Chapter 28 Quantum Mechanics of Atoms

Lecture PowerPoint
Chapter 28
Physics: Principles with
Applications, 6 th edition
Giancoli
Chapter 28
Quantum Mechanics of
Atoms
© 2005 Pearson Prentice Hall
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Units of Chapter 28
• Quantum Mechanics – A New Theory
• The Wave Function and Its Interpretation; the
Double-Slit Experiment
• The Heisenberg Uncertainty Principle
• Philosophic Implications; Probability versus
Determinism
• Quantum-Mechanical View of Atoms
• Quantum Mechanics of the Hydrogen Atom;
Quantum Numbers
Units of Chapter 28
• Complex Atoms; the Exclusion Principle
• The Periodic Table of Elements
• X-Ray Spectra and Atomic Number
• Fluorescence and Phosphorescence
• Lasers
28.1 Quantum Mechanics – A New Theory
Quantum mechanics incorporates wave-particle
duality, and successfully explains energy
states in complex atoms and molecules, the
relative brightness of spectral lines, and many
other phenomena.
It is widely accepted as being the fundamental
theory underlying all physical processes.
28.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
An electromagnetic wave has oscillating
electric and magnetic fields. What is oscillating
in a matter wave?
Relation between intensity, electric field, and
photon number, N:
I ∝ E
2 ∝ N
E 2 is proportional to the usually large photon
number.
1

28.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
This role is played by the wave function, .
The square of the wave function at any point is
proportional to the number of electrons
expected to be found there.
For a single electron, the square of the wave
function is the probability of finding the
electron at that point.
28.2 The Wave Function and Its
Interpretation; the Double-Slit Experiment
For example: the interference pattern is observed
after many electrons have gone through the slits.
If we send the electrons through one at
a time, we cannot predict the path any
single electron will
take, but we can
predict the overall
distribution.
28.3 The Heisenberg Uncertainty Principle
Fundamental: the position can only be
measured to about one wavelength.
But according to De Broglie:
28.3 The Heisenberg Uncertainty Principle
Combining, we find the combination of
uncertainties:
λ =
h
p
λ ⋅ p = h
This is called the Heisenberg uncertainty
principle.
It tells us that the position and momentum
cannot simultaneously be measured with
precision.
28.3 The Heisenberg Uncertainty Principle
This relation can also be written as a relation
between the uncertainty in time intervals and
the uncertainty in energy:
28.3 The Heisenberg Uncertainty Principle
Example: measure position of an electron to
10 -10 m. What is the velocity uncertainty?
∆p
h
∆v
= ≈ = 7⋅10
m m ∆x
6
m/s
This says that if an energy state only lasts for a
limited time, its energy will be uncertain. It also
says that conservation of energy can be
violated if the time is short enough.
What is the position uncertainty for ∆p=mc?
h
∆x
≈ =
∆p
h
mc
= 2.5⋅10
−12
m
2

28.3 The Heisenberg Uncertainty Principle
Example: for what time am I allowed to create
out of nothing two optical photons?
∆E
= 2 h⋅
f ≈ 6⋅10
−19
J
28.4 Philosophic Implications; Probability
versus Determinism
The world of Newtonian mechanics is a
deterministic one. If you know the forces on an
object and its initial velocity, you can predict
where it will go.
∆t
≈
h
∆E
≈ 4⋅10
−15
s
Quantum mechanics is very different – you can
predict what masses of electrons will do, but
have no idea what any individual one will.
28.5 Quantum-Mechanical View of Atoms
Since we cannot say exactly where an
electron is, the Bohr picture of the atom, with
electrons in neat orbits, cannot be correct.
Quantum theory
describes an electron
probability distribution;
this figure shows the
distribution for the
ground state of
hydrogen:
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
There are four different quantum numbers
needed to specify the state of an electron in
an atom.
1. Principal quantum number n gives the total
energy:
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
2. Orbital quantum number l gives the angular
momentum; l can take on integer values from 0
to n - 1.
(28-3)
3. The magnetic quantum number, m l , gives
one component of the electron’s angular
momentum, and can take on integer values
from –l to +l.
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
This plot indicates the
quantization of angular
momentum direction for l = 2.
The other two components
of the angular momentum
are undefined.
3

28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
4. The spin quantum number, m s , which for an
electron can take on the values +½ and -½. The
need for this quantum number was found by
experiment; spin is an intrinsically quantum
mechanical quantity, although it mathematically
behaves as a form of angular momentum.
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
This table summarizes the four quantum
numbers.
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
Lowest state has no angular momentum, that
is no orbit in classical sense.
Why does it not fall into the nucleus?
Remember Heisenberg’s uncertainty principle!
The electron cannot be localized so well.
2
2
2
e m 2 p h
k = E = v = ≈
2 2
r 2 2m
2π
mr
2 2
2 2 4
1 2π
ke 2π
k e
≈ E ≈ = 13.6 eV
2
2
r h
h
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
The angular momentum quantum numbers do
not affect the energy level much, but they do
change the spatial distribution of the electron
cloud.
28.6 Quantum Mechanics of the Hydrogen
Atom; Quantum Numbers
“Allowed” transitions between energy levels
occur between states whose value of l differ by
one:
This is a quantum-mechanical variant of
momentum conservation because the photon
also has angular momentum.
28.7 Complex Atoms; the Exclusion
Principle
Complex atoms contain more than one electron,
so the interaction between electrons must be
accounted for in the energy levels. This means
that the energy depends on both n and l.
A neutral atom has Z electrons, as well as Z
protons in its nucleus. Z is called the atomic
number.
4

28.7 Complex Atoms; the Exclusion
Principle
In order to understand the electron
distributions in atoms, another principle is
needed. This is the Pauli exclusion principle:
No two electrons in an atom can occupy the
same quantum state.
The quantum state is specified by the four
quantum numbers; no two electrons can have
the same set.
28.7 Complex Atoms; the Exclusion
Principle
This chart shows the occupied – and some
unoccupied – states in He, Li, and Na.
28.8 The Periodic Table of the Elements
28.8 The Periodic Table of the Elements
We can now understand the organization of the
periodic table.
Electrons with the same n are in the same shell.
Electrons with the same n and l are in the same
subshell.
The exclusion principle limits the maximum
number of electrons in each subshell to 2(2l + 1).
Each value of l is
given its own letter
symbol.
28.8 The Periodic Table of the Elements
Electron configurations are written by giving
the value for n, the letter code for l, and the
number of electrons in the subshell as a
superscript.
For example, here is the ground-state
configuration of sodium:
28.8 The Periodic Table of the Elements
This table shows the configuration
of the outer electrons only.
5

28.8 The Periodic Table of the Elements
Atoms with the same number of electrons in
their outer shells have similar chemical
behavior. They appear in the same column of
the periodic table.
Noble gases: outer subshell is full.
Halogens: one electron missing in outer shell.
Alkalis: one electron in s-subshell.
28.9 X-Ray Spectra and Atomic Number
The effective charge that an electron “sees”
is the charge on the nucleus shielded by
inner electrons. Only the electrons in the first
level see the entire nuclear charge.
The energy of a level is proportional to Z 2 , so
the wavelengths corresponding to
transitions to the n = 1 state in high-Z atoms
are in the X-ray range.
28.9 X-Ray Spectra and Atomic Number
Inner electrons can be ejected by high-energy
electrons. The resulting X-ray spectrum is
characteristic of the element.
This example is for
Molybdene.
28.9 X-Ray Spectra and Atomic Number
Measurement of these spectra allows
determination of inner energy levels, as well
as Z, as the wavelength of the shortest X-rays
is inversely proportional to Z 2 .
c
E = h f = h
λ
−15
E = 3⋅10
J = 2⋅10
4
eV
28.9 X-Ray Spectra and Atomic Number
The continuous part of the X-ray spectrum
comes from electrons that are decelerated by
interactions within the material, and therefore
emit photons. This radiation is called
bremsstrahlung (“braking radiation”).
28.10 Fluorescence and Phosphorescence
If an electron is excited to a higher energy
state, it may emit two or more photons of
longer wavelength as it returns to the lower
level.
6

28.10 Fluorescence and Phosphorescence
Fluorescence occurs when the absorbed
photon is ultraviolet and the emitted photons
are in the visible range.
Phosphorescence occurs when the electron is
excited to a metastable state; it can take
seconds or more to return to the lower state.
Meanwhile, the material glows.
28.11 Lasers
A laser produces a narrow, intense beam of
coherent light. This coherence means that, at
a given cross section, all parts of the beam
have the same phase.
The top figure shows absorption of
a photon. The bottom picture shows
stimulated emission – if the atom is
already in the excited state, the
presence of another photon of the
same frequency can stimulate the
atom to make the transition to the
lower state sooner. These photons
are in phase.
28.11 Lasers
To obtain coherent light from stimulated
emission, two conditions must be met:
1. Most of the atoms must be in the excited
state; this is called an inverted population.
28.11 Lasers
2. The higher state must be metastable,
meaning that the probability of spontaneous
decay is small. This means that transitions
occur through stimulated emission rather than
spontaneously.
28.11 Lasers
The laser beam is narrow, only spreading due
to diffraction, which is determined by the size
of the end mirror.
28.11 Lasers
A metastable state can also be created through
interactions between two sets of atoms, such as
in a helium-neon laser.
An inverted population
can be created by
exciting electrons to a
state from which they
decay to a metastable
state. This is called
optical pumping.
7

28.11 Lasers
Lasers are used for a wide variety of
applications: surgery, machining, surveying,
reading bar codes, CDs, and DVDs, and so on.
This diagram shows how a CD is read.
Summary of Chapter 28
• Quantum mechanics is the basic theory at the
atomic level; it is statistical rather than
deterministic
• Heisenberg uncertainty principle:
• Electron state in atom is specified by four
numbers, n, l, m l , and m s
Summary of Chapter 28
• n, the principal quantum number, can have any
integer value, and gives the energy of the level
• l, the orbital quantum number, can have values
from 0 to n – 1
• m l , the magnetic quantum number, can have
values from –l to +l
• m s , the spin quantum number, can be +½ or -½
• Energy levels depend on n and l, except in
hydrogen. The other quantum numbers also
result in small energy differences.
Summary of Chapter 28
• Pauli exclusion principle: no two electrons in
the same atom can be in the same quantum
state
• Electrons are grouped into shells and
subshells
• Periodic table reflects shell structure
• X-ray spectrum can give information about
inner levels and Z of high-Z atoms
8