Chapter 28
Objectives
1. Distinguish between the Bohr model and the quantum
mechanical model of the atom.
2. State two forms of the Heisenberg uncertainty principle and
explain how the principle predicts an inherent unpredictability in
nature.
3. Use the uncertainty principle to compute the minimum uncertainty
of a molecule's momentum or position.
4. Name the four quantum numbers required to describe the state
of an electron in an atom. State the symbol used to represent each
quantum number.
5. Given a value for the principle quantum number, list the range of
values for the other quantum numbers.
6. State the Pauli exclusion principle. Use this principle to
determine the maximum number of electrons that fill the energy levels
of atoms where n = 1 or n = 2.
7. Given the atomic number of a particular element, write the
electronic configuration for the ground state of the atom.
8. Use a periodic table to identify an element whose outer electronic
configuration is given.
9. Describe two ways x-ray photons can be produced.
10. State the Bragg equation. Use the equation to determine either
the wavelength of the x-rays incident on a crystal or the distance
between the layers of atoms which make up the crystal.
11. Determine the cut-off frequency and wavelength of an x-ray
photon produced by accelerating electrons through a known potential
difference.
12. Determine the wavelength of a Kα x-ray of known energy. In
addition, use Moseley's equation to determine the atomic number of
the element which produced this x-ray.
Outline
28-1 Quantum Mechanics–A New Theory
28-2 The Wave Function and Its Interpretation; the Double-Slit
Experiment
28-3 The Heisenberg Uncertainty Principle
*28-4 Philosophic Implications; Probability versus Determinism
28-5 Quantum-Mechanical View of Atoms
28-6 Quantum Mechanics of the Hydrogen Atom; Quantum Numbers
28-7 Complex Atoms; the Exclusion Principle
28-8 The Periodic Table of Elements
*28-9 X-Ray Spectra and Atomic Number
*28-10
Fluorescence and Phosphorescence
*29-11
Lasers
*28-12
Holography
Summary
Chapter 28 extends the quantum model to a more comprehensive
theory that is able to explain the structure and predict the spectra of
more complex atoms. The Heisenberg uncertainty principle is
presented. Quantum numbers are defined. The periodic table is built
using the Pauli exclusion principle. The chapter ends with a discussion
of various quantum phenomena, including lasers and holography.
Major Concepts
By the end of the chapter, students should understand each of the
following and be able to demonstrate their understanding in problem
applications as well as in conceptual situations.
 The quantum model
 Wave functions
 Schrödinger equation
 Electron probability clouds
 Heisenberg uncertainty principle
 The quantum mechanical model of hydrogen
 Quantum numbers
 Pauli exclusion principle
 Periodic table
 Lasers
Lecture Note:
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 uncertainty. We
h
h
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, . . .
total energy and radius of the "orbit."
This quantizes the
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..
1
. The electron can be thought of
2
(roughly) as spinning with its angular momentum up or down. Even
with zero external magnetic field, some lines are seen to be multiplets
ms or s: spin quantum number = 
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.
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.