Download Lecture Notes V: Spin, Pauli Exclusion Principle, Symmetric

From One to Many Electrons…
Electron Spin
Pauli Exclusion Principle
Symmetric and Antisymmetric
Wave Functions
Slides adapted from: http://web.mst.edu/~pringle/Phys107
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.
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: ???
Ordinary Zeeman effect: classical
magnetic dipole in B field
Let’s think about electrons and magnetism:
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 P222 that the electron
will experience a deflection.
   
-
   
   
   
Orbital angular momentum of electron
gives rise to magnetic dipole moment:
!" !"
µ = IA
I = ev / 2π r = eL / mr
2
Torque, energy of magnetic
dipole in an external
magnetic field:
Normal Zeeman effect: States with same n
different l (magnetic dipole moment) will have
different energies
!
"! "!
τ = IA×B
or
! "! "!
τ =µ×B
"! "!
U = −µ • B
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.
If the silver atom did behave “like” a dipole, it 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. Also what
should happen for µL = 0.
Classical expectation if there is some
other source of magnetic dipole: 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!
Failure of another classical prediction, leads to discovery
of new quantum mechanical property: Spin
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:
But how does the electron obtain a magnetic moment if it has
zero angular momentum and therefore is not a "current loop”?
This binary (one or the other, but only two choices) property is
the electron spin.
Spin, as in a spinning top?
“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.”
Not exactly….
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 due to its motion about nucleus—
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 our heads, 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.”
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 ( s +1) h =
h.
4
capital S
lowercase s
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
1
Sz = ms h = ± h .
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 of the electron’s
spin angular momentum vector...
up
down
There is nothing special about the z-axis: projection of electron
spin along any axis takes on two possible values.
Exclusion Principle
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.
Pauli won the 1945 Nobel Prize for discovering the exclusion principle
(nominated for the prize by Einstein).
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!
"State" refers to the four quantum numbers n, ℓ, mℓ, ms.
Obviously, all electrons have the same s.
An even more general statement reads:
“No two fermions in the same potential can exist in the same
quantum state.”
A fermion obeys the Pauli exclusion principle; a boson does
not. Examples of fermions: electron, proton, neutron.
Examples of bosons: photon, pion. It is found experimentally
that all bosons have integer spin. All fermions have half-oddinteger spin.
Pauli is perhaps most famous among physicists for the “Pauli Effect*”: catastrophic
failure of experiments when he walked into a lab!!
*Sources: W. Cropper, Great Physicists, Oxford, 2001, p.256-7; G. Gamow, Thirty
Years That Shook Physics, Heinemann, 1966, p.64.
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:
ψ(1,2,3,...n) = ψ(1) ψ(2) ψ(3) ... ψ(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
ψ(1,2)
2
=
ψ(2,1)
2
.
Recall that the magnitude of the wave function squared
(probability density) is related to what we measure.
The equation above implies
ψ(2,1) = ψ(1,2)
symmetric
or ψ(2,1) = - ψ(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
ψI = ψa (1) ψb (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
ψII = ψa (2) ψb (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 ψI and ψII apart. One is just as “good” as the
other. Both ψI and ψII are equally likely to describe our
system.
ψI = ψa (1) ψb (2)
ψII = ψa (2) ψb (1)
How do we know which to pick???
Probabilitistic interpretation of the wave
function to the rescue:
We say that the system spends half of its time in state I and
half in state II.
ψI = ψa (1) ψb (2)
ψII = ψa (2) ψb (1)
Our system's wave function should therefore be
constructed of equal parts of ψI and ψII.
While this approach may seem like nonsense on a macroscopic
level; however, it is correct on the quantum level.
There are two ways to construct our system's total wave
function ψ out of equal parts of ψI and ψII.
Symmetric:
1
ψS =
⎡⎣ψa (1) ψb (2) + ψa (2) ψb (1) ⎤⎦
2
particle 1 in state a
particle 2 in state b
Antisymmetric:
particle 2 in state a
particle 1 in state b
1
ψA =
⎡⎣ψa (1) ψb (2) - ψa (2) ψb (1) ⎤⎦
2
Exchanging particles 1 and 2 changes the sign of ψA but not
the sign of ψS.
1
ψS =
⎡⎣ψa (1) ψb (2) + ψa (2) ψb (1) ⎤⎦
2
1
ψA =
⎡⎣ψa (1) ψb (2) - ψa (2) ψb (1) ⎤⎦
2
Let’s put both particles (1 and 2) in the same state, say a.
1
2
ψS =
ψa (1) ψa (2)
⎡⎣ψa (1) ψa (2) + ψa (2) ψa (1) ⎤⎦ =
2
2
1
ψA =
⎡⎣ψa (1) ψa (2) - ψa (2) ψa (1) ⎤⎦ = 0
2
PS = 2 ψ a (1)* ψ a (2)* ψ a (1) ψ a (2) ≠ 0
PA = 0
What?
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 we’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 …which comes first: wave function
antisymmetry or Pauli’s exclusion principle?
Pauli “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.
See the following link for why the Pauli principle is responsible
for the fact that matter doesn’t just bunch up and “implode”!
http://antwrp.gsfc.nasa.gov/apod/ap030219.html
Implications for multi-atomic systems:
I) Molecules
Consider Hydrogen Molecule with just one electron, in ground
state
- A single electron sees two potential wells from two protons
- Recall: for single well, if ψ is a solution to SE, then so is –ψ.
Two unique ways of combining single-well wave functions
to get solution to SE for double well:
Notes:
1)  Talking about 1 electron here, so sym/
anti-sym is NOT w.r.t. exchange of
electrons (i.e., not related to Pauli
Exclusion Principle).
2)  Difference in energies of ψs and ψA is
small
3)  Protons move closer together, more likely
to find electron between protons in ψs
à ψs is a “bonding orbital”; electron pulls
protons closer together.
Band Theory of Solids
The essential feature of the band theory is that the allowed
energy states for electrons are nearly continuous over
certain ranges, called energy bands, with forbidden energy
gaps between the bands.
•  Consider initially the known wave functions of
two hydrogen atoms far enough apart so that
they do not interact.
31
Band Theory of Solids
•  Interaction of the wave functions occurs as the atoms get closer:
Symmetric
Antisymmetric
•  An atom in the symmetric state has a nonzero probability of being
halfway between the two atoms, while an electron in the
antisymmetric state has a zero probability of being at that location.
32
Band Theory of Solids
•  In the symmetric case the binding energy is slightly
stronger resulting in a lower energy state.
–  Thus there is a splitting of all possible energy levels (1s,
2s, and so on).
•  When more atoms are added (as in a real solid),
there is a further splitting of energy levels. With a
large number of atoms, the levels are split into
nearly continuous energy bands, with each band
consisting of a number of closely spaced energy
levels.
33
Kronig-Penney Model
•  An effective way to understand the energy gap in
semiconductors is to model the interaction between
the electrons and the lattice of atoms.
•  Kronig and Penney (1931) model: an electron experiences
an infinite one-dimensional array of finite potential wells.
•  Each potential well models attraction to an atom in the
lattice, so the size of the wells must correspond roughly to
the lattice spacing.
34
Implications for multi-atomic systems:
Solids and the band structure of electronic states in crystals
Electronic characteristics of solids:
1) Insulators
2) Semi-Conductors
3) Conductors
Now imagine many single-electron atoms next to each other –
Not just two, but many, possible multi-well single-electron wave
functions corresponding to a particular single-well state (i.e.,
1S), AND all very close in energy, forming a “Band”.
Pauli Principle à only two electrons can have the same spatial
wave function, so band fills up from lowest to highest energy.
For a regular array (crystal) of multi-electron atoms forming a
solid, outer shell à band that may or not be filled
Implications for multi-atomic systems:
Solids and the band structure of electronic states in crystals
Example: imagine an array of Sodium atoms (forming a metal):
(Sodium has 11 electrons: 1s2 2s2 2p6 3s1 )
Valence and Conduction Bands
•  The band structures of insulators and semiconductors
resemble each other qualitatively. Normally there exists in
both insulators and semiconductors a filled energy band
(referred to as the valence band) separated from the
next higher band (referred to as the conduction band) by
an energy gap.
•  If this gap is at least several electron volts, the material is
an insulator. It is too difficult for an applied field to
overcome that large an energy gap, and thermal
excitations lack the energy to promote sufficient numbers
of electrons to the conduction band.
37
Smaller energy gaps create semiconductors
•  For energy gaps smaller than about 1 electron volt,
it is possible for enough electrons to be excited
thermally into the conduction band, so that an
applied electric field can produce a modest current.
The result is a semiconductor.
38
Implications for multi-atomic systems:
Solids and the band structure of electronic states in crystals
Electronic characteristics of solids:
1) Insulators
2) Semi-Conductors
3) Conductors