Short description: Discrete energy state of an electron in an atom

An energy level is a discrete value of energy that an electron can have in an atom. These levels arise from the quantization of the electron’s motion and are one of the fundamental concepts of quantum mechanics.

Discrete energy levels in an atom; transitions between them produce spectral lines.

Description

A quantum mechanical system that is spatially confined can only possess certain discrete values of energy, known as energy levels. This differs from classical systems, where energy may vary continuously. The concept is especially important for electrons bound to atomic nuclei, but also applies to molecular vibrations, rotations, and nuclear states.

Electrons in atoms can only occupy specific energy levels. These levels are determined by solving the Schrödinger equation for electrons bound to a nucleus. The allowed states correspond to standing-wave solutions of the electron wavefunction.^[1]

When an electron moves between energy levels it must absorb or emit energy, usually in the form of a photon. These transitions produce atomic spectral lines and form the basis of spectroscopy.

Atomic energy levels

In atoms, the allowed electron energies are associated with shells and orbitals. The shells correspond to the principal quantum numbers ${{{1}}}$ and are often labeled K, L, M, and so forth.

Each shell can contain only a limited number of electrons. The maximum number is approximately given by:

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Thus the first shell holds two electrons, the second eight, and the third eighteen.^[2]

The lowest possible energy configuration of an atom is called the ground state. Higher states are called excited states. An atom may contain several excited electrons simultaneously.

Quantization

Quantized energy levels arise because matter behaves as both particles and waves. Electrons confined near a nucleus form standing wave patterns. Only certain wavelengths satisfy the boundary conditions of the system, leading to discrete allowed energies.

For hydrogen-like atoms containing one electron, the energy levels are approximately given by:

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where $R \infty$ is the Rydberg constant, $Z$ is the atomic number, and $n$ is the principal quantum number.

The related Rydberg formula for emitted or absorbed wavelengths is:

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These relations successfully explain the spectral lines of hydrogen and hydrogen-like ions.

Electron interactions

In multi-electron atoms, electron–electron interactions alter the simple hydrogenic energy levels. Inner electrons partially shield the positive nuclear charge, producing an effective nuclear charge $Z eff$ .

An approximate expression becomes:

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These effects help determine electron configurations and atomic structure.

Fine and hyperfine structure

Small corrections to energy levels arise from relativistic effects and spin interactions.

Fine structure results from relativistic kinetic-energy corrections and spin–orbit coupling.
Hyperfine structure results from interactions between electron spin and nuclear spin.

These small splittings are measurable with high-resolution spectroscopy.

External field effects

Zeeman effect

External magnetic fields split atomic energy levels through interactions with magnetic moments associated with orbital and spin angular momentum.

The interaction energy is:

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This splitting produces the Zeeman effect.

Stark effect

External electric fields can also shift and split atomic energy levels, producing the Stark effect.

Molecular energy levels

Chemical bonds in molecules also produce quantized energy states. Molecular systems possess:

electronic energy levels
vibrational energy levels
rotational energy levels

The total molecular energy may be written as:

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Different molecular orbitals may be bonding, antibonding, or non-bonding.^[3]

Energy level transitions

Electrons may transition between energy levels by absorbing or emitting photons. The photon energy equals the difference between the initial and final states:

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where $h$ is the Planck constant, $f$ is frequency, and $λ$ is wavelength.

These transitions form the basis of atomic and molecular spectroscopy. Depending on the energy involved, transitions may emit or absorb radio waves, infrared radiation, visible light, ultraviolet radiation, or X-rays.

History

The first evidence for quantized atomic energy levels came from the observation of spectral lines in sunlight by Joseph von Fraunhofer and William Hyde Wollaston.

In 1913, Niels Bohr proposed quantized electron orbits in the Bohr model of the atom. Modern quantum mechanical explanations based on the Schrödinger equation were developed in the 1920s by Erwin Schrödinger and Werner Heisenberg.^[4]

Crystalline materials

In crystalline solids, electrons occupy energy bands rather than isolated levels. These bands arise because many closely spaced atomic energy levels overlap within the crystal lattice.

Important band-related quantities include: