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Hydrogen Energy Levels Calculator

Enter the principal quantum number n to find the energy level.

How it is calculated

The tool uses the Bohr-model formula En = −13.6 ÷ n², with E in electronvolts (eV) and n the principal quantum number. The energy in joules comes from multiplying by 1.602176634 × 10−19 J per eV. The binding (ionization) energy is the absolute value of En, the energy needed to free the electron from that level. The value 13.6 eV is a rounded constant, so results are accurate to about three significant figures, even though four decimals are shown. Only whole numbers n = 1, 2, 3, … are accepted; zero, negatives and fractions such as 2.5 give no result. The model applies to hydrogen-like one-electron atoms only when scaled by Z², which this tool does not do.

Worked example

For the third level, n = 3. Then n² = 9, so E = −13.6 ÷ 9 = -1.5111 eV. In joules, −1.5111 × 1.602176634 × 10−19 = −2.4211e-19 J. The binding energy is the absolute value, 1.5111 eV: that is how much energy would ionize an electron sitting in n = 3.

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Common mistakes

  • Entering the wavelength, the transition or the number of levels instead of n. The tool takes one level and does not compute energy differences between two levels.
  • Dropping the minus sign when comparing: the energy is negative for a bound electron, and only the binding energy is positive.
  • Typing a non-integer such as 2.5 or 0, which is rejected, since n must be a whole number of at least 1.
  • Applying the result to helium ions or other atoms; the formula here is for hydrogen (Z = 1) and would need a Z² factor otherwise.

About this calculator

The hydrogen energy levels calculator computes the energy of a hydrogen atom's electron in orbital n using the Bohr model formula Eₙ = −13.6 eV / n², where n is the principal quantum number (1, 2, 3, …). The negative sign reflects that the electron is bound to the nucleus; n = 1 is the ground state at −13.6 eV, and as n increases the levels get closer together and approach zero, the point at which the electron is no longer bound (ionization).

The energy difference between two levels, ΔE = 13.6 eV × (1/n₁² − 1/n₂²), predicts the exact wavelengths of hydrogen's spectral lines — the Lyman series (transitions to n = 1, ultraviolet), Balmer series (to n = 2, visible light), and Paschen series (to n = 3, infrared) all follow directly from this formula. Physics and chemistry students, and anyone studying atomic spectroscopy use this calculator to find a given level's energy in electronvolts or joules and check it against the Bohr model before moving on to the full quantum-mechanical treatment.

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