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Chem 3502/5502
Physical Chemistry II (Quantum Mechanics)
3 Credits
Fall Semester 2009
Laura Gagliardi
Lecture 16, October 16, 2009
Solved Homework
(Homework for grading is also due today)
Given that for a hydrogenic atom
H
=
T
+
V
=
1
2
"
2
!
!
1
where we have explicitly written the proper kinetic and potential energy operators. We
are reminded/given that
H
=
!
Z
2
2
n
2
and
r
!
1
=
Z
n
2
so it is trivial to determine
T
=
Z
2
2
n
2
Thus, the expectation value of the kinetic energy is
−
1/2 times the expectation value of
the potential energy. This relationship of <
T
> =
−
(1/2)<
V
> is a manifestation of what is
known as the quantum mechanical virial theorem, and it holds true for
all
wave functions
where the potential energy term in the Hamiltonian operator depends only on
r
–1
to one or
more nuclei.
Atomic Spectroscopy
The hydrogenic (oneelectron) atom has 3 quantum numbers associated with each
wave function. Two of these are from the spherical harmonics, and we already know the
selection rules on the spherical harmonics:
!
l
= ±
1
and
!
m
l
=
0,
±
1
(161)
without derivation we will simply accept that the selection rule for
Δ
n
is that
absorption/emission is allowed for
any
change in
n
(note that
n
must change from one
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value to another, or else
E
, which depends only on
n
, fails to change, and then there is no
opportunity to absorb or emit a photon in the first place!)
The figure on the next page illustrates allowed transitions in the spectrum of a
hydrogenic atom. Note that, although the figure is terribly complicated in the sense that
many, many transitions are allowed, an actual measured spectrum would be relatively
simple, because of the very small number of different
Δ
E
values. A single photon
frequency is associated with each
Δ
E
, and thus there is a single
ν
for every
n
= 1
→
n
= 2
transition, and a single
ν
for every
n
= 2
→
n
= 3 transition, irrespective of the actual
orbitals involved at the particular principal quantum number level.
Truth is, of course, life is not quite that simple. It's only that simple if you use a
lowresolution spectrometer. If you look more carefully, or you make the experiment a
bit more complicated, suddenly you find a lot of new lines in the spectrum (different
photon frequencies). Let's start with the most profound complication.
Electron Spin
In 1922, Stern and Gerlach did the following experiment.
1)
Heat a block of silver until it vaporizes (whoa.
..)
2)
Arrange the pressure in the experimental system such that the gas of silver atoms
collimates into a "beam" that passes through the poles of a magnet.
3)
Observe where the silver atoms strike a target behind the magnet.
Here's what Stern and Gerlach expected. The silver atom can be thought of as being like a
very big hydrogen atom. That's because all of the electrons but one completely fill
principal quantum number levels 1 to 3 and the 4s, 4p, and 4d levels. So, for the one
remaining electron in the 5s orbital, it’s a little bit like being around a nucleus of unit
positive charge since “underneath” it, it sees 47 protons and 46 electrons that are
spherically symmetric. With that picture, it should be clear that the total angular
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