ManyElectron Atoms

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Many-Electron Atoms electron spin Pauli
exclusion principle symmetric and
antisymmetric wave functions
Had she taken a bullfighter I would have
understood but an ordinary chemist....Wolfgang
Pauli, explaining his deep depression after his
wife left him for (gasp!) a chemist.
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Chapter 7 Many-Electron Atoms
We spend only 2 days on chapter 7!
7.1 Electron Spin
Electron spin is the cause of fine structure in
spectral lines, and the anomalous Zeeman effect
("extra" and "missing" splittings of spectral
lines in the presence of weak magnetic fields).
Electron spin is also of critical importance in
magnetism.
You were exposed to the Zeeman effect at the
end of chapter 6. The anomalous Zeeman effect
involves even more splittings of spectral lines
that cant be explained by the normal Zeeman
effect.
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Spectral lines (absorption or emission) are
caused by photons absorbed or emitted when
electrons change their energy state.
Changes in the principal quantum number n cause
the most noticeable changes .
However, changes in other quantum numbers also
give rise to changes in electron energies. Such
changes typically involve less energy, and result
in a "splitting" of the primary lines.
1s?2p so selection rules are not violated!
The ordinary Zeeman effect.
http//csep10.phys.utk.edu/astr162/lect/light/zeem
an-split.html
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Not all splittings can be explained by the
quantum theory developed in chapter 6. It turns
out we need another quantum number -- spin.
Anomalous Zeeman effect.
How can one look happy when he is thinking about
the anomalous Zeeman effect?Pauli, 1923
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Lets think about electrons and magnetism for a
moment.
If you shoot an electron through a region of
space with no magnetic field, the electron will
experience no deflection (assuming no
gravitational forces).
If you shoot an electron through a region of
space with a nonzero magnetic field, you know
from Physics 24 that the electron will experience
a deflection.
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A silver atom has 47 protons and electrons. It
has a single outermost 5s electron, and this 5s
electron has zero orbital angular momentum. The
single electron acts sort of like a lone
electron (it sees a 47 proton nucleus shielded
by 46 electrons, so it is sort of like hydrogen.
The 5s electron has l0 and so it (the outer
electron) should not interact with an external
magnetic field.
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However, the silver atom is like a dipole, and
a dipole should be deflected by an external
magnetic field.
If one uses an oven to heat silver to boiling
and makes a beam of silver atoms, the silver atom
dipoles should have randomly oriented (in space)
dipole moments.
A magnetic field should deflect the beam of
silver atoms in all directions.
With these thoughts in mind, lets consider the
Stern-Gerlach experiment, in which silver atoms
were shot through a magnetic field.
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The Stern-Gerlach experiment (1924)
With field off, atoms go straight through.
Classical expectation with field on, atoms will
deflect in all directions. (The funny shape
is due to the magnet geometry.)
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Lets see what really happens
Experimental result.
Dang! Another classical prediction down the
tubes. What happened?
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See http//hyperphysics.phy-astr.gsu.edu/hbase/spi
n.htmlc6 for a more detailed discussion.
Evidently the silver 5s electron has some binary
property.
It can be this kind of electron
Or it can be this kind of electron
All electrons together do this
This binary (one or the other, but only two
choices) property is the electron spin.
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OK, so he says electrons have spin
OK, so he says electrons have spin aha, like
this
A spinning ball of charge is equivalent to a
current loop, which would produce a magnetic
moment, so the electron would interact with an
external magnetic field.
No! No! No!
No! No! No! Not an electron! See Example
7.1
But before we discard this classical picture of
electron as spinning ball of charge, lets think
about it for a minute.
The picture suggests the electron has an
intrinsic angular momentum, associated with the
spin and independent of the orbital angular
momentumthis is in fact the case.
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The picture also explains the intrinsic
magnetic moment of an electron.
So it is OK to keep this picture
in your head, and even use it to help explain
spin, but just remember that it is ultimately
wrong.
However, this statement is correct
The electron spin gives rise to an intrinsic
angular momentum, associated with the spin and
independent of the orbital angular momentum. It
also gives rise to the intrinsic magnetic moment
of an electron.
Or--chicken and egg--you might wish to say the
electron has an intrinsic (built in) angular
momentum, which manifests itself as the spin.
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The electrons orbital angular momentum is
quantized, and so is its spin angular momentum.
The spin quantum number s which describes the
spin angular momentum of an electron has a single
value, s½.
All electrons have the same s!
Just as is the case with the orbital quantum
number l and orbital angular momentum L, the spin
angular momentum is given by
capital S
lowercase s
Ill make the difference obvious on an exam!
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All electrons have the same spin angular momentum
S (magnitude!).
S (3/4)½ h is the magnitude of the electron
spin angular momentum.
Aha! There are only two possible values of the
z-component of the spin angular momentum. Now we
understand the Stern-Gerlach experiment!
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There are exactly two possible orientations (see
fig. 7.2) of the electrons spin angular momentum
vector...
up
down
also, up is not quite up! down is not quite down!
I find this exceedingly strange!
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You can calculate the spin magnetic moment of an
electron, and its z component (equations 7.3 and
7.4). Because we skipped corresponding section on
magnetism in Chapter 6, we will not go into
further detail here, and I will not hold you
repsonsible for it on exams or quizzes.
7.2 Exclusion Principle
This is a very brief, but very important section.
In 1925 Wolfgang Pauli postulated the (Pauli)
exclusion principle, which states that no two
electrons in one atom can exist in the same
quantum state.
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Here are a couple of alternate ways to express
the exclusion principle
No two electrons in the same atom can have the
same four quantum numbers (n, l, ml, ms).
Generalizing no two electrons in the same
potential can exist in the same quantum state.
(Vital to the understanding of solid state
physics.)
In 1925, only three quantum numbers were known
(n, l, ml). Pauli realized there needed to be a
fourth.
Pauli was a boy genius mathematician. After high
school he began publishing papers on relativity.
He won the 1945 Nobel Prize for discovering the
exclusion principle (he was nominated for the
prize by Einstein).
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"State" refers to the four quantum numbers n, l,
ml, ms. Obviously, all electrons have the same s.
On the surface, the exclusion principle is very
simple, but it is extremely important. We will
come back to it many times in this course.
An even more general statement reads
No two fermions in the same potential can exist
in the same quantum state.
Before long youll know what a fermion is.
Pauli is perhaps most famous among physicists for
the Pauli Effect. You will not be quizzed or
tested on the following two slides about this
effect.
Sources W. Cropper, Great Physicists, Oxford,
2001, p.256-7 G. Gamow, Thirty Years That Shook
Physics, Heinemann, 1966, p.64.
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Pauli's awkwardness in the lab was legendary and
some physicists haved termed it the Pauli
Effect, a phenomenon much dreaded by
experimentalists. According to this physical law,
Pauli could cause, by his mere presence,
laboratory accidents and experimental
catastrophes of all kinds.
Pauli was such a good theoretical physicist that
something usually broke in the lab whenever he
merely stepped across the threshold.
There were well-documented instances of Pauli's
appearance in a laboratory causing machines to
break down, vacuum systems to spring leaks, and
glass apparatus to shatter.
Otto Stern is said to have forbidden Pauli to
enter his institute for fear of such
malfunctions.
Stern-Gerlach
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Read here to see how Pauli's destructive spell
became so powerful that he was credited with
causing an explosion when he was not even within
immediate surroundings.
Corollary of the Pauli Effect some physicists
tried to play a practical joke on him to
demonstrate the Pauli effect. They made an
elaborate device to bring a chandelier crashing
down when Pauli arrived at a reception.
But when Pauli appeared, naturally the Pauli
effect went into effect and a pulley jammed. The
chandelier failed to come down.
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7.3 Symmetric and Antisymmetric Wave Functions
We are about to study many-particle systems
(many-electron atoms and many-atom systems). It
is important to understand the different kinds of
wave functions such systems can have.
In this section, the abstract mathematics of
quantum mechanics leads us to some interesting
results, including the Pauli exclusion principle.
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For a system of n noninteracting identical
particles, the total wave function of the system
can be written as a product of individual
particle wave functions
Electrons, because they satisfy the Pauli
exclusion principle, dont like each other and
are actually rather good at being
noninteracting. In a few minutes, we will see
that there is a different take on this idea
If the particles are identical, it shouldn't make
a difference to our measurements if we exchange
any two (or more) of them. (Should it?)
Looks the same as before to me!
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For a two particle system, we express this
interchangeability mathematically as
Keep in mind that the magnitude of the wave
function squared is related to what we measure.
The equation just above implies
symmetric
antisymmetric
If the wave function does not change sign upon
exchange of particles, it is said to be
symmetric.
If the wave function does not change sign upon
exchange of particles, it is said to be
symmetric. If it does change sign, it is said to
be antisymmetric.
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Remember, we can't directly measure the wave
function, so we don't know what its sign is,
although, as you will see in a minute, we can
tell if the wave function changes sign upon
exchange of particles.
This discussion can be extended to any number of
particles. If the total wave function of a
many-particle system doesn't change sign upon
exchange of particles, it is symmetric. If it
does change sign, it is antisymmetric.
Now let's take these ideas another step further,
and consider two identical particles (1 and 2)
which may exist in two different states (a and
b).
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If particle 1 is in state a and particle 2 is in
state b then
1 in state a
2 in state b
is the wave function of the system.
If particle 2 is in state a and particle 1 is in
state b then
2 in state a
1 in state b
is the wave function of the system.
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But we cant tell particles 1 and 2 apart
(remember, they are identical).
So we cant tell ?I and ?II apart. One is just
as good as the other. Both ?I and ?II are
equally likely to describe our system.
me!
no, me!
No, youre both equally good.
So if theyre equally likely, it could be
either. How do I know which to pick?
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In quantum mechanics, instead of throwing up our
hands in despair at this uncertainty, we invoke
Schrödingers cat.
We say that the system spends half of its time in
state I and half in state II.
Our system's wave function should therefore be
constructed of equal parts of ?I and ?II.
Our system's wave function should therefore be
constructed of equal parts of ?I and ?II. (Live
cat.)
Our system's wave function should therefore be
constructed of equal parts of ?I and ?II. (Live
cat, dead cat.)
Perhaps this approach is nonsense on a
macroscopic level however, it is correct on the
quantum level.
http//www.ruthannzaroff.com/wonderland/Cheshire-C
at.htm
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There are two ways to construct our system's
total wave function ? out of equal parts of ?I
and ?II.
Symmetric
particle 2 in state a particle 1 in state b
particle 1 in state a particle 2 in state b
Antisymmetric
Exchanging particles 1 and 2 changes the sign of
?A but not the sign of ?S.
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Lets put both particles (1 and 2) in the same
state, say a.
Huh?
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If individual particle wave functions are
antisymmetric, then if we try to put both
particles in the same state, we get P0.
There is zero probability of finding the system
in such a state.
The system cannot exist in such a state.
Does this remind you of anything youve seen
recently?
In fact, electrons obey the Pauli exclusion
principle because their wave functions in a
system are antisymmetric.
How do we know electron wave functions are
antisymmetric? Because electrons obey the Pauli
exclusion principle!
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Chicken and egg againwhich comes first, the wave
function bit, or Paulis exclusion principle?
Paulis discovered the exclusion principle in
1925.
Heisenberg formulated matrix mechanics in 1925
and Schrödinger discovered his equation in 1926.
However, all of these discoveries are
consequences of the wave nature of matter.
Paulis exclusion principle is a logical
consequence of the wave nature of matter. Giving
it a name like the Pauli exclusion principle
makes it sound like it is something outside the
framework of quantum mechanics, but it is not.
If it werent for Pauli, wed all implode. See
here http//antwrp.gsfc.nasa.gov/apod/ap030219.ht
ml
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Simple-minded experimentalist that I am, I find
this really fascinating. Abstract quantum
mechanics has led to something concretely
demanded by experiment.
Half integral spin particles (s1/2, 3/2, etc.)
have antisymmetric wave functions and are called
fermions.
Electrons in a system are described by
antisymmetric wave functions which change sign
upon exchange of pairs of them.
Other examples are neutrons (neutrons??--you
should ask how they can have a spin if they have
no charge) and protons. They are also fermions.
Only one fermion in a system can have a given set
of quantum numbers!
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What about particles having symmetric wave
functions?
Integral spin particles (s0,1,2, etc) have
symmetric wave functions, and are called bosons.
Photons in a cavity are described by symmetric
wave functions which do not change sign upon
exchange of pairs of them.
Other examples are alpha particles and nuclei
with integral spins.
There is no restriction on how many bosons in the
same system can have the same set of quantum
numbers.