Energy Level
An energy level is one of the discrete, quantized values of energy that a bound quantum system (such as an electron in an atom or a molecule) is permitted to have. Unlike classical systems where energy can take any continuous value, quantum mechanics constrains bound particles to specific allowed states, each characterized by a set of quantum numbers. Transitions between energy levels result in the absorption or emission of photons with energies exactly equal to the difference between the two levels, producing the characteristic spectral lines used in atomic spectroscopy.
Key Formula
Eₙ = −(Z² × 13.6 eV) / n²
LaTeX: E_n = -\frac{Z^2 \cdot 13.6\,\text{eV}}{n^2}
| Symbol | Meaning | Unit |
|---|---|---|
| Eₙ | Energy of the nth level | eV |
| Z | Atomic number (number of protons) | dimensionless |
| n | Principal quantum number (1, 2, 3, …) | dimensionless |
| 13.6 eV | Ground state energy magnitude of hydrogen | eV |
Worked Example
Problem
Calculate the energy of the n = 2 level for a He⁺ ion (Z = 2, one electron, hydrogen-like).
Solution
Step 1: Identify values. Z = 2 (helium), n = 2 Step 2: Apply the formula. E₂ = −(Z² × 13.6) / n² E₂ = −(4 × 13.6) / 4 E₂ = −54.4 / 4 E₂ = −13.6 eV
Answer
E₂ = −13.6 eV for He⁺ at n = 2. (Compare: for H, E₂ = −3.4 eV — helium's stronger nuclear charge pulls the electron to lower energy.)
Energy Levels of Hydrogen-Like Atoms (eV)
| n | H (Z=1) | He⁺ (Z=2) | Li²⁺ (Z=3) | Designation |
|---|---|---|---|---|
| 1 | −13.60 | −54.40 | −122.40 | K shell (ground state) |
| 2 | −3.40 | −13.60 | −30.60 | L shell |
| 3 | −1.51 | −6.04 | −13.60 | M shell |
| 4 | −0.85 | −3.40 | −7.65 | N shell |
| ∞ | 0.00 | 0.00 | 0.00 | Ionization limit |
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Related Terms
Bohr Model
The Bohr model, proposed by Niels Bohr in 1913, describes the hydrogen atom as having electrons orbiting the nucleus in discrete, quantized circular orbits with specific allowed energies. Electrons can jump between orbits by absorbing or emitting photons whose energy equals the difference between the two energy levels, explaining the discrete spectral lines of hydrogen. While superseded by quantum mechanics, the Bohr model correctly predicts hydrogen's spectral series and introduced the revolutionary idea of quantized atomic energy levels.
Ground State
The ground state is the lowest possible energy state of a quantum mechanical system, such as an atom, molecule, or nucleus, in which all quantum numbers take their minimum allowed values consistent with the Pauli Exclusion Principle. A system in the ground state is thermodynamically stable and does not spontaneously emit radiation. The ground state energy of hydrogen is −13.6 eV, and the ground state represents the reference level from which excitation energies of higher states are measured.
Excited State
An excited state is any quantum state of an atom, molecule, or nucleus in which one or more particles occupy energy levels higher than the ground state, having absorbed energy from a photon, collision, or thermal source. Excited states are inherently unstable — atoms typically remain in an excited state for about 10⁻⁸ seconds (nanosecond timescale) before spontaneously returning to a lower energy state by emitting a photon. The controlled management of excited states is fundamental to lasers (population inversion), fluorescence microscopy, and phosphorescence.
The term "energy level" emerged from early quantum theory in the 1910s–1920s. "Energy" derives from the Greek energeia (activity, operation), used in physics since Thomas Young (1800). "Level" comes from the Latin libella (a balance/plumb line), reflecting the horizontal lines used in energy diagrams.