23.1 Emission/Absorption Spectroscopy and Energy Levels 953
which allows the second version of this equation to be written. The integration is over
all values of all of the coordinates of the system, abbreviated by the symbolq. The
integral in Eq. (23.1-8) is thezcomponent of thetransition dipole momentfor states
nandj.
For two states that have a nonzero transition dipole moment, a transition between
them is predicted to occur with the absorption or emission of a photon. Such a transition
is called anallowed transition. A transition between two states that have a zero transition
dipole moment is predicted not to occur, and is called aforbidden transition. A rule that
tells which transitions are allowed is called aselection rule. The selection rules that we
give in this chapter are generally obtained with approximate wave functions using first-
order perturbation theory. Most of the selection rules are therefore not exactly obeyed.
Forbidden transitions frequently do occur, but generally with lower probabilities than
allowed transitions.
Inspection of Eq. (23.1-8) shows that the transition dipole moment is replaced by its
complex conjugate if the wave functions for the initial and final states are switched. This
means that incident radiation will induce transitions in either direction with the same
probability. A transition that raises the energy of the atom or molecule corresponds to
absorptionof radiation, whereas one that lowers the energy corresponds tostimulated
emissionof radiation.
We observe the emission or absorption from a system of many atoms or molecules.
Absorption and stimulated emission occur simultaneously if both the upper and lower
states are populated. Absorption is observed if the lower-energy state has a greater
population than the higher-energy state, and stimulated emission is observed if the
higher-energy state has a greater population. Stimulated emission has the same wave-
length as the incident radiation, moves in the same direction, and is in phase with the
incident radiation. That is, the crests and troughs of its waves coincide with those of
the waves of the incident radiation. With many atoms or molecules emitting radiation
stimulated by the same wave, a strong beam of unidirectional radiation can result with
all of its waves in phase. Such radiation is said to becoherent. This kind of radiation
is emitted by lasers, which amplify electromagnetic radiation by adding radiation to
an incident beam by stimulated emission. “Laser” is an acronym for “light amplifi-
cation by stimulated emission of radiation.” In a laser, some means must provide a
nonequilibrium excess population in the upper energy state.
Transitions resulting in the emission of photons can also occur in the absence of
stimulating radiation. This is calledspontaneous emission. The probability of such
transitions is also proportional to the square of the transition dipole moment, but is
independent of the intensity of any radiation. Because there is no inducing radiation to
specify a direction and phase, radiation that is spontaneously emitted by a system of
many molecules is emitted in all directions and is not coherent. Inemissionspectroscopy
spontaneously emitted radiation from excited atoms or molecules is observed.
Inabsorption spectroscopythe attenuation of an incident beam is observed. The
amount of absorption of radiation depends on three things: the intensity of the radiation,
the inherent probability that the transition will take place, and the numbers of molecules
in the initial state and in the final state. In a system of many atoms or molecules at
thermal equilibrium, the number of atoms or molecules occupying a state of energyE
is proportional to the Boltzmann factor of Eq. (22.5-1):(Number of molecules in a state with energyE) ∝e−E/kBT (23.1-9)wherekBis Boltzmann’s constant (equal to 1. 3807 × 10 −^23 JK−^1 ) andTis the abso-
lute temperature. The symbol∝stands for “is proportional to.”