Download Atomic Structure, angular momentum, electron orbitals

PHYS274 Spring 2017
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• Quiz
• Homework #9 due 4/5
– Atomic Structure
– Chap 41 in its entirety
1
The hydrogen atom: Quantum numbers
• The Schrödinger equation for
the hydrogen atom is best
solved using coordinates (r, θ,
ϕ) rather than (x, y, z) (see
Figure at right).
• The stationary states are
labeled by three quantum
numbers: n (which describes
the energy), l (which
describes orbital angular
momentum), and ml (which
describes the z-component of
orbital angular momentum).
2
Q29.1
A quantum-mechanical system initially
in its ground level absorbs a photon and ends up in its
first excited state. The system then absorbs a second
photon and ends up in the second excited state.
For which of the following systems does the second
photon have a longer wavelength than the first one ?
A)
B)
C)
D)
A harmonic oscillator
A hydrogen atom
A particle in a box
Impossible for any system by the Heisenberg
uncertainty principle
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Q29.1
A quantum-mechanical system initially
in its ground level absorbs a photon and ends up in its
first excited state. The system then absorbs a second
photon and ends up in the second excited state.
For which of the following systems does the second
photon have a longer wavelength than the first one ?
A)
B)
C)
D)
A harmonic oscillator
A hydrogen atom
A particle in a box
Impossible for any system by the Heisenberg
uncertainty principle
4
Q29.1
A quantum-mechanical system initially
in its ground level absorbs a photon and ends up in its
first excited state. The system then absorbs a second
photon and ends up in the second excited state.
For which of the following systems does the second
photon have a longer wavelength than the first one ?
A)
B)
C)
D)
A harmonic oscillator
A hydrogen atom
A particle in a box
Impossible for any system by the Heisenberg
uncertainty principle
5
Q29.1
A quantum-mechanical system initially
in its ground level absorbs a photon and ends up in its
first excited state. The system then absorbs a second
photon and ends up in the second excited state.
For which of the following systems does the second
photon have a longer wavelength than the first one ?
A)
B)
C)
D)
A harmonic oscillator
A hydrogen atom
A particle in a box
Impossible for any system by the Heisenberg
uncertainty principle
6
The hydrogen atom: Results
This result agrees
with the Bohr model
!
Here l=0,1,2,….n-1
This result does not agree
with the Bohr model.
Question: Why ? What
happens for n =1 ?
Here m=0,±1, ±2,…. ±l
The Bohr model does not include
this part at all.
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Q29.2
8
Q29.2
9
Q29.3
10
Particle in a three-dimensional box
• For a particle enclosed in a cubical box with sides of length L
(see Figure below), three quantum numbers nX, nY, and nZ label
the stationary states (states of definite energy).
• The three states shown here are degenerate: Although they have
different values of nX, nY, and nZ, they have the same energy E.
11
Q29.3
Compare (1,1,1) to
(2,1,1) or (1,2,1) or (1,1,2)
gives 6 / 3 =2 A
12
Q29.4
For a particle confined inside a 3-D box, the energies are given
by
Which of the following sets of states are degenerate ?
A. (1,1,1),(2,2,2),(3,3,3)
B. (1,2,1),(2,1,2),(1,1,2)
C. (3,1,1),(1,1,3),(3,1,1)
D. (1,1,1),(0,0,0),(-1,1,0)
Hint: Degenerate
means the
energies are
equal.
13
Q29.4
For a particle confined inside a 3-D box, the energies are given
by
Which of the following sets of states are degenerate ?
A. (1,1,1),(2,2,2),(3,3,3)
B. (1,2,1),(2,1,2),(1,1,2)
C. (3,1,1),(1,1,3),(3,1,1)
D. (1,1,1),(0,0,0),(-1,1,0)
Hint: Degenerate
means the
energies are
equal.
14
The hydrogen atom: Results
This result agrees
with the Bohr model
!
Here l=0,1,2,….n-1
This result does not agree
with the Bohr model.
Question: Why ? What
happens for n =1 ?
Here m=0,±1, ±2,…. ±l
The Bohr model does not include
this part at all.
15
The hydrogen atom: Results
16
The hydrogen atom: Quantum states
• Table 41.1 (below) summarizes the quantum states of the
hydrogen atom.
• For each value of the quantum number n, there are n
possible values of the quantum number l. For each value
of l, there are 2l + 1 values of the quantum number ml.
17
What do these letters s, p, d, f mean ?
For atomic structure, we distinguish the
orbital angular momentum states as
follows:
s-wave: l=0
p-wave: l=1
d-wave: l=2
f-wave: l=3
For the principal quantum numbers, in x-ray
spectroscopy we use the old labeling:
K-shell: n=1
L-shell: n=2
M-shell: n=3
N-shell: n=4
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Function follows form
• The way the electrons distribute determines atomic behavior
19
Example of counting hydrogen states
How many distinct (n,l,ml) states of the hydrogen
atom with n=3 are there ? What are their energies ?
Answer:
n=3 l=0,1,2 (s,p and d waves
are possible)
For l=0, there is one state. For
l=1, ml=-1,0,1 (3 states)
l=2, ml=-2,1,0,1,2 (5 states)
So all together there are 1+3+5= 9
states of the hydrogen atom in n=3
E=-13.6eV/9=-1.51 eV
All 9 states are degenerate.
20
The hydrogen atom: Degeneracy
• Hydrogen atom states
with the same value of n
but different values of l
and ml are degenerate
(have the same energy).
• The figure on the right
shows the states with l =
2 and different values of
ml. The orbital angular
momentum has the same
magnitude L for each
these states, but has
different values of the zcomponent Lz.
21
Clicker atom on 3-D hydrogen atom
This illustration shows the
possible orientations of the
angular momentum vector in a
hydrogen atom state with l = 2.
For a given value of Lz,
A. the angular momentum vector can point in any
direction tangent to the cone for that value of Lz.
B. the electron orbits along the corresponding red circle,
so the orbit may or may not have the nucleus at its
center.
C. both A. and B. are true.
D. neither A. nor B. is true.
22
Clicker question on 3-D hydrogen atom
This illustration shows the
possible orientations of the
angular momentum vector in a
hydrogen atom state with l = 2.
For a given value of Lz,
A. the angular momentum vector can point in any
direction tangent to the cone for that value of Lz.
B. the electron orbits along the corresponding red circle,
so the orbit may or may not have the nucleus at its
center.
C. both A. and B. are true.
D. neither A. nor B. is true.
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Angular momentum in an excited state of hydrogen
Consider the n=4 states of hydrogen. (a) What is the
maximum magnitude L of the orbital angular momentum ?
(b)What is the minimum angle between the L vector and
the z-axis
Answer:
n=4 l=0,1,2,3 (s,p,d and f waves are
possible)
So l=3 is the maximum possible
24
The hydrogen atom: Probability distributions I
• States of the hydrogen atom with l = 0 (zero orbital angular
momentum) have spherically symmetric wave functions that
depend on r but not on θ or ϕ. These are called s states. The figure
(below) shows the electron probability distributions for three of
these states.
25
The hydrogen atom: Probability distributions II
• States of the hydrogen atom with nonzero orbital angular
momentum, such as p states (l = 1) and d states (l = 2), have wave
functions that are not spherically symmetric. The figure (below)
shows the electron probability distributions for several of these
states, as well as for two spherically symmetric s states.
26
Clicker question on the hydrogen atom
This illustration shows radial
probability distribution
functions for three hydrogen
atom wave functions, plotted
versus r/a (r = distance from
the center of the atom and a =
0.0529 nm). It follows that
A. an electron in a 4p state is always farther from the
center of the atom than is an electron in a 2p state.
B. an electron in a 2p state can be found at the atom’s
center.
C. a 3p state has three units of orbital angular
momentum.
D. none of the above is true.
27
Clicker question on the hydrogen atom
This illustration shows radial
probability distribution
functions for three hydrogen
atom wave functions, plotted
versus r/a (r = distance from
the center of the atom and a =
0.0529 nm). It follows that
A. an electron in a 4p state is always farther from the
center of the atom than is an electron in a 2p state.
B. an electron in a 2p state can be found at the atom’s
center.
C. a 3p state has three units of orbital angular
momentum.
D. none of the above is true.
28
Clicker question on the hydrogen atom
The Bohr model and the Schrödinger equation both make
predictions about the hydrogen atom. For which of the
following quantities are the predictions different?
A. the energy of the lowest (n = 1) energy level
B. the difference in energy between the n = 2 and n = 1
energy levels
C. the orbital angular momentum of the electron in the
lowest (n = 1) energy level
D. more than one of A., B., and C.
E. none of A., B., or C.—the predictions are identical
for all of these
29
A41.5
The Bohr model and the Schrödinger equation both make
predictions about the hydrogen atom. For which of the
following quantities are the predictions different?
A. the energy of the lowest (n = 1) energy level
B. the difference in energy between the n = 2 and n = 1
energy levels
C. the orbital angular momentum of the electron in the
lowest (n = 1) energy level
D. more than one of A., B., and C.
E. none of A., B., or C.—the predictions are identical
for all of these
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For next time
• Atomic concepts
 Read material in advance
 Review spherical coordinates, angular momentum
• Homework #9 available, due 4/5
Don’t overdo it and see you in April
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