Download Ch.41- Orbital angular momentum, counting states

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Summary of Midterm II
Histogram Content
27-37 1
38-48 5
49-59 5
60-70 11
71-81 12
82-92 15
93-103 7
Lowest Score
27
Highest Score
99
Average
73.5
Total Number of Scores
56
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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).
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The hydrogen atom: Results
-13.6eV
En =
n2
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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The hydrogen atom: Results
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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.
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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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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
-13.6eV
En =
n2
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E=-13.6eV/9=-1.51 eV
All 9 states are degenerate.
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 five states with
l = 2 and different values
of ml. The orbital angular
momentum has the same
magnitude L for each
these five states, but has
different values of the zcomponent Lz.
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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.
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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
cos(qmin ) = Lz / L
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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.
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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.
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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.
Copyright © 2012 Pearson Education Inc.
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.
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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
Copyright © 2012 Pearson Education Inc.
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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