Atomic energy levels

Discrete states and the meaning of energy

A bound electron has negative total energy when zero is defined as a free electron infinitely far from the atom. More negative means more tightly bound. Ionization raises the electron to the continuum at or above zero.

For hydrogen, the nonrelativistic level energy is Eₙ = −13.6 eV/n². Multi-electron atoms do not follow this simple formula because electrons repel and shield one another.

Photons connect one state to another

A transition occurs only when the energy difference matches the energy exchanged:

Absorption moves an atom to a higher state; emission occurs when it falls to a lower state. The wavelength therefore measures an energy difference, not an absolute orbital energy by itself.

Shell, subshell and level are not synonyms

In hydrogen, states with the same principal quantum number n are nearly degenerate in the simplest model. In multi-electron atoms, penetration and shielding split subshell energies: for a given shell, s is usually more penetrating and lower in energy than p, then d, then f.

Fine structure, spin–orbit coupling and external electric or magnetic fields can split levels further. A real atomic spectrum is therefore richer than a Bohr diagram.

Excitation, ionization and lifetimes

An excited state is still bound if its energy remains below the ionization limit. Excited atoms usually decay, often by photon emission, but different transitions can have very different lifetimes because quantum selection rules affect how strongly states couple to light.

Metastable states can persist much longer than an intuitive “electron falls immediately” picture suggests. Lasers exploit controlled populations of excited states and stimulated emission.

Photon calculator

This conversion uses hc ≈ 1239.84 eV·nm. A 2.0 eV gap corresponds to visible red light near 620 nm.

Hydrogen levels from Coulomb attraction

The 1/n² pattern is specific to a one-electron Coulomb system. Hydrogen-like ions such as He⁺ and Li²⁺ follow the same structure with a stronger nuclear charge:

He⁺ therefore has a ground-state binding energy four times that of H. The Z² scaling disappears as a simple formula once multiple electrons introduce shielding and mutual repulsion.

Degeneracy, splitting and selection rules

Two quantum states are degenerate when they have the same energy. In ideal nonrelativistic hydrogen, states with the same n share the same energy even when their orbital angular momentum differs. Real atoms contain additional interactions that split these degeneracies.

Spin–orbit coupling links an electron’s spin with its orbital motion. External magnetic fields produce Zeeman splitting; electric fields produce Stark splitting. High-resolution spectroscopy resolves these small energy differences.

Not every pair of levels gives a strong spectral line. Electromagnetic transition probabilities obey selection rules derived from symmetry and angular momentum. A level can exist even if a direct optical transition to another state is weak or forbidden.

Multi-electron atoms and photoelectron spectroscopy

For many-electron atoms, an orbital label such as 3p or 4s does not carry one universal energy independent of the rest of the atom. Penetration, shielding and electron correlation alter the energies as the configuration changes.

Photoelectron spectroscopy removes electrons with known photon energy and measures their kinetic energy. Energy conservation gives the electron binding energy. Peaks from different subshells provide experimental evidence for shell and subshell structure.

This is a useful bridge between diagrams and measurement: an “energy level” is not just a line drawn in a textbook; it has observable consequences in the energy required to excite or remove electrons.

Level diagrams for real atoms

In sodium, the outer electron occupies a 3s state outside a neon-like core. Excitation to 3p gives the strong yellow sodium D lines near 589 nm, but the two closely spaced D lines already reveal fine-structure splitting that a single 3s→3p arrow misses.

Helium is more complex because two electrons can couple their spins into singlet or triplet families. States with the same apparent orbital occupancy can have different total spin and different energies. This is one reason multi-electron level diagrams are organized by spectroscopic terms rather than by n alone.

The lesson is practical: a horizontal line in an energy diagram labels a many-electron quantum state. It should not automatically be interpreted as “one electron sitting at that height.”

Thresholds versus lines

A bound-to-bound transition produces a discrete spectral line. Bound-to-free absorption begins at an ionization threshold and can continue over a range of photon energies because the ejected electron can carry different kinetic energies.

Photoionization therefore produces a continuum beyond threshold rather than one single wavelength. Measuring the threshold gives the ionization energy of the initial state, while the kinetic-energy distribution carries additional information about the electronic structure.

Exercises

Hydrogen excitation

How much energy is required to excite H from n=1 to n=2?

Emission wavelength

A transition releases 3.10 eV. Estimate λ.

Ionization

What distinguishes excitation from ionization?