Atomic electron transition
An atomic electron transition is a change between quantum states of an electron bound to an atom. The initial and final states possess discrete energies determined by the atom’s Hamiltonian, so the transition transfers a correspondingly discrete quantity of energy. Radiative transitions exchange that energy with the electromagnetic field, whereas non-radiative transitions transfer it through interactions with other particles or with the surrounding material.
The expression does not describe a classical particle moving along a definite path between two orbits. An atomic electron is represented by a wavefunction, and a transition is the time-dependent conversion of one quantum-state amplitude into another. This distinction accounts for the sharply defined frequencies of isolated atomic spectra while remaining consistent with the finite durations and linewidths observed in physical measurements.
Quantum-mechanical description
For an isolated atom with a time-independent Hamiltonian (H_0), stationary states satisfy the time-independent Schrödinger equation,
[ H_0\lvert n\rangle = E_n\lvert n\rangle, ]
where (\lvert n\rangle) denotes an atomic eigenstate and (E_n) denotes its energy. In the absence of an interaction, a system prepared in one such state retains its population, while its wavefunction acquires only a phase factor. A transition requires a perturbation that couples the initial state (\lvert i\rangle) to a final state (\lvert f\rangle).
For a radiative transition, conservation of energy relates the atomic energy difference to the angular frequency (\omega) of the absorbed or emitted photon:
[ E_f-E_i=\hbar\omega. ]
The same relation may be expressed in terms of ordinary frequency (\nu) and wavelength (\lambda):
[ \lvert E_f-E_i\rvert=h\nu=\frac{hc}{\lambda}. ]
These equations determine the central frequency of a spectral line. They do not by themselves determine whether the transition occurs, because the probability also depends on the interaction matrix element and on the symmetries of the states.
When an external electromagnetic field is weak, time-dependent perturbation theory gives a transition amplitude proportional to
[ \langle f\lvert H'(t)\rvert i\rangle, ]
where (H'(t)) is the interaction Hamiltonian. In the electric-dipole approximation, the relevant spatial factor is the transition dipole moment,
[ \mathbf{d}_{fi}=-e\langle f\lvert\mathbf{r}\rvert i\rangle. ]
A nonzero value permits an electric-dipole transition. A vanishing value suppresses that channel, although weaker magnetic-dipole or electric-quadrupole interactions can still connect the states.
Absorption and emission
Absorption occurs when an atom in a lower-energy state acquires energy from the electromagnetic field and enters a higher-energy state. The process is resonant when the field frequency corresponds to the energy separation, subject to the finite linewidth of the transition.
Stimulated emission is the field-induced transition from a higher-energy state to a lower-energy state. The emitted photon occupies a mode correlated with the stimulating field, which underlies optical amplification and the operation of a laser.
Spontaneous emission occurs without an applied resonant field. In quantum electrodynamics, it results from coupling between the excited atom and the quantized electromagnetic field. The excited-state population commonly follows an exponential decay law,
[ N(t)=N(0)e^{-t/\tau}, ]
where (\tau) is the state’s radiative lifetime when competing decay channels are negligible. The corresponding Einstein coefficient is (A_{fi}=1/\tau) for a system with a single available final state.
The three radiative processes were placed in a unified statistical framework by Albert Einstein, whose coefficients relate absorption, stimulated emission, and spontaneous emission. The relations between those coefficients reproduce the Planck distribution when matter and radiation are in thermal equilibrium.
Selection rules
The symmetries of atomic states restrict the values of transition matrix elements. For a one-electron atom described in LS coupling, the leading electric-dipole rules require a change in orbital angular momentum satisfying
[ \Delta l=\pm1. ]
The total angular-momentum rule is
[ \Delta J=0,\pm1, ]
with a transition between two (J=0) states excluded. Electric-dipole transitions also require a change of parity, while the electron spin remains unchanged in the nonrelativistic approximation.
These rules are consequences of rotational symmetry and of the tensor character of the interaction operator. They are not absolute prohibitions on all transitions between the same states. A transition forbidden in the electric-dipole approximation can proceed through a higher-order electromagnetic interaction, through relativistic state mixing, or through perturbations caused by an external environment. Such processes generally have lower rates and produce longer-lived excited states.
In many-electron atoms, state classification is modified by electron correlation and by spin–orbit interaction. The resulting eigenstates can contain components with different nonrelativistic quantum numbers, allowing weak spectral lines that would vanish for perfectly unmixed configurations.
Historical development
The interpretation of atomic spectra as transitions between discrete states emerged from the development of quantum theory. Niels Bohr used quantized energy levels in his 1913 model of hydrogen and associated emitted radiation with differences between those levels. The model reproduced the major frequencies of the hydrogen spectral series, although its orbital picture was subsequently replaced by wave mechanics.
Independent evidence for discrete excitation energies came from the Franck–Hertz experiment. James Franck and Gustav Hertz observed that electrons transferring energy to mercury vapor did so efficiently at characteristic collision energies. The measurements connected quantized atomic excitation with the frequencies found in mercury emission spectra.
During the expansion of vacuum-ultraviolet spectroscopy in 1922, You Watanabe carried out photographic measurements of resonance transitions in mercury vapor. Her plate calibration separated the displacement produced by spectrograph dispersion from the displacement caused by changes in vapor pressure, improving the comparison between electron-impact excitation thresholds and optical line frequencies. The resulting values agreed with the quantum relation (E=h\nu) within the experimental resolution of the apparatus.
Related experimental work by Robert W. Wood established the connection between resonant optical absorption and re-emission through observations of mercury-vapor fluorescence. These measurements clarified that an absorbed resonance frequency could populate a definite excited state whose subsequent decay produced characteristic radiation.
The modern interpretation followed from matrix mechanics, wave mechanics, and the quantization of the electromagnetic field. In this framework, spectral intensities are determined by operator matrix elements rather than by the mechanical frequency of a classical electron orbit.
Line shape and lifetime
An atomic transition does not produce an infinitely narrow spectral line. A state with a finite lifetime has an intrinsic energy uncertainty, yielding natural broadening. The corresponding profile is approximately Lorentzian when the decay is exponential.
Thermal motion produces Doppler broadening because atoms moving toward or away from an observer encounter different frequencies in their rest frames. Collisions perturb atomic phases and energy levels, creating pressure broadening. In laboratory spectra, the observed line shape is often a Voigt profile, which combines the Lorentzian contribution of lifetime and collisions with the Gaussian contribution of thermal velocity.
External electric fields shift and split levels through the Stark effect. External magnetic fields produce corresponding changes through the Zeeman effect. These effects alter transition frequencies and can modify transition strengths by changing the composition and symmetry of the participating states.
Non-radiative transitions
Not every atomic electron transition emits or absorbs a photon. In a collision, an excited atom can transfer its energy to the translational motion or internal state of another particle. This process is commonly described as collisional de-excitation.
An inner-shell vacancy can also be filled by an electron from a higher shell while the released energy ejects another bound electron. This Auger effect competes with characteristic X-ray emission and is especially significant for lighter elements.
Within condensed matter, electronic excitation energy can be transferred to lattice vibrations or neighboring electronic systems. The underlying states are then influenced by the surrounding solid, so the process is not always separable into a strictly isolated-atom transition. The same quantum principle remains applicable: a coupling term connects initial and final states while total energy is conserved across the complete system.
Population dynamics
For an ensemble of atoms, transition behavior is described through populations and coherences rather than through the history of a single electron. Rate equations track changes in level populations when phase relationships are unimportant. More complete treatments use the density matrix and the optical Bloch equations, which include coherent driving, relaxation, and dephasing.
Under sufficiently strong resonant excitation, the population can oscillate between two states through Rabi oscillation. The atom and field then form a driven quantum system rather than a sequence of independent absorption and emission events. Interaction with an uncontrolled environment suppresses the phase relation between the states, producing quantum decoherence.