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Physics of Condensed Matter I
1100-4INZ`PC
Faculty of Physics UW
[email protected]
Summary of the lecture
1. Quantum mechanics
2. Optical transitions
3. Lasers
4. Optics
5. Molecules
6. Properties of molecules
7. Crystals
8. Crystalography
9. Solid state
10. Electronic band structure
11. Effective mass approximation
12. Electrons and holes
13. Carriers
14. Transport
15. Optical properties of solids
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Classical and quantum universe
Hydrogen atom:
Eigenstates of L:
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Hydrogen atom:
Eigenstates of L:
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Real functions:
4
Hydrogen atom:
Eigenstates of L:
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Spherical harmonics Ylm:
5
Eigenstate:
Stan 1s
E.g. hydrogen wavefunction
Quantum numbers!
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Electric field
Stark effect of hydrogen atom
electric field E

H '  pE  ez E z
dipole moment p
Eigenfunctions of hydrogen atom for 2p state:
𝜓200 , 𝜓21−1 , 𝜓210 , 𝜓211
′
෡𝑖𝑗
෡ 𝜓𝑗
Perturbation 𝐻
= 𝜓𝑖 𝐻′
http://pl.wikibooks.org/wiki/Mechanika_kwantowa/Rachunek_zaburzeń_dla_równania_Schrödingera_niezależnego_od_czasu
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Electric field
electric field E

dipole moment p
Eigenfunctions of hydrogen atom for 2p state:
𝜓200 , 𝜓21−1 , 𝜓210 , 𝜓211
′
෡𝑖𝑗
෡ 𝜓𝑗
Perturbation 𝐻
= 𝜓𝑖 𝐻′
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Electric field
electric field E

dipole moment p
Atom in ligand field
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Electric field
electric field E

dipole moment p
Atom in ligand field
Crystal Field Theory (CFT)
(magnetic properties as well as color.)
http://www.tutorvista.com/content/chemistry/chemistry-iv/coordination-compounds/crystal-field-splitting.php
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Magnetic field and spin
Magnetic field:
𝐻 ′ = −𝑚𝐵
Here 𝑚 is the magnetic moment
clasically:
𝑚 = 𝐼 𝑆Ԧ =
𝑒 2
𝑒
𝑒
𝜋𝑟 =
𝜋𝑟 2 = 𝑟𝑣 [Am2]
𝑇
2𝜋𝑟/𝑣
2
m
r
v
𝐿 = 𝑟Ԧ × 𝑚0 𝑣Ԧ
𝐿෠ = 𝐿෠ 𝑥 , 𝐿෠ 𝑦 , 𝐿෠ 𝑧
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Magnetic field:
𝐻 ′ = −𝑚𝐵
clasically:
𝑚 = 𝐼 𝑆Ԧ =
𝑒 2
𝑒
𝑒
𝜋𝑟 =
𝜋𝑟 2 = 𝑟𝑣 [Am2]
𝑇
2𝜋𝑟/𝑣
2
m
thus:
𝑒
𝜇𝐵
𝑚=−
𝐿=− 𝐿
2𝑚0
ℏ
Bohr magneton 𝜇𝐵 =
r
v
ℏ𝑒
2𝑚0
𝜇𝐵 = 9,274009994(57)×10−24 J/T
𝜇𝐵
′
𝐻 = −𝑚𝐵 =
𝐿෠ 𝐵
ℏ
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𝜇𝐵 =
ℏ𝑒
2𝑚0
𝐿෠ = 𝐿෠ 𝑥 , 𝐿෠ 𝑦 , 𝐿෠ 𝑧
16
Magnetic field:
𝐻 ′ = −𝑚𝐵 =
for 𝐵 = 0,0, 𝐵𝑧
we have: 𝐻 ′ =
𝜇𝐵
𝐿෠ 𝐵
ℏ
𝜇𝐵
𝐿෠ 𝐵
ℏ 𝑧 𝑧
= 𝜇𝐵 𝐵𝑧 𝑚 where 𝑚 = −𝑙, −𝑙 + 1, … 𝑙 − 1, 𝑙
m
Here 𝑚 is the quantum number |𝑛, 𝑙, 𝑚ۧ
r
𝑚=1
the base: |𝑙, 𝑚ۧ
v
𝑙=1
𝑚=0
𝑚 = −1
𝐵=0
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𝐵≠0
17
Magnetic field:
𝐻 ′ = −𝑚𝐵 =
for 𝐵 = 0,0, 𝐵𝑧
we have: 𝐻 ′ =
𝜇𝐵
𝐿෠ 𝐵
ℏ
𝜇𝐵
𝐿෠ 𝐵
ℏ 𝑧 𝑧
= 𝜇𝐵 𝐵𝑧 𝑚 where 𝑚 = −𝑙, −𝑙 + 1, … 𝑙 − 1, 𝑙
m
Here 𝑚 is the quantum number |𝑛, 𝑙, 𝑚ۧ
r
𝑚=1
the base: |𝑙, 𝑚ۧ
v
𝑙=1
𝑚=0
𝑚 = −1
𝐵=0
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𝐵≠0
18
Stern-Gerlach experiment (1922 r.)
http://hyperphysics.phy-astr.gsu.edu
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What is the „spin”?
• What is „mass”?
Mariusz Pudzianowski http://www.pudzian.pl/
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• What is „momentum”?
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• What is „charge”?
http://www.chaseday.com
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http://www.floridahistory.com/[email protected]
• Spin?
Sebastian Münster, Cosmographia in 1544
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Disney
23
http://geomag.usgs.gov/
The history
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The history
Magnetyt (z 1750 r.) typowy ferryt
i magnes z ziem rzadkich. Każdy z
nich o gęstości energii 1J.
http://www.azom.com/details.asp?ArticleID=637
http://www.tcd.ie/Physics/Schools/what/materials/magnetism/seven.html
A lodestone magnet from the 1750's and typical ferrite and rare earth used in modern
appliances. Each of these produce about 1J of energy.
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How magnets work?
http://lucy.troja.mff.cuni.cz/~tichy/elektross/magn_pole/stac_mp.html
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How magnets work?
http://lucy.troja.mff.cuni.cz/~tichy/elektross/magn_pole/stac_mp.html
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How magnets work?
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How magnets work?
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How magnets work?
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How magnets work?
Moving charges
produce a magnetic
field…
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How magnets work?
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How magnets work?
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How magnets work?
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How magnets work?
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How magnets work?
These smal magnets
are electrons
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How magnets work?
These smal magnets
are electrons
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Skąd się biorą magnesy?
A więc płynie jakiś prąd?
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How magnets work?
The movement of
electrons around the
nucleus?
The spin of
electrons around it’s
own axis?
So, there is a current?
Internal property of
electrons?
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How magnets work?
The spin of
electrons around it’s
own axis?
The movement of
electrons around the
nucleus?
So, there is a current?
electrons?
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!
internal angular momentum
SPIN
40
Albert Einstein - Johannes Wander de Haas,
Berlin 1914,
internal angular momentum
electrons
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!
http://www.ptb.de/en/publikationen/jahresberichte/jb2005/nachrdjahres/s23e.html
41
projections of the spin on axis
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projections of the spin on
ANY of axes has only TWO
values:
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http://hyperphysics.phy-astr.gsu.edu43
Stern-Gerlach experiment (1922 r.)
http://hyperphysics.phy-astr.gsu.edu
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Spin, spin-orbit interaction
Spin operators 𝑆መ𝑥 , 𝑆መ𝑦 , 𝑆መ𝑧 , 𝑆መ 2
𝜓 𝑟,
Ԧ 𝑆𝑧 = 𝜓 𝑟Ԧ 𝜒 𝑆𝑧
Spinor
𝑆መ𝑥 , 𝑆መ𝑦 = 𝑖ℏ𝑆መ𝑧 , etc.
Pauli matrices: 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑧
1
1 0
𝑆መ𝑥 = ℏ𝜎𝑥 = ℏ
2
2 1
1
0
𝑆መ𝑦 =
1
1 0 −𝑖
ℏ𝜎 = ℏ
2 𝑦 2 𝑖 0
𝑆መ𝑥 =
1
1 1
ℏ𝜎𝑧 = ℏ
2
2 0
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0
−1
projections of the spin on the axis 𝑧
1
0
𝜒↑ =
, 𝜒↓ =
0
1
45
𝜇𝐵
𝐻 =
𝐿෠ + 𝑔𝑆መ 𝐵
ℏ
′
𝑔-factor for the agreement with
experiments
1
1 0
2
2 1
1
0
𝑆መ𝑦 =
1
1 0 −𝑖
ℏ𝜎 = ℏ
2 𝑦 2 𝑖 0
𝑆መ𝑥 =
1
1 1
ℏ𝜎𝑧 = ℏ
2
2 0
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0
−1
1
0
𝜒↑ =
, 𝜒↓ =
0
1
46
𝜇𝐵
𝐻 =
ℏ
′
experiments
1
1 0
2
2 1
1
0
𝑆መ𝑦 =
1
1 0 −𝑖
ℏ𝜎 = ℏ
2 𝑦 2 𝑖 0
𝑆መ𝑥 =
1
1 1
ℏ𝜎𝑧 = ℏ
2
2 0
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0
−1
𝑔 = −2.00231930436182 ± 0.00000000000052
1
0
𝜒↑ =
, 𝜒↓ =
0
1
47
QED – Quantum ElectroDynamics
𝑔 = −2.00231930436182 ± 0.00000000000052
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𝜇𝐵
𝐻 =
ℏ
′
experiments
መ the base |𝑗, 𝑚𝑗 ൿ
Total angular momentum operator 𝐽መ = 𝐿෠ + 𝑆,
෡=𝑀
෡𝐿 + 𝑀
෡𝑆 = −𝑔𝐿 𝜇𝐵 𝐿෠ −𝑔𝑆 𝜇𝐵 𝑆መ
Total magnetic moment 𝑀
ℏ
ℏ
=1
=2
෡ ≠ 𝐽መ - magnetic anomaly of spin
𝑀
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෡𝑆𝑂 = 𝜆𝐿෠ 𝑆መ with the base |𝑛, 𝑙, 𝑠, 𝑚𝑙 , 𝑚𝑠 ۧ
Spin-orbit interaction 𝐻
For 𝑠-states 𝐿෠ = 0 ⇒ 𝐿෠ 𝑆መ = 0
෡𝑆𝑂 = 𝜆𝐿෠ 𝑆መ = 𝜆
𝐻
fine-structure constant
𝐸𝑆𝑂 =
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1 2
1
2
2
𝐽 − 𝐿 − 𝑆 = 𝜆 𝐿𝑧 𝑆𝑧 + 𝐿+ 𝑆− + 𝐿− 𝑆+
2
2
𝑍𝛼 2 1
𝜆 = ℎ𝑐 𝐴 =
𝑅𝑦 = ℎ𝑐𝑅∞
2 𝑟3
𝑚𝑒 𝑒 4
2
𝑒
1
𝑅∞ = 8𝜀2 ℎ3 𝑐
0
𝛼=
≈
4𝜋𝜀0 ℏ𝑐 137.037
𝑅∞ = 1,097 × 107 m-1
න 𝜓∗ 𝐻𝑆𝑂 𝜓 𝑑𝑉
𝑍
=
2 137
2
න 𝜓∗
𝐿෠ 𝑆መ
𝜓 𝑑𝑉
𝑟3
50
1 2
1
2
2
𝐽 − 𝐿 − 𝑆 = 𝜆 𝐿𝑧 𝑆𝑧 + 𝐿+ 𝑆− + 𝐿− 𝑆+
2
2
1 𝑍𝑒 2
𝑔𝑠
𝐿෠ 𝑆መ
=
2 4𝜋𝜖0 2𝑚2 𝑐 2 𝑟 3
1
𝑍3
1
=
𝑟3
𝑛3 𝑎𝐵3 𝑙 𝑙 + 1 𝑙 + 1
2
2
ℏ
𝐿෠ 𝑆መ =
𝑗 𝑗 + 1 − 𝑙 𝑙 + 1 − 𝑠(𝑠 + 1)
2
𝐻
e.g. for 𝜓210 we get
𝐸𝑆𝑂 =
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1
𝑟3
=
1 𝑍 3
24 𝑎0
4
and for general 𝑛 (principal quantum number)
𝑍
𝑗 𝑗 + 1 − 𝑙 𝑙 + 1 − 𝑠(𝑠 + 1)
2𝑙(𝑙 + 1/2)(𝑙 + 1)
2 137 2 𝑎03 𝑛3
51
𝐿ത 𝑆ҧ =
1 2
1
ҧ𝐽 − 𝐿ത2 − 𝑆ҧ 2 = 𝐿𝑧 𝑆𝑧 + 𝐿+ 𝑆− + 𝐿− 𝑆+
2
2
2𝑃
3/2
𝐿෠ = 1; 𝑆መ =

3
1
2
2𝑃
1/2
2
3
the base: |𝑛, 𝑙, 𝑠, 𝑗, 𝑚𝑗 ൿ
shortly: |𝑗, 𝑚𝑗 ൿ
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Fine structure
The fine structure means the splitting of the spectral lines of atoms due to electron spin and
relativistic corrections to the Schrödinger equation. We got corrections to the value of
energy levels.
• Kinetic energy relativistic correction
• Spin-orbit coupling
• Darwin term
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Fine structure
energy levels.
• Darwin term
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Fine structure
energy levels.
𝐸=
The correct description of the atom requires taking into account
𝑝Ԧ2 𝑐 2 + 𝑚02 𝑐 4 the relativistic effects which lead to the Dirac Hamiltonian.
The square root can be expanded into a series:
𝐸 = 𝑚0
𝑐2
𝑝2
𝑝4
1+
−
+⋯
2𝑚02 𝑐 2 2𝑚04 𝑐 4
thus kinetic energy can be expressed as
and the Hamiltonian is:
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= 𝑚0
𝑐2
𝑝2
𝑝4
+
−
+⋯
2𝑚0 8𝑚03 𝑐 2
𝑝2
𝑝4
𝐸𝐾 = 𝐸 − 𝑚0 𝑐 =
−
+⋯
2𝑚0 8𝑚03 𝑐 2
2
𝜕
ℏ2 2
ℏ4
𝑖ℏ Ψ 𝑟,
Ԧ𝑡 = −
𝛻 + V 𝑟Ԧ −
𝛻 4 Ψ 𝑟,
Ԧ𝑡
3
2
𝜕𝑡
2𝑚
8𝑚0 𝑐
55
Fine structure
𝜕
ℏ2 2
ℏ4
𝑖ℏ Ψ 𝑟,
Ԧ𝑡 = −
𝛻 + V 𝑟Ԧ −
𝛻 4 Ψ 𝑟,
Ԧ𝑡
3
2
𝜕𝑡
2𝑚
8𝑚0 𝑐
Using perturbation theory one can find correction due to the relativistic mass change for
each principal quantum number and corresponding energy 𝐸𝑛.
𝑚0
79
𝑚=
For instance the electron speed of 1𝑠 in gold Au 𝑣 = 53% 𝑐!
𝑣2
1− 2
𝑐
2
2𝑍2
𝐸
4𝑛
𝛼
𝑛
3
𝑛
2
Δ𝐸𝑛′ = −
−
3
=
−
𝐸
−
𝑒
1
𝑛
2
4
1 4
2𝑚𝑐 𝑙 + 1
2𝑛
𝛼
=
≈
𝑙+
4𝜋𝜖0 ℏ𝑐 137
2
2
2015-11-06
Tadeusz Stacewicz
Correction from this perturbation:
• Small for light atoms
• Rapidly decreases with the principal quantum number 𝑛
• Significant for large 𝑍
𝑚𝑐 2 𝛼 2 𝑍 2
𝐸𝑛 = −
2 𝑛2
56
Fine structure
energy levels.
• Darwin term
𝐻
e.g. for 𝜓210 we get
𝐸𝑆𝑂 =
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1
𝑟3
1 2
1
𝐽 − 𝐿2 − 𝑆 2 = 𝜆 𝐿𝑧 𝑆𝑧 + 𝐿+ 𝑆− + 𝐿− 𝑆+
2
2
1 𝑍𝑒 2
𝑔𝑠
𝐿෠ 𝑆መ
=
2 4𝜋𝜖0 2𝑚2 𝑐 2 𝑟 3
1 𝑍 3
= 24 𝑎
0
4
𝑍
2 137 2 𝑎03 𝑛3
and for general 𝑛 (principal quantum number)
𝑗 𝑗 + 1 − 𝑙 𝑙 + 1 − 𝑠(𝑠 + 1)
2𝑙(𝑙 + 1/2)(𝑙 + 1)
57
Fine structure
energy levels.
• Darwin term
Darwin term is the non-relativistic
expansion of the Dirac equation:
3
𝛽𝑚𝑐 2 + 𝑐
෍ 𝛼𝑛 𝑝𝑛
𝑛
𝜓 𝑟,
Ԧ 𝑡 = 𝑖ℏ
𝜕𝜓 𝑟,
Ԧ𝑡
𝜕𝑡
Negative 𝐸 solutiuons to the equation
→ antimatter
https://pl.wikibooks.org/wiki/Mechanika_kwantowa/Relatywistyczna_teoria_kwant%C3%B3w_Diraca
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58
Fine structure
energy levels.
• Darwin term
Darwin term is the non-relativistic expansion of the Dirac equation:
3
𝛽𝑚𝑐 2
෍ 𝛼𝑛 𝑝𝑛
+𝑐
𝑛
0
𝛼𝑥 = 0
0
1
1 0 0
0
0
𝛽= 0 1 0
0 0 −1
0
0 0 0 −1
0
0
1
0
0
1
0
0
1
0
0
0
𝜕𝜓 𝑟,
Ԧ𝑡
𝜓 𝑟,
Ԧ 𝑡 = 𝑖ℏ
𝜕𝑡
0
𝛼𝑦 = 0
0
𝑖
0 0 −𝑖
0 𝑖 0
−𝑖 0 0
0 0 0
𝛼, 𝛽 – matrices 4 × 4
0
𝛼𝑧 = 0
1
0
0 1 0
0 0 −1
0 0 0
−1 0 0
https://pl.wikibooks.org/wiki/Mechanika_kwantowa/Relatywistyczna_teoria_kwant%C3%B3w_Diraca
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Fine structure
energy levels.
• Darwin term
Darwin term is the non-relativistic expansion of the Dirac equation:
ℏ2
𝑍𝑒 2
𝐻𝐷𝑎𝑟𝑤𝑖𝑛 =
4𝜋
𝛿 𝑟Ԧ
8𝑚2 𝑐 2
4𝜋𝜖0
ℏ2
𝑍𝑒 2
𝐻𝐷𝑎𝑟𝑤𝑖𝑛 =
4𝜋
𝜓 𝑟Ԧ = 0 2
2
2
8𝑚 𝑐
4𝜋𝜖0
Only for s-orbit, because: 𝜓 𝑟Ԧ = 0 = 0 for 𝑙 > 0
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Fine structure
energy levels.
𝐸𝑛2
4𝑛
𝛼 2𝑍2
𝑛
3
′
Δ𝐸𝑛 = −
−3 = −
−
2
4 𝐸𝑛
1
1 4
2𝑚𝑐
2𝑛
𝑙+
𝑙+
2
2
• Darwin term
𝑍4
𝑗 𝑗 + 1 − 𝑙 𝑙 + 1 − 𝑠(𝑠 + 1)
𝐸𝑆𝑂 =
2𝑙(𝑙 + 1/2)(𝑙 + 1)
2 137 2 𝑎03 𝑛3
Total effect:
𝐸𝐷𝑎𝑟𝑤𝑖𝑛
ℏ2
𝑍𝑒 2
=
4𝜋
8𝑚2 𝑐 2
4𝜋𝜖0
𝜓 𝑟Ԧ = 0
2
2𝑍2
𝛼
𝑛
3
Δ𝐸𝑛′ = − 2 𝐸𝑛
−
1
𝑛
𝑗+2 4
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Fine structure
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Tadeusz Stacewicz
Paul Dirac calculations:
62
Fine structure
Paul Dirac (Nobel 1933) calculations:
Bohr
3P3/2 and 3D3/2
Dirac
2015-11-06
Tadeusz Stacewicz
2S1/2 and 2P1/2
63
Fine structure
Willis Lamb (Nobel 1955) calculations:
2S1/2 and 2P1/2
QED – Quantum ElectroDynamics – Lamb shift due to the interaction of the atom with virtual
photons emitted and absorbed by it. In quantum electrodynamics the electromagnetic field is
quantized and therefore its lowest state cannot be zero (𝐸𝑚𝑖𝑛 =
Coulomb potential.
2015-11-06
ℏ𝜔
),
2
which perturbs
64
2015-11-06
http://backreaction.blogspot.com/2007/12/hydrogen-spectrum-and-its-fine.html
Fine structure
65
http://backreaction.blogspot.com/2007/12/hydrogen-spectrum-and-its-fine.html
Fine structure
10.2 eV
Lamb shift – (Willis Lamb) QED
size of the proton!
(hydrogen vs muonic hydrogen)
[about 1 GHz]
2015-11-06
for Hydrogen the order
of 0.000045 eV
(magnetic field of
electron 𝐵 ≈ 0.4T)
Hyperfine interaction: interaction
with nuclear moment
[21 cm (1420 MHz) for atomic H]
66
Fine structure
The Maxwell wave function of the photon M. G. Raymer and Brian J. Smith, SPIE conf. Optics and Photonics 2005
2015-11-06
67

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