Many-Electron Atoms

electron spin

Pauli exclusion principle

symmetric and antisymmetricwave 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 7Many-Electron Atoms

7.1 Electron Spin

We spend only 2 days on chapter 7!

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).

*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.

Electron spin is also of critical importance in magnetism.

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.

Spectral lines (absorption or emission) are caused by photons absorbed or emitted when electrons change their energy state.

http://csep10.phys.utk.edu/astr162/lect/light/zeeman-split.html

The “ordinary” Zeeman effect.

1s2p so selection rules are not violated!

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…

Dang! Another classical prediction down the tubes. What happened?

Experimental result.

Evidently the silver 5s electron has some binary “property.”

See http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html#c6 for a more detailed discussion.

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…

“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

“OK, so he says electrons have spin… aha, like this:”

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 anintrinsic (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

3= s s+1 = .

4S

capital S 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.

s

1m =± .

2

z s

1 = m = ± .

2S

Aha! There are only two possible values of the z-component of the spin angular momentum. Now we understand the Stern-Gerlach experiment!

Just as the space quantization of L is specified by mℓ, the space quantization of S is described by ms.

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.)

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).

In 1925, only three quantum numbers were known (n, ℓ, mℓ).

Pauli realized there needed to be a fourth.

"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 FunctionsWe 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

2 2 (1,2) = (2,1) .

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) or (2,1) = - (1,2) .

If the wave function does not change sign upon exchange of particles, it is said to be symmetric.

symmetric 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 b = (1) (2)

is the wave function of the system.

1 in state a 2 in state b

If particle 2 is in state a and particle 1 is in state b then

II a b = (2) (1)

is the wave function of the system.

2 in state a 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.

me!

no, me!

No, you’re both equally “good.”

I a b = (1) (2)

II a b = (2) (1)

I a b = (1) (2)

II a b = (2) (1)

“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.

Our system's wave function should therefore be constructed of equal parts of I and II.

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

I a b = (1) (2) II a b = (2) (1)

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.)

Symmetric:

Antisymmetric:

S a b a b

1

2 = (1) (2) + (2) (1)

A a b a b

1

2 = (1) (2) - (2) (1)

particle 1 in state aparticle 2 in state b

particle 2 in state aparticle 1 in state b

Exchanging particles 1 and 2 changes the sign of A but not the sign of S.

There are two ways to construct our system's total wave function out of equal parts of I and II.

S a b a b

1

2 = (1) (2) + (2) (1)

A a b a b

1

2 = (1) (2) - (2) (1)

Let’s put both particles (1 and 2) in the same state, say a.

S a a a a a a

1 2

2 2 = (1) (2) + (2) (1) = (1) (2)

A a a a a

1

2 = (1) (2) - (2) (1) =0

* *S a a a a2 P = (1) (2) (1) (2) 0

AP = 0 Huh?

SP 0 AP = 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?

If it weren’t for Pauli, we’d all implode. See here: http://antwrp.gsfc.nasa.gov/apod/ap030219.html

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.

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.

Simple-minded experimentalist that I am, I find this really fascinating. Abstract quantum mechanics has led to something concretely demanded by experiment.

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!

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.

What about particles having symmetric wave functions?