Hydrogen Atom Energy Levels Calculator

Calculator Form

Formula Used

For the hydrogen atom, the Bohr-model energy at level n is:

E_n = -13.6 / n² eV

Transition energy is:

ΔE = E_f - E_i

Photon energy magnitude is |ΔE|. The calculator then uses:

λ = hc / |ΔE|

ν = |ΔE| / h

ṽ = 1 / λ

r_n = a₀ n²

v_n = αc / n

These equations are ideal for hydrogen and hydrogen-like introductory calculations. Fine structure, relativistic corrections, and external field effects are not included.

How to Use This Calculator

Enter the initial principal quantum number nᵢ.

Enter the final principal quantum number n_f.

Select the preferred energy output unit.

Select the wavelength unit for photon results.

Choose how many levels to display in the graph.

Choose how many levels to list in the detailed table.

Set the decimal places for displayed answers.

Press Calculate Energy Levels to show results above the form.

Use the export buttons to download the current result as CSV or PDF.

Example Data Table

Transition	Type	Photon Energy (eV)	Wavelength (nm)	Series
2 → 1	Emission	10.2000	121.568	Lyman
3 → 2	Emission	1.8889	656.470	Balmer
4 → 2	Emission	2.5500	486.275	Balmer
5 → 2	Emission	2.8560	434.174	Balmer

FAQs

1. What formula does this calculator use?

It uses the Bohr-model relation E_n = -13.6/n² eV for hydrogen. It also computes transition energy, wavelength, frequency, wavenumber, orbital radius, and electron speed from standard constants.

2. Why are hydrogen energy levels negative?

Negative energy means the electron is bound to the nucleus. Zero energy represents a free electron at infinite separation, so bound states appear below that reference.

3. What does the principal quantum number n mean?

The principal quantum number labels the electron’s main energy level. Larger n values place the electron farther from the nucleus and closer to zero energy.

4. Does this work for every atomic detail?

It is excellent for ideal hydrogen energy-level calculations. It does not include fine structure, Lamb shift, Zeeman splitting, or multi-electron atomic effects.

5. What is the difference between emission and absorption?

Emission occurs when the electron drops to a lower level and releases a photon. Absorption occurs when the electron moves upward after taking in photon energy.

6. Why can wavelength appear unavailable?

If the initial and final levels are identical, there is no transition. Without a transition, no photon is produced or absorbed, so wavelength is not applicable.

7. How is the spectral series identified?

The calculator uses the lower involved level to name the series. For example, transitions connected to n = 2 are labeled Balmer, while those tied to n = 1 are Lyman.

8. Which units can I export?

The export includes the currently displayed result values and units. Energy can be shown in eV, J, or kJ/mol, while wavelength can be shown in nm, Å, or m.