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Final Exam
Tuesday, May 8, 2012
Starting at 8:30 a.m.,
Hoyt Hall.
------------------Duration: 2h 30m
Chapter 39
Quantum Mechanics of
Atoms
Units of Chapter 39
39-1 Quantum-Mechanical View of Atoms
39-2 Hydrogen Atom: Schrödinger Equation and
Quantum Numbers
39-3 Hydrogen Atom Wave Functions
39-4 Complex Atoms; the Exclusion Principle
39-5 The Periodic Table of Elements
39-7 Magnetic Dipole Moments; Total Angular
Momentum
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:
where
37-11 The Bohr Model
In each of its orbits, the
electron would have a
definite energy:
Z2
En = −(13.6 eV) 2 ,
n
n = 1, 2, 3,...
37-10 Atomic Spectra: Key to the
Structure of the Atom
A portion of the complete spectrum of hydrogen
is shown here. The lines cannot be explained by
the Rutherford theory.
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  gives the
angular momentum;  can take on integer
values from 0 to n - 1.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
3. The magnetic quantum number, m gives
the direction of the electron’s angular
momentum, and can take on integer values
from –  to + .
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
This plot indicates the
quantization of angular
momentum direction
for  = 2.
The other two
components of the
angular momentum are
not defined.
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.
39.7 Magnetic Dipole Moments; Total
Angular Momentum
A magnetic dipole moment is associated to the
electron orbital motion:

e 
µ=−
L
2m
where L is the quantized angular momentum.
The magnitude of the dipole moment is then:

L = l(l +1) 
→

e
µ =
l(l +1)
2m
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:
e
µz = −
Lz
2m
Because Lz is also quantized
e
Lz = ml  → µ z = −
ml = −µ B ml
2m
with µB the Bohr magneton
39.7 Magnetic Dipole Moments; Total
Angular Momentum
An atom placed in a magnetic field will
have its energy levels shifted depending
on the value of ml .
This is called the Zeeman effect:
z
B
y
x
 
U = − µ ⋅ B = µ B B ml
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
In a magnetic field, the energy levels of an
atom split depending on ml (Zeeman effect).
Zeeman Effect
The Zeeman effect
appears as the splitting
of spectral lines in a
strong magnetic field.
In this case the upper
level, with l = 1, splits
into three different
levels
39.7 Magnetic Dipole Moments; Total
Angular Momentum
In the Stern-Gerlach experiment, atoms are
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 are quantized,
corresponding to the quantized values of
the magnetic moment.
39.7 Magnetic Dipole Moments
For H atoms in the ground state we see only two
lines: The orbital momentum is zero (l = 0),
therefore the effect must be due to an intrinsic
momentum of the electron.
µ z = − µ B ml
µ z = − g µ B ms
Stern-Gerlach experiment
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 in hydrogen.
How many different states are possible
in hydrogen for an electron whose
principal quantum number is n = 3?
39.2 Hydrogen Atom: Quantum
Numbers
Problem 39-9: Show that for the hydrogen atom
the number of different electron states of a
given value of n is 2n2.
Solution: For a given n, 0 ≤ l ≤ (n-1). Since for
each l the number of possible states is
2(2l + 1), the number of possible states
for a given n is as follows:
(
)
⎛n n−1 ⎞
2
2
2l
+
1
=
4
l
+
2
=
4
+
2n
=
2n
∑
∑ ∑ ⎜ 2 ⎟
l=0
l=0
l=0
⎝
⎠
n−1
(
)
n−1
n−1
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 a
hydrogen electron in each of the
hydrogen atom states of Example 39–1.
39.2 Hydrogen Atom: Schrödinger
Equation and Quantum Numbers
Selection Rules:
“Allowed” transitions between energy levels
occur between states whose value of  differ
by one:
Other, “forbidden,” transitions also occur but
with much lower probability.
39.2 Hydrogen Atom: Quantum
Numbers
Problem 39-10:
A hydrogen atom is excited in a 5d state.
(a)  Name all the states to which the atom is
dipole-allowed to decay trough emission of a
photon.
(b)  How many different wavelengths are there
(ignoring fine structure)?
39.3 Hydrogen Atom Wave Functions
The ground state wave function of the hydrogen
atom is:
n = 1 l = 0 ml = 0 →
ψ100 =
2
3
0
r
e
−r r0
1
4π
r0 = 0.0529 nm
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:
2
2
P(r)dr = ψ dV = ψ 4πr 2 dr
P(r) = 4πr ψ 100
2
2
4 2 −2r r0
= 3r e
r0
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 (r0 =
0.0529 nm) of the nucleus.
Spherical Coordinates
x = r sin θ cosϕ
y = r sin θ sin ϕ
z = r cosθ
∂2 f ∂2 f ∂2 f
+ 2 + 2 =
2
∂x
∂y
∂z
39.3 Hydrogen Atom Wave Functions
n =1
n=2
n=2
n=2
n=2
l=0
l=0
l =1
l =1
l =1
ml = 0
ml = 0
ml = +1
ml = 0
ml = −1
2
ψ 100 =
r
1 ⎛
r ⎞ − r 2r0
1−
e
⎟
3 ⎜
2r0 ⎠
2r0 ⎝
ψ 200 =
ψ 211 =
e
3
0
1
4π
− r r0
1
2 6r03
ψ 210 =
ψ 21−1 =
1
4π
⎛ r ⎞ − r 2r0
⎜⎝ r ⎟⎠ e
3
sin θ eiϕ
8π
⎛ r ⎞ − r 2r0
⎜⎝ r ⎟⎠ e
3
cosθ
4π
⎛ r ⎞ − r 2r0
⎜⎝ r ⎟⎠ e
3
sin θ e−iϕ
8π
1
2 6r03
1
2 6r03
0
0
0
Comparison of 1s and 2s States
The plot of the radial
probability density for
the 2s state has two
peaks
The highest value of P
corresponds to the most
probable value
39.3 Hydrogen Atom Wave Functions
Probability distributions for n = 2 and = 1 (the
distributions for m = +1 and m = -1 are the same),
and the radial distribution for all n = 2 states.
(
2
2
P(r) dr = ψ 210 + ψ 211 + ψ 21−1
2
)
4πr 2 dr
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 .
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
This table shows the configuration of the
outer electrons only.
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  are
in the same subshell.
The exclusion principle limits the
maximum number of electrons in each
subshell to 2(2 + 1).
39.5 The Periodic Table of the Elements
Electron configurations are written by giving
the value for n, the letter code for  , and the
number of electrons in the subshell as a
superscript.
For example, here is the ground state
configuration of sodium (Na):
1s2 2s2 2p6 3s1
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) 1s2 2s2 2p6 3s3;
(b) 1s2 2s2 2p6 3s2 3p5 4s2;
(c) 1s2 2s2 2p6 2d1 ?
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.
39.5 The Periodic Table of the Elements
Example: Let us apply the exclusion principle to the infinitely
high square well. Let there be five electrons confined
to this rigid box whose width is l. Each energy level can have a maximum of two
electrons, since the only quantum numbers are n
and ms. Thus the lowest energy level will have
two electrons in the n = 1 state, two electrons in
the n = 2 state, and 1 electron in the n = 3 state.
2
2
h
h
E = 2E1 + 2E2 + E3 = ⎡ 2 1 + 2 22 + 1 3 ⎤
= 19
⎢⎣
⎥⎦ 8ml 2
8ml 2
()
2
( ) ()
2