# Hydrogen-Like Atom Calculator

URL: https://speedycalculator.com/calculators/physics/hydrogen-like-atom/
Last updated: 2026-10-30

> The hydrogen-like atom calculator finds the energy, orbit radius and electron speed of level n in a one-electron atom or ion of nuclear charge Z, with an optional reduced-mass correction.

## Inputs

- Atomic number Z
- Principal quantum number n
- Nuclear mass (u, 0 = infinite)

## Results

- Reduced mass ratio μ/mₑ
- Energy E_n
- Orbit radius r_n
- Electron speed
- Ionisation energy from this level

## Formula

```
E_n = −13.6057 eV·Z²/n²·(μ/mₑ),  r_n = a₀·n²/Z·(mₑ/μ),  v_n = Z·α·c/n
```

## Explanation

Ions like He⁺, Li²⁺ and U⁹¹⁺ that have only one electron behave just like hydrogen but with a stronger nucleus: energies grow with Z², orbits shrink with Z and electrons move faster.

### How to use the Hydrogen-Like Atom Calculator

- **Atomic number Z**
- **Principal quantum number n**
- **Nuclear mass (u, 0 = infinite)**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Enter the atomic number, the level n and the nuclear mass (for the reduced-mass correction).

### Inputs explained

FieldWhat to enterAtomic number ZA numberPrincipal quantum number nA numberNuclear mass (u, 0 = infinite)A number

### What each result means

- **Energy E_n** = -13.605693×[Atomic number Z]^2÷[Principal quantum number n]^2×[Reduced mass ratio μ÷mₑ]
- **Orbit radius r_n** = 5.29177210903e-11×[Principal quantum number n]^2÷[Atomic number Z]÷[Reduced mass ratio μ÷mₑ]
- **Electron speed** = [Atomic number Z]×0.0072973525693×299792458÷[Principal quantum number n]
- **Ionisation energy from this level** = 13.605693×[Atomic number Z]^2÷[Principal quantum number n]^2×[Reduced mass ratio μ÷mₑ]

### Example values

How the main result changes when **Atomic number Z** changes and everything else stays at the example values above:

Atomic number ZEnergy E_n (eV)1-13.60381.5-30.60862-54.41532.5-85.02393-122.43454-217.6613

## Worked example

Inputs — Atomic number Z: 2; Principal quantum number n: 1; Nuclear mass (u, 0 = infinite): 4.0026.

Results — Reduced mass ratio μ/mₑ = 0.99986296; **Energy E_n = -54.415314 eV**; Orbit radius r_n = 26.462487 pm; Electron speed = 4,375.382527 km/s; Ionisation energy from this level = 54.415314 eV.

## Key terms

- **Bohr radius**: a₀ = 52.9 pm.

## FAQ

**What is a hydrogen-like ion?**
Any atom stripped to a single electron.

**Why does energy grow with Z²?**
Stronger nuclear attraction and smaller orbits.

**What is the reduced-mass correction?**
The nucleus also moves slightly, lowering energies by ~0.05% for hydrogen.

**Are relativistic effects included?**
No – they matter for heavy ions.


## Sources

- [OpenStax – University Physics Volume 3](https://openstax.org/details/books/university-physics-volume-3)
- [NIST – Fundamental physical constants](https://physics.nist.gov/cuu/Constants/)