Hydrogen Atom Energy Levels Calculator
Calculates hydrogen-like atom energy levels, orbital radii, and spectral wavelengths from the Schrödinger equation. Ideal for students and researchers studying atomic structure and spectroscopy.
About this calculator
The Schrödinger equation for a hydrogen-like atom yields exact analytical solutions. The energy of level n is E_n = −13.6 × Z² / n² eV, where Z is the atomic number and n is the principal quantum number. The most probable orbital radius (Bohr radius scaled) is r_n = a₀ × n² / Z, where a₀ = 5.292 × 10⁻¹¹ m. Spectral emission wavelengths follow the Rydberg formula: 1/λ = R_∞ × Z² × (1/n_f² − 1/n_i²), where R_∞ = 1.0974 × 10⁷ m⁻¹. The ionization energy is simply the magnitude of the ground-state energy: 13.6 × Z² eV. These formulas predict the hydrogen spectrum to extraordinary precision and form the foundation of modern atomic physics.
How to use
To find the energy of the n = 2 level in hydrogen (Z = 1): E = −13.6 × 1² / 2² = −13.6 / 4 = −3.4 eV (≈ −5.45 × 10⁻¹⁹ J). For the orbital radius: r = 5.292 × 10⁻¹¹ × 4 / 1 = 2.117 × 10⁻¹⁰ m (≈ 2.12 Å). For a photon emitted from n = 3 to n = 2 (Hα line): 1/λ = 1.0974 × 10⁷ × 1 × (1/4 − 1/9) = 1.524 × 10⁶ m⁻¹, giving λ ≈ 656 nm — the red line of the Balmer series. Enter your Z, n, and calculation type to get results immediately.
Frequently asked questions
How do hydrogen atom energy levels relate to spectral emission lines?
When an electron transitions from a higher energy level n_i to a lower level n_f, it emits a photon whose energy equals the difference between the two levels. This energy difference determines the photon's wavelength via E = hc/λ. The Balmer series (transitions to n=2) falls in the visible range, the Lyman series (to n=1) in the UV, and the Paschen series (to n=3) in the infrared. Each series produces a characteristic set of sharp spectral lines that uniquely identifies the element.
Why does the energy formula use a negative sign for hydrogen energy levels?
The negative sign reflects that the electron is in a bound state — it requires energy input to escape the nucleus. The zero of energy is defined as the electron at infinite separation from the nucleus (n → ∞). As the electron moves closer and becomes more tightly bound, its total energy decreases below zero. The ground state (n = 1) has the most negative energy, −13.6 eV for hydrogen, meaning you need at least 13.6 eV to ionize a ground-state hydrogen atom.
Can this calculator be used for helium or other atoms beyond hydrogen?
This calculator applies to hydrogen-like (one-electron) ions, where you simply substitute the atomic number Z. For example, singly ionized helium (He⁺, Z = 2) has a ground-state energy of −13.6 × 4 = −54.4 eV. However, for multi-electron atoms like neutral helium or heavier elements, electron-electron repulsion makes the Schrödinger equation analytically unsolvable, and numerical or approximate methods are required. The hydrogen-like model still gives useful estimates for inner-shell electrons in heavier atoms.