Download Chapter 39 Quantum Mechanics of Atoms

Chapter 39
Quantum Mechanics of
Atoms
Units of Chapter 39
• Quantum-Mechanical View of Atoms
• Hydrogen Atom: Schrödinger Equation and
Quantum Numbers
• Hydrogen Atom Wave Functions
• Complex Atoms; the Exclusion Principle
• The Periodic Table of Elements
• X-Ray Spectra and Atomic Number
• Magnetic Dipole Moments; Total Angular
Momentum
Units of Chapter 39
• Fluorescence and Phosphorescence
• Lasers
• Holography
39.1 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:
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Potential energy for the hydrogen atom:
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
The time-independent Schrödinger
equation in three dimensions is then:
Equation 39-1 goes here.
where
Equation 39-2 goes here.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
There are four different quantum numbers
needed to specify the state of an
electron in an atom.
1. The principal quantum number n gives
the total energy.
2. The orbital quantum number l gives the
angular momentum; l can take on
integer values from 0 to n - 1.
Equation 39-3 goes here.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
3. The magnetic quantum number, ml,
gives the direction of the electron’s
angular momentum, and can take on
integer values from – l to + l.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Figure 39-3 goes
here.
This plot indicates the
quantization of angular
momentum direction
for l = 2. The other
two components of the
angular momentum are
undefined.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
In a magnetic field, the energy levels
split depending on ml .
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
4. The spin quantum number, ms,
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.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
This table summarizes the four quantum
numbers.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Conceptual Example 39-1: Possible
states for n = 3.
How many different states are possible
for an electron whose principal
quantum number is n = 3?
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Example 39-2: E and L for n = 3.
Determine (a) the energy and (b) the
orbital angular momentum for an
electron in each of the hydrogen atom
states of Example 39–1.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
“Allowed” transitions between energy levels
occur between states whose value of l differ by
one:
Other, “forbidden,” transitions also occur but
with much lower probability.
39.3 Hydrogen Atom Wave Functions
The wave function of the ground state
of hydrogen has the form:
The probability of finding the electron
in a volume dV around a given point
is then |ψ|2 dV.
39.3 Hydrogen Atom Wave Functions
The ground state is spherically symmetric;
the probability of finding the electron at a
distance between r and r + dr from the
nucleus is:
39.3 Hydrogen Atom Wave Functions
Example 39-3: Most probable
electron radius in hydrogen.
Determine the most probable
distance r from the nucleus at
which to find the electron in the
ground state of hydrogen.
39.3 Hydrogen Atom Wave Functions
Example 39-4: Calculating probability.
Determine the probability of finding
the electron in the ground state of
hydrogen within two Bohr radii of the
nucleus.
39.3 Hydrogen Atom Wave Functions
This figure shows the three probability
distributions for n = 2 and l= 1 (the distributions
for ml = +1 and m l = -1 are the same), as well as
the radial distribution for all n = 2 states.
39.4 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.
39.4 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.
39.4 Complex Atoms; the Exclusion
Principle
This chart shows the occupied – and some
unoccupied – states in He, Li, and Na.
39.5 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).
39.5 The Periodic Table of the Elements
Each value of l is
given its own letter
symbol.
39.5 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 groundstate
configuration of sodium:
1s22s22p63s1
39.5 The Periodic Table of the Elements
This table shows the configuration of the
outer electrons only.
39.5 The Periodic Table of the Elements
Conceptual Example 39-5: Electron
configurations.
Which of the following electron
configurations are possible, and which
are not:
(a) 1s22s22p63s3;
(b) 1s22s22p63s23p54s2;
(c) 1s22s22p62d1?
39.5 The Periodic Table of the Elements
Atoms with the same number of
electrons in their outer shells have
similar chemical behaviors. They appear
in the same column of the periodic table.
The outer columns – those with full,
almost full, or almost empty outer shells
– are the most distinctive. The inner
columns, with partly filled shells, have
more similar chemical properties.
39.6 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 Z2, so
the wavelengths corresponding to
transitions to the n = 1 state in high-Z atoms
are in the X-ray range.
39.6 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
molybdenum.
39.6 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 Z2.
39.6 X-Ray Spectra and Atomic Number
Example 39-6: X-ray wavelength.
Estimate the wavelength for an n = 2 to
n = 1 transition in molybdenum (Z = 42).
What is the energy of such a photon?
39.6 X-Ray Spectra and Atomic Number
Example 39-7: Determining atomic
number.
High-energy photons are used to
bombard an unknown material. The
strongest peak is found for X-rays
emitted with an energy of 66 keV. Guess
what the material is.
39.6 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”).
39.6 X-Ray Spectra and Atomic Number
Example 39-8: Cutoff wavelength.
What is the shortest-wavelength X-ray
photon emitted in an X-ray tube
subjected to 50 kV?
39.7 Magnetic Dipole Moments; Total
Angular Momentum
If we consider the electron to be moving
in a circle of radius r at speed v around
the nucleus, the magnetic dipole moment
is given by:
where the angular momentum L = mvr.
This semiclassical calculation is also
valid quantum mechanically as long as
the angular momentum is quantized.
39.7 Magnetic Dipole Moments; Total
Angular Momentum
The z component of the dipole moment,
where z is defined to be the direction of an
external magnetic field, is given by:
.
The Bohr magneton is defined as:
39.7 Magnetic Dipole Moments; Total
Angular Momentum
Now we can write the z component of
the magnetic dipole moment as:
An atom placed in a magnetic field will
have its energy levels shifted
depending on the value of ml; this is
the Zeeman effect.
39.7 Magnetic Dipole Moments; Total
Angular Momentum
In the Stern-Gerlach experiment, atoms
were sent through a nonuniform
magnetic field. This field deflects
atoms differently depending on their
magnetic dipole moments. Classically,
one would expect a continuum of
deflection angles.
39.7 Magnetic Dipole Moments; Total
Angular Momentum
Instead, the angles were quantized,
corresponding to the quantized values of
the magnetic moment.
Figure 39-14 goes here.
39.7 Magnetic Dipole Moments; Total
Angular Momentum
The total angular momentum (the vector
sum of the orbital and spin angular
momenta) is also quantized:
The state of an electron can then
be written in the form nLj, such as
2P3/2 (n = 2, l= 1, j = 3/2) and 1S1/2
(ground state of hydrogen).
39.8 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.
Figure 39-15 goes here.
39.8 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 longer to return to the lower state.
Meanwhile, the material glows.
39.9 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.
(a) shows absorption of a photon.
(b) 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.
39.9 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.
Figure 39-18 goes here.
39.9 Lasers
2. The higher state must be a metastable
state, so that once the population is inverted,
it stays that way. This means that transitions
occur through stimulated emission rather
than spontaneously.
Figure 39-19 goes here.
39.9 Lasers
The laser beam is narrow, only spreading due
to diffraction, which is determined by the size
of the end mirror.
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.
39.9 Lasers
A metastable state can also be created through
interactions between two sets of atoms, such as
in a helium—neon laser.
39.9 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.
39.10 Holography
Holograms are created using the coherent
light of a laser. The beam is split, allowing the
film to record both the intensity and the
relative phase of the light. The resulting
image, when illuminated by a laser, is threedimensional.
White-light holograms are made with a laser
but can be viewed in ordinary light. The
emulsion is thick, and contains interference
patterns that make the image somewhat
three-dimensional.
Summary of Chapter 39
• The electron state in an atom is specified by
four numbers; n, l, m l, and ms.
•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.
• ml, the magnetic quantum number, can have
values from –l to +l.
• ms, the spin quantum number, can be +½ or -½.
Summary of Chapter 39
• Energy levels depend on n and l, except in
hydrogen. The other quantum numbers also
result in small energy differences.
•Pauli exclusion principle: no two electrons in
the same atom can be in the same quantum
state.
• Electrons are grouped into shells and
subshells.
• The periodic table reflects shell structure.
• The X-ray spectrum can give information about
inner levels and Z of high-Z atoms.