Transcript Many-Electron Atoms

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.
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 can’t be explained
by the normal Zeeman effect.
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/zeeman-split.html
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
Let’s 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.
   
-
   
   
   
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 ℓ=0 and so it (the outer electron) should
not interact with an external magnetic field.
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, let’s consider the Stern-Gerlach
experiment, in which silver atoms were “shot” through a
magnetic field.
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.)
Let’s see what really happens…
Experimental result.
Dang! Another classical prediction down the tubes.
What happened?
See http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html#c6 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.
“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, let’s 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 momentum—this is in fact the case.
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.
The electron’s 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, s=½.
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
S = s  s +1
capital S
3
=
.
4
lowercase s
I’ll make the difference obvious on an exam!
All electrons have the same spin angular momentum S
(magnitude!).
S = (3/4)½ ħ is the magnitude of the electron spin angular
momentum.
Just as the space quantization of L is specified by mℓ, the
space quantization of S is described by ms.
1
ms = ± .
2
S z = ms = ±
1
2
.
Aha! There are only two possible values of the z-component of the spin
angular momentum. Now we understand the Stern-Gerlach experiment!
There are exactly two possible orientations (see fig. 7.2) of the
electron’s spin angular momentum vector...
up
down
I find this exceedingly strange!
also, up is not quite up!
down is not quite down!
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.
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).
"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 you’ll 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.
“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
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.”
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.
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:
(1,2,3,...n) = (1) (2) (3) ... (n) .
Electrons, because they satisfy the Pauli exclusion principle,
don’t “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!
For a two particle system, we express this interchangeability
mathematically as
(1,2)
2
=
(2,1)
2
.
Keep in mind that the magnitude of the wave function squared
is related to what we measure.
The equation just above implies
(2,1) = (1,2)
symmetric
or (2,1) = - (1,2) .
antisymmetric
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.
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).
If particle 1 is in state a and particle 2 is in state b then
I = a (1) b (2)
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
II = a (2) b (1)
2 in state a
is the wave function of the system.
1 in state b
But we can’t tell particles 1 and 2 apart (remember, they are
identical).
So we can’t tell I and II apart. One is just as “good” as the
other. Both I and II are equally likely to describe our
system.
I = a (1) b (2)
me!
II = a (2) b (1)
no, me!
No, you’re both equally “good.”
“So if they’re equally likely, it could be either. How do I know
which to pick?”
In quantum mechanics, instead of throwing
up our hands in despair at this uncertainty,
we invoke Schrödinger’s cat.
We say that the system spends half of its time in state I and
half in state II.
I = a (1) b (2)
II = a (2) b (1)
Our system's wave function should therefore be
constructed of equal parts of I and II. (Live
cat.)dead cat.)
cat,
Perhaps this approach is nonsense on a macroscopic level;
however, it is correct on the quantum level.
http://www.ruthannzaroff.com/wonderland/Cheshire-Cat.htm
There are two ways to construct our system's total wave
function  out of equal parts of I and II.
Symmetric:
1
S =
 a (1) b (2) +  a (2) b (1) 
2
particle 1 in state a
particle 2 in state b
Antisymmetric:
particle 2 in state a
particle 1 in state b
1
A =
 a (1) b (2) -  a (2) b (1) 
2
Exchanging particles 1 and 2 changes the sign of A but not
the sign of S.
S =
1
 a (1) b (2) +  a (2) b (1) 
2
A =
1
 a (1) b (2) -  a (2) b (1) 
2
Let’s put both particles (1 and 2) in the same state, say a.
S =
1
2

(1)

(2)
+

(2)

(1)
=
 a (1)  a (2)
 a

a
a
a
2
2
A =
1
 a (1)  a (2) -  a (2)  a (1)  = 0
2
PS = 2  a (1)*  a (2)*  a (1)  a (2)  0
PA = 0
Huh?
PS  0
PA = 0
If individual particle wave functions are antisymmetric, then if
we try to put both particles in the same state, we get P=0.
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 you’ve 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!
Chicken and egg again…which comes first, the wave function
bit, or Pauli’s exclusion principle?
Pauli’s “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.
Pauli’s 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 weren’t for Pauli, we’d all implode. See here:
http://antwrp.gsfc.nasa.gov/apod/ap030219.html
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 (s=1/2, 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!
What about particles having symmetric wave functions?
Integral spin particles (s=0,1,2, 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.