hydrogen?
The ionization energy of hydrogen is 13.606 eV which is the energy required to remove the electron from the ground state n=1 to infinity. For hydrogen-like atoms the ionization energy scales with Z squared so He+ has 54.42 eV and Li2+ has 122.45 eV.
What are the Lyman and Balmer series?
The Lyman series consists of spectral lines from electron transitions to the n=1 energy level producing ultraviolet light. The Balmer series involves transitions to n=2 producing visible light lines at 656 nm red 486 nm cyan and 434 nm blue. Higher series Paschen n=3 and Brackett n=4 produce infrared light.
How does atomic number Z affect energy levels?
Energy levels scale with Z squared meaning a helium nucleus Z=2 produces energy levels four times deeper than hydrogen Z=1. The Bohr radius becomes half as large and transition wavelengths become one-quarter as long. This scaling applies to all one-electron ions and atoms.
What is the Bohr radius?
The Bohr radius a0 is 52.92 picometers or 0.5292 angstroms and represents the most probable distance between the proton and electron in hydrogen's ground state. For energy level n the radius scales as n squared times a0 divided by Z so for hydrogen n=2 the radius is 211.7 pm.
What is the orbital velocity of an electron in hydrogen?
The orbital velocity of an electron in hydrogen's ground state n=1 Z=1 is approximately 2.188 million meters per second which is about 1/137 of the speed of light. This speed decreases with increasing n as v proportional to Z divided by n.
Can this calculator be used for exotic atoms?
Yes this calculator works for any hydrogen-like system with a single electron including muonic atoms where a muon replaces the electron and Rydberg atoms with very large n values up to 20 atomic number Z up to 118 and any transition between levels. Results use the non-relativistic Bohr model which is highly accurate for low Z.