Last week of Course

1
Last week of Course

• Today -- Atoms, Molecules, Solids
• States with many electrons filled according to
  the Pauli exclusion principle

• Next time Consequences of quantum mechnanics
• Metals, insulators, semiconductors,
  superconductors, lasers, . .

Final Exam Monday Mar. 9
Review session Sunday Mar. 8
Extra office hours (Wed.,Sun.,Mon)
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Building Atoms, Molecules and solids
3
Overview

• Nuclear Spin and MRI
• Atomic Configurations
• States in atoms with many electrons filled
  according to the Pauli exclusion principle

• Molecular Wavefunctions origins of covalent
  bonds
• Example H H ? H2

• Electron energy bands in Solids
• States in atoms with many electrons filled
  according to the Pauli exclusion principle

4
Nuclear Spin and MRI Example

• Magnetic resonance imaging (MRI) depends on the
  absorption of electromagnetic radiation by the
  nuclear spin of the hydrogen atoms in our bodies.
  The nucleus is a proton with spin ½, so in a
  magnetic field B there are only two possible spin
  directions with definite energy. The energy
  difference between these states is DE2mpB, with
  mp 1.41 x 10-26 J /Tesla.

Question 1 The person to be scanned by an MRI
machine is placed in a strong magnetic field,
with B1 T being a typical value. What is the
energy difference between spin-up and spin-down
proton states in this field?
Question 2 What is the frequency f of photons
that can be absorbed by this energy difference?
5
Nuclear Spin and MRI Example

• Magnetic resonance imaging (MRI) depends on the
  absorption of electromagnetic radiation by the
  nuclear spin of the hydrogen atoms in our bodies.
  The nucleus is a proton with spin ½, so in a
  magnetic field B there are only two possible spin
  directions with definite energy. The energy
  difference between these states is DE2mpB, with
  mp 1.41 x 10-26 J /Tesla.

Question 1 The person to be scanned by an MRI
machine is placed in a strong magnetic field,
with B1 T being a typical value. What is the
energy difference between spin-up and spin-down
proton states in this field?
Solution
DE 2mpB 2 x (1.41 x 10-26 J/T) x 1 T
2.82 x 10-26 J
2.82 x 10-26 J x 1 eV/ 1.6 x 10-19 J 1.76
x 10-7 eV
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Nuclear Spin and MRI Example

• Magnetic resonance imaging (MRI) depends on the
  absorption of electromagnetic radiation by the
  nuclear spin of the hydrogen atoms in our bodies.
  The nucleus is a proton with spin ½, so in a
  magnetic field B there are only two possible spin
  directions with definite energy. The energy
  difference between these states is DE2mpB, with
  mp 1.41 x 10-26 J /Tesla.

Question 2 What is the frequency f of photons
that can be absorbed by this energy difference?
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Act 1
We just saw that radio-wave photons with energy
1.7 x 10-7 eV can cause a nuclear spin to flip.
What therefore must be the angular momentum of
each photon?
(a). 0 (b). h/2 (c). h
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Act 1 - Solution
We just saw that radio-wave photons with energy
1.7 x 10-7 eV can cause a nuclear spin to flip.
What therefore must be the angular momentum of
each photon?
(a). 0 (b). h/2 (c). h
Initial angular mom. Final ang.
momentum Szelectron(?) Szphoton
Szelectron(?) Szphoton Szelectron(?) -
Szelectron(?) h/2 (- h/2) h
Conclusion Photons carry angular momentum. In
particular, they can carry h or -h around
their propagation axis, corresponding to right
and left circular polarization. (photon oddity
no m0 state around that axis) Linear
polarization is a superposition of equal parts
right and left. Particles with an intrinsic spin
of n are called bosons.
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FYI UIUC Prof. Slichter wins National Medal of
Science (2007)
Professor Slichter was recognized "for
establishing nuclear magnetic resonance as a
powerful tool to reveal the fundamental molecular
properties of liquids and solids. His inspired
teaching has led genera-tions of physicists and
chemists to develop a host of modern
techno-logies in condensed matter physics,
chemistry, biology, and medicine."
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FYI Recent Breakthrough Detection of a single
electron spin!

• (Nature July 14, 2004) -- IBM scientists achieved
  a breakthrough in nanoscale magnetic resonance
  imaging (MRI) by directly detecting the faint
  magnetic signal from a single electron buried
  inside a solid sample.

Raffi Budakian (started at UIUC Fall 05)
Next step detection of single nuclear spin
(660x smaller).
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Pauli Exclusion Principle
Lets start building more complicated atoms
to study the Periodic Table. For atoms with many
electrons (e.g., carbon 6, iron 26, etc.) -
what energies do they have?
From spectra of complex atoms, Wolfgang Pauli
(1925) deduced a new rule
Pauli Exclusion Principle
No two electrons can be in the same quantum
state, i.e. in a given atom they cannot have the
same set of quantum numbers n, l, ml, ms --
i.e., every atomic orbital with n,l,ml can hold 2
electrons (??)

• Therefore, electrons do not pile up in the lowest
  energy state, i.e, the (1,0,0) orbital.
• They are distributed among the higher energy
  levels according to the Pauli Principle.
• Particles that obey the Pauli Principle are
  called fermions

Note More generally, no two identical fermions
(any particle with spin of h/2, 3h/2, etc.) can
be in the same quantum state.
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Filling the atomic orbitals according to the
Pauli Principle
is valid only for one electron in the Coulomb
potential of Z protons. The energy levels shift
as more electrons are added, due to
electron-electron interactions. Nevertheless,
this hydrogenic diagram helps us keep track of
where the added electrons go.
l label orbitals (2l1) 0 s
1 1 p 3 2
d 5 3 f
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Z atomic number number of protons
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Act 2 Pauli Exclusion Principle
1. Which of the following states (n,l,ml,ms)
is/are NOT allowed?
(a). (2, 1, 1, -1/2) (b). (4, 0, 0, 1/2) (c).
(3, 2, 3, -1/2) (d). (5, 2, 2, 1/2) (e). (4, 4,
2, -1/2)
2. Which of the following atomic electron
configurations violates the Pauli Exclusion
Principle?
(a). 1s2, 2s2, 2p6, 3d10 (b). 1s2, 2s2, 2p6,
3d4 (c). 1s2, 2s2, 2p8, 3d8 (d). 1s1, 2s2, 2p6,
3d5 (e). 1s2, 2s2, 2p3, 3d11
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Act 2 Pauli Exclusion Principle - Solution
1. Which of the following states (n,l,ml,ms)
is/are NOT allowed?
2. Which of the following atomic electron
configurations violates the Pauli Exclusion
Principle?
(a). 1s2, 2s2, 2p6, 3d10 (b). 1s2, 2s2, 2p6,
3d4 (c). 1s2, 2s2, 2p8, 3d8 (d). 1s1, 2s2, 2p6,
3d5 (e). 1s2, 2s2, 2p3, 3d11
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Act 2 Pauli Exclusion Principle - Solution
1. Which of the following states (n,l,ml,ms)
is/are NOT allowed?
2. Which of the following atomic electron
configurations violates the Pauli Exclusion
Principle?
(a). 1s2, 2s2, 2p6, 3d10 (b). 1s2, 2s2, 2p6,
3d4 (c). 1s2, 2s2, 2p8, 3d8 (d). 1s1, 2s2, 2p6,
3d5 (e). 1s2, 2s2, 2p3, 3d11
2(2l 1) 6 allowed electrons
2(2l 1) 10 allowed electrons
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Filling Procedure for Atomic Orbitals--example
Bromine
Due to electron-electron interactions, the
hydrogen levels fail to give us the correct
filling order as we go higher in the periodic
table. The actual filling order is given in the
table below. Electrons are added by proceeding
along the arrows shown. Bromine is an element
with Z 35. Find its electronic configuration
(e.g., 1s2 2s2 2p6 ).
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As you learned in chemistry, the various
behaviors of all the elements (and all the
molecules made up from them) is all due to the
way the electrons organize themselves, according
to quantum mechanics.
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Act 3 Pauli Exclusion Principle Part 2
The Pauli exclusion principle applies to all
fermions in all situations (not just to electrons
in atoms). Consider electrons in a 2-dimensional
infinite square well potential. 1. How many
electrons can be in the first excited states,
i.e., next lowest after the ground states?
(a). 1 (b). 2 (c). 3 (d). 4 (e). 5
2. If there are 4 electrons in the well, what is
the energy of the most energetic one (ignoring
e-e interactions, and assuming the total energy
is as low as possible)?
(a). (h2/8mL2) x 2 (b). (h2/8mL2) x 5 (c).
(h2/8mL2) x 10
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Act 3 Pauli Exclusion Principle Part 2 -
Solution
The Pauli exclusion principle applies to all
fermions in all situations (not just to electrons
in atoms). Consider electrons in a 2-dimensional
infinite square well potential. 1. How many
electrons can be in the first excited states?
(a). 1 (b). 2 (c). 3 (d). 4 (e). 5
The ground state has quantum numbers (1, 1), and
can hold 2 electrons, i.e., (1,1,1/2)
(1,1,-1/2) The first excited state has two
degenerate single-electron states (2,1) and
(1,2), each of which can have 2 electrons
(2,1,1/2), (2,1,-1/2), (1,2,1/2), (1,2,-1/2)
2. If there are 4 electrons in the well, what is
the energy of the most energetic one (ignoring
e-e interactions, and assuming the total energy
is as low as possible)?
(a). (h2/8mL2) x 2 (b). (h2/8mL2) x 5 (c).
(h2/8mL2) x 10
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Act 3 Pauli Exclusion Principle Part 2 -
Solution
The Pauli exclusion principle applies to all
fermions in all situations (not just to electrons
in atoms). Consider electrons in a 2-dimensional
infinite square well potential. 1. How many
electrons can be in the first excited state?
(a). 1 (b). 2 (c). 3 (d). 4 (e). 5
The ground state has quantum numbers (1, 1), and
can hold 2 electrons, i.e., (1,1,1/2)
(1,1,-1/2) The first excited state has two
degenerate single-electron states (2,1) and
(1,2), each of which can have 2 electrons
(2,1,1/2), (2,1,-1/2), (1,2,1/2), (1,2,-1/2)
2. If there are 4 electrons in the well, what is
the energy of the most energetic one (ignoring
e-e interactions, and assuming the total energy
is as low as possible)?
The ground state has 2 electrons. The next two
are in the first excited state, the energy of
which is (h2/8mL2) x (12 22) 5(h2/8mL2)
(a). (h2/8mL2) x 2 (b). (h2/8mL2) x 5 (c).
(h2/8mL2) x 10
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Bonding between atoms-- How can two neutral
objects bind together? H H ? H2
Continuum of free electron states.
Lets represent the atom in space by its Coulomb
potential centered on the proton (e)
In this picture the potential energy of the two
protons in an H2 molecule look something like
this
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Particle in a box-- Finite square well
potential(analogue to Molecular binding)
If this were the atomic potential,
Bound states
then this would be the molecular potential
Again, we dont know exactly what the energy
levels are, although in 1-D we could solve the
equation exactly if we had to. For now we settle
for a qualitative understanding...
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Just consider the ground state
Lets say that the lowest energy level is about
equal to that of an infinite well
L 1 nm
For convenience, we are going to plot the
electronic wavefunction with the energy level as
a baseline
yA
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Molecular Wavefunctions and Energies
Atomic Wavefunctions
yA
Molecular Wavefunctions 2 atomic states
? 2 molecular states
When the wells are far apart, atomic functions
dont overlap. The single electron can be in
either well with equal probability, and E 0.4
eV.
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Molecular Wavefunctions and Energies
Wells far apart
(Degenerate states)
Wells closer together
Atomic states are beginning to overlap and
distort. y2even and y2odd are not the same (note
center point). The degeneracy is broken Eeven
lt Eodd (why?)
Yeven no zero-crossings, Yodd one zero-crossing
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Act 4 Symmetric vs. Antisymmetric states
What will happen to the energy of yeven as the
two wells come together (i.e., as d is reduced)?
Hint think of the limit as d ? 0
(a). Eeven increases (b). Eeven decreases (c).
Eeven stays the same
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Act 4 Symmetric vs. Antisymmetric states
What will happen to the energy of yeven as the
two wells come together (i.e., as d is reduced)?
Hint think of the limit as d ? 0
(a). Eeven increases (b). Eeven decreases (c).
Eeven stays the same
As d becomes very small the curvature in yeven is
reduced, reducing the energy. What does this
mean for the two atoms?
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Energies as a function of distance between wells
When the wells just touch (becoming one well) we
can solve for the energies easily
2L 2 nm
yeven
(n 1 state)
yodd
(n 2 state)
As the wells are brought the even state always
has lower kinetic energy (smaller curvature) .
The odd state stays at about the same energy
(increase due to larger curvature and decrease
due to thinner barrier).

• f 0.4 0.1 eV 0.3 eV
• splitting between even and odd states.
• (mainly ground state lowering)

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Exclusion Principle and bonding
Up to now we have considered only the energy of
the states when theres a single electron.

• What happens if there is more than one electron?

2 electrons Lowest energy for both in lowest
state spin up and down
3 electrons One electron must go in higher
energy (odd) state
4 electrons Both even and odd states are
filled
gt 4 electrons Must start filling the higher
states of the wells
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Molecular Wavefunctions and Energies -- back to
Coulomb Potential
Molecular states
Remember These are single-electron states.
To understand the bonding, we must see how
electrons fill these states following the Pauli
Exclusion Principle
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Energies as a function of distance between atoms

• The even and odd states behave similarly to the
  square well considered before -- but in addition
  there is repulsion between the nuclei that
  prevents them from coming too close.
• Schematic picture for the total energy of the
  nuclei and electrons

• H2 molecules -- since there are 2 electrons,
  both go in the lowest energy state to form a
  covalent bond -- a strongly bound molecule!

• In contrast, consider two He atoms. He is a
  rare gas atom with 2 electrons that fill the
  lowest atomic state. When two He atoms are
  brought together, there are 4 electrons that fill
  bonding and anti-bonding states no bonding
  which is correct for rare gas atoms !

• The rest is chemistry --- very important and
  far too many possibilities for us to consider
  here!

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Electron states in a crystal (1)
Again start with simple atomic state
What do these crystal states look like? --
approximately linear combinations of atomic
orbitals.
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Electron states in a crystal (2)
Kruse Energy bands
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The in between" states
Key points
1. The wavevector k has N possible values from k
p/L to k ? p/a.
k ?p/a is the maximum range since cos and sin
repeat and give the same function for k outside
this range
2. For a crystal N is very large (the number of
atoms!) and the states approach a continue of
energies between the lowest and highest energies
? a band of energies.
3. Because there are N L/a states one state
per atom a band has exactly enough states to
hold 2 electron per atom (spin up and down).
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Electron Wavefunctions and Energy Band
Highest energy wavefunction
Energy
Closely spaced energy levels form a band a
continuum of energies between the max and min
energies
Lowest energy wavefunction
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Conclusions

• Atomic Configurations
• States in atoms with many electrons filled
  according to the Pauli exclusion principle

• Molecular Wavefunctions origins of covalent
  bonds
• Example H H ? H2

• Electron energy bands in Solids
• Continuous range of energies for allowed
  states of an electron in a crystal
• A Band Gap is a range of energies where there
  are no allowed states
• Bands are filled according to the Pauli
  exclusion principle

• Bands and band gaps are properties of waves in
  periodic systems
• Light waves propagating through many layers
• Electron waves in crystals

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Supplement Example Problem 1
What is the electronic structure of lithium (3
electrons)? That is, what quantum numbers do the
electrons have?
Solution
The guiding principle is to find the lowest
energy. This involves (for atoms without too
many electrons) putting the electrons into the
smallest possible n state, because energy depends
only on n (to a good approximation).
As you saw in Act 1, the first two electrons have
n 1. This forces them to have ? 0 and m?
0. All electrons have s 1/2, so it is not
listed. ms is always 1/2 or -1/2. The first
two electrons can have n 1, but the third must
have n 2. ? 0 has lower energy than ? 1,
because of effects of those n1 electrons. ms
doesnt affect the energy, (symmetry) so either
value is OK.
Electron 1 (n, ?, m?, ms) (1,0,0,1/2) Electro
n 2 (n, ?, m?, ms) (1,0,0,-1/2) Electron
3 (n, ?, m?, ms) (2,0,0, 1/2)
Whenever an atom has a single electron in a
higher energy state (n value) than the others,
that electron is not tightly bound, and the atom
can easily lose it. This kind of atom is
chemically very reactive. All of the alkali
metals (group IA) have this electronic
configuration.
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Supplement Example Problem 2
A hydrogen atom is in the n 3, ? 2, m? -2
state. To what states can the electron fall when
it emits a photon? Which are the strongest (most
likely) transitions?
Solution
Final state quantum numbers n 1, 2 ? 0, 1 m?
any Final states with strongest transition ?
1 n 2 m? -1

• Remember the selection rules
• Dn is negative here (conservation of energy)
• D? ? 0 (conservation of angular momentum)
• ml any value
• D? 1 for strongest transition (dipole allowed).
• 1 requires n ? 2.
• Dm? -1, 0, or 1.
• Dm? 1, only possibility consistent with initial
  m?.

In a different atom, it may be possible to have
a Dn 0 transition, and still conserve energy
(e.g., go from 3s ? 3p).