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Chapter 28
QUANTUM MECHANICS
By the 1920's, the Bohr model of the atom was in trouble. It did not work well for atoms other
than hydrogen, and why it worked when it did was not clear. Schrodinger, Heisenberg, and
others invented quantum mechanics to describe the atomic realm.
Schrodinger's wave equation describes the motion of particles in a manner akin to Newton's
F = ma. Given energy and potentials, matter waves of amplitude  are solutions to the equation.
We interpret the square of the amplitude as proportional to the probability a particle is at a
specific point.
Heisenberg advanced his uncertainty principle: The product of the uncertainty in position and
momentum of a particle cannot be made vanishingly small, but always exceed some minimum
h
h
uncertainty. We write ΔxΔp 
. Similarly, it was found Et 
.
2π
2
"To measure is to disturb," describes this principle.
Note the probabilistic interpretation above. Einstein rejected this and for thirty years tried to
show there was an underlying determinism to nature. He failed. Bohr and his followers were
correct -- nature does play dice with the universe.
Quantum mechanics view atoms as surrounded by electron "clouds," corresponding to the
probability distribution of the electrons that emerges as a solution to the Schrodinger equation.
Various quantum numbers emerge when we solve this equation, which can be interpreted as
follows:
n: the principal quantum number =1, 2, 3, . . . This quantizes the total energy and radius of the
"orbit."
ℓ: the orbital quantum number = 0, 1, 2, . . . (n - 1). This is related to the angular momentum of
the electron and affects the shape of the orbit. Spherical orbits have ℓ = 0. These values are also
written s,p,d,... so the lowest energy spherical orbit of an electron orbiting a hydrogen nucleus
would have n = 1, ℓ = 0 or 1s.
mℓ or m: the magnetic quantum number = -ℓ to ℓ. This gives the orientation of the orbit relative
to an external magnetic field. This is evident in the Zeeman effect where a spectral line splits
into several lines when the atom is in a strong external magnetic field..
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. The electron can be thought of (roughly) as spinning with
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its angular momentum up or down. Even with zero external magnetic field, some lines are seen
to be multiplets of lines very closely space, the so called fine structure; these lines are accounted
for by spin in Dirac's modification of Schrodinger's equation.
ms or s: spin quantum number = 
There are various selection rules pertaining to how elections change orbits and produce spectral
lines. For instance, if Δ  1 the transition is forbidden and occurs with very low probability.
The photon carries away the angular momentum lost in the allowed transition as spin.
For complex atoms, we invoke the Pauli exclusion principle: no two electrons in an atom can
occupy the same quantum state. The order in which electrons fill various energy levels and
which levels are filled determine an atom's chemical properties and explain the periodic table.
In column one, the alkali metals have one outer (valence) electron that is easily lost.
In column seven, the halogens are one electron short of filling a shell or subshell and so they
want to combine chemically to get that electron.
In column eight, the noble gasses have filled shells and are loathe to form compounds.
The transition elements, the lanthanides (rare earths), and the actinides are similar due to
incomplete inner subshells.
Electronic Structure of Atom: Applications
X-ray spectra of heavy atoms revealed details of their electronic structure. Electrons are
accelerated and collide with a target, occasionally knocking inner electron loose. When another
electron drops to the n = 1 level in these atoms, the emitted radiation is in the x-ray (high energy
photon) region of the EM spectrum. There is also a continuum component of x-ray spectra due
to bremsstrahlung or braking radiation as the bombarding electrons lose part of their energy
colliding with the target.
Florescence occurs when a UV photon excites an electron to a higher energy level. The electron
can drop in steps from this level, emitting visible light photons. This is the basis of florescent
lights. Phosphorescent materials are excited to a metastable state. Normally, electrons decay to
lower orbits in 10-8s but in those states, the average decay may take seconds. Some take much
longer, giving a long lasting glow after the material is energized.
Lasers
Invented around 1960, the laser (an acronym for light amplification by stimulated emission of
radiation) is based on a property predicted almost 50 years earlier by Einstein. When an atom
has an electron in an excited, metastable state, an incoming photon of just the right energy can
stimulate the electron to drop to a lower level. The photon it emits is the same energy as, and is
emitted in the same direction and in phase with, the incident photon. If these two photons cause
a similar reaction in other atoms, the light is amplified.
A laser beam is nearly monochromatic (all photons of nearly the same energy). The photons all
travel in the same direction; their crests and troughs are "in step". We say the light is coherent.
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