is to create a population inversion of level 3 compared to level 4. Rather
than trying to create this directly, two new states, 1 and 2, are introduced.
An electron is first pumped from level 1 to level 2. The electronic levels are
poised such that an electron in level 2 can make a radiationless transition
to level 3. The laser transition is from level 3 to level 4. The electrons in
level 4 are not stable but relax back to level 1. By repeated pumping, the
electrons are cycled in order to maintain the desired populationinversion.
Lasers take advantage of population inversions to perform Light Amplifica-
tion using Stimulated Amplification of Radiation. A population inversion
in a dye, which is created by a pumping radiation, experiences repeated
exposure to a light that bounces between two mirrors and has the energy
matching the transition between levels 3 and 4. As the stimulated processes
occur, the light intensity at that frequency increases until a equilibrium
is reached. A small portion of the light is allowed to escape to be used for
an experiment. Since the original designs, many other configurations have
been developed including laser diodes that can be used for small-scale
electronics. In addition to the continuous wave design, lasers can be
operated in a pulse mode that allows the excitation of a sample at a specific
wavelength for times of less than 1 fs.
Selection rules
When an electron undergoes a transition from one state to another the
quantum numbers describing the electronic state change. Since a photon
carries angular momentum (s=1), only certain changes in quantum num-
bers, representing the angular momentum of the electrons, are allowed
to change. For an electron in the hydrogen atom, the electron is uniquely
described by the four quantum numbers, n, l, ml, and ms. When an electron
makes a transition, the change in angular momentum must exactly com-
pensate for the angular momentum of the photon. Since the photon has
s=1, then the change in the quantum number lmust be also 1. Thus
some transitions are forbidden, such as a change from a d orbital (l=2)
to an s orbital (l=0).
These selection rules can be derived explicitly from the wavefunctions
derived from Schrödinger’s equation and the definition of the transition
dipole moment:
(14.25)
μ
π
ππ
3 ()=− *
∞
∫∫Re rRrrYnflf nili lfm fl4
3
2
0020d∫∫ YY 10 li m i,lsinθθφdd
μψ ψτ 3 =−∫ *(fiez) d
μψμψτ 3 =∫ *fid