Physics

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Hydrogen Energy Levels Calculator.

Calculate a hydrogenic atom's energy at a principal quantum level.

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Results update as you type.

Energy: -3.401423

Energy

0.000000
Energy: -0

Energy

0.000000

Educational SI models with assumptions stated in each description; verify engineering decisions independently.

Use Cases

Physics Homework and Study

Quickly verify energy level calculations for hydrogen-like atoms in quantum mechanics assignments or exam prep.

Example: Check the energy of He+ (Z=2) at n=3: E = -13.6 * (4/9) ≈ -6.04 eV.

Spectroscopy and Research

Estimate transition energies between levels for hydrogenic ions in laboratory or astrophysical contexts.

Example: Find the energy difference between n=2 and n=1 for hydrogen (Z=1).

Frequently Asked Questions

What is a hydrogenic atom?
A hydrogenic atom is any atom or ion with only one electron, such as hydrogen (Z=1), He+ (Z=2), or Li2+ (Z=3). Its energy levels depend only on the nuclear charge Z and the principal quantum number n.
How is the energy level calculated?
The energy is calculated using the Bohr model formula: E = -13.6 eV * (Z^2 / n^2). You input the atomic number Z and the principal quantum number n, and the calculator returns the energy in electron volts (eV).
Why is the energy negative?
Negative energy indicates a bound state. The electron is bound to the nucleus, and energy must be added to ionize the atom (bring the energy to zero). More negative means more tightly bound.

Tips & Common Mistakes

Tips

  • Enter the atomic number Z as a positive integer (e.g., 1 for hydrogen, 2 for helium ion).
  • Enter the principal quantum number n as a positive integer (n = 1, 2, 3, ...).
  • Remember that the energy is inversely proportional to n squared, so higher n means less negative (higher) energy.
  • Use the result to compute transition energies by subtracting energy levels.

Common Mistakes to Avoid

  • Using Z=1 for all atoms; remember to use the actual nuclear charge for ions.
  • Entering n=0 or negative values; n must be a positive integer.
  • Forgetting that the energy is negative; a positive result indicates an error in input.

Last updated: August 13, 2026