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3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fund of Mat Sci: Bonding – Lecture 5/6
THE HYDROGEN ATOM
Comic strip removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last Time
• Metal surfaces and STM• Dirac notation• Operators, commutators, some postulates
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Mon Oct 3
• Study: 18.4, 18.5, 20.1 to 20.5.• Read – before 3.014 starts next week:
22.6 (XPS and Auger)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Second Postulate
• For every physical observable there is a corresponding Hermitian operator
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hermitian Operators1. The eigenvalues of a Hermitian operator are real
2. Two eigenfunctions corresponding to different eigenvaluesare orthogonal
3. The set of eigenfunctions of a Hermitian operator is complete
4. Commuting Hermitian operators have a set of common eigenfunctions
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The set of eigenfunctions of a Hermitianoperator is complete
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Third Postulate
• In any single measurement of a physical quantity that corresponds to the operator A, the only values that will be measured are the eigenvalues of that operator.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Position and probability
Graphs of the probability density for positions of a particle in a one-dimensional hard box according to classical mechanics removed for copyright reasons.
Graph of the probability density for positions of a particle in a one-dimensional hard box removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 554, figure 15.2.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 555, figure 15.3.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Quantum double-slit
Source: Wikipedia
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Quantum double-slit
Above: Thomas Young's sketch of two-slit diffraction of light. Narrow slits at A and B act as sources, and waves interfering in various phases are shown at C, D, E, and F. Source: Wikipedia
Image of the double-slit experiment removed for copyright reasons.
See the simulation at http://www.kfunigraz.ac.at/imawww/vqm/movies.html:
"Samples from Visual Quantum Mechanics": "Double-slit Experiment."
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Fourth Postulate
• If a series of measurements is made of the dynamical variable A on an ensemble described by Ψ, the average (“expectation”) value is
ΨΨΨΨ
=A
Aˆ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Deterministic vs. stochastic
• Classical, macroscopic objects: we have well-defined values for all dynamical variables at every instant (position, momentum, kinetic energy…)
• Quantum objects: we have well-defined probabilities of measuring a certain value for a dynamical variable, when a large number of identical, independent, identically prepared physical systems are subject to a measurement.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spherical Coordinates
sin cossin sincos
x ry rz r
θ ϕθ ϕθ
===
z
θ
0
φ
P
y
r = r
x
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3-d Integration
Diagram of an infinitesimal volume element in spherical polarcoordinates removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 1006, figure B.4.
Angular Momentum
Classical Quantum
L r p= ×r r r
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Commutation Relation
2 2 2 2
2 2 2
ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ, , , 0
ˆ ˆ ˆ, 0
x y z
x y z
x y z
L L L L
L L L L L L
L L i L
= + +
⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ = ≠⎣ ⎦ h
Angular Momentum in Spherical Coordinates
22 2
2 2
ˆ
1 1ˆ sinsin sin
zL i
L
ϕ
θθ θ θ θ ϕ
∂= −
∂
⎛ ⎞∂ ∂ ∂⎛ ⎞= − +⎜ ⎟⎜ ⎟∂ ∂ ∂⎝ ⎠⎝ ⎠
h
h
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Simultaneous eigenfunctions of L2, Lz
( ) ( )( ) ( ) ( )2 2
ˆ , ,ˆ , 1 ,
m mz l l
m ml l
L Y m Y
L Y l l Y
θ ϕ θ ϕ
θ ϕ θ ϕ
=
= +
h
h
( ) ( ) ( ),m ml l mY θ ϕ θ ϕ= Θ Φ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spherical Harmonics in Real Form
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
An electron in a central potential (I)2
2 2
2 22
2 2 2
2 22
2 2 2 2 2
ˆ ( ) needs to be in spherical coordinates2
1 1 1ˆ sin ( )2 sin
ˆ1ˆ ( )2
sin
e
e
e
LH r
H V rm
H r V rm r r
V rm r r r
r r
r
rϑ
ϑ ϑ ϑ ϑ ϕ
= − ∇ + ∇
⎡ ⎤
⎡ ⎤∂ ∂
∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= − + + +⎢ ⎜ ⎟ ⎜ ⎟ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦
⎛ ⎞= − − +⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎣ ⎦
h
h
h
h
r*
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
An electron in a central potential (II)2 2
22 2
ˆ1ˆ ( )2 2e e
d d LH r V rm r dr dr m r
⎛ ⎞= − + +⎜ ⎟⎝ ⎠
h
( ) ( ) ( , )r R r Yψ ϑ ϕ=r
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
An electron in a central potential (III)
2 22
2 2
1 ( 1) ( ) ( ) ( )2 2 nl nl nl
e e
d d l lr V r R r E R rm r dr dr m r
⎡ ⎤+⎛ ⎞− + + =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
h h
What is the V(r) potential ?
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
2
1
-1
-2
1 2 3 4 5 6
Vcentripetal(r)
1010r(m)
Veff(r) VCoulomb(r)
1018
v(r)
(J)
Figure by MIT OCW.
The Radial Wavefunctions
for Coulomb V(r)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
2
2
1
10
0 3 4
R10
r
r
r
r
-0.20
0
0
0.20.4
0.040.080.12
0.6
2
r
r
r
6 10
2 6 100
0
00
R20
R21
R30
R31
R32
4
4
8
8
12
12
16
16
4 8 12 16
-0.1
0.2
0.4
-0.04
0.04
0.08
0.02
0.04
Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for
atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Radial Density 2
2
1
10
0 3 4
R10
00
0.2
0.4
1 2 3 4
r2R220
r2R210
r2R221
r2R230
r2R231
r2R232
rr
r
r
r
r
r
r
r
-0.20
0
0
0.20.4
0.040.080.12
0.6
2
r
r
r
6 10
2 6 10
00
0.10.05
0.05
0.15
2 6 10
0
00
0
0
00
02 6 10
R20
R21
R30
R31
R32
0.1
4
4
8
8
12
12
16
16
4
4
8
8
12
12
16
16
00
4 8 12 16
r0 4 8 12 16
-0.1
0.2
0.4
0.04
0.08
-0.04
0.04
0.08
0.02
0.04
0.4
0.8
0
0.4
0.8
Radial functions Rnl(r) and radial distribution functions r2R2nl(r) for
atomic hydrogen. The unit of length is aµ = (m/µ) a0, where a0 is thefirst Bohr radius.
0.15
Z
Thickness dr
y
X
r
Figure by MIT OCW.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Three Quantum Numbers
• Principal quantum number n (energy, accidental degeneracy)
• Angular momentum quantum number l (L2)l=0,1,…,n-1 (a.k.a. s, p, d… orbitals)
• Magnetic quantum number m (Lz )m=-l,-l+1,…,l-1,l
( ) ( )2 2 2 2
2 2 20 0
13.6058 eV 1 Ry8n
e Z Z ZEa n n nπε
= − = − = −
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Emission and absorption lines
Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Balmer lines in a hot star
Courtesy of the Department of Physics and Astronomy at the University of Tennessee. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
XPS in Condensed Matter
Diagram of Moon composition as seen in X-rays, removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Grand Table
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Solutions in the central Coulomb Potential: the Alphabet Soup
Table of orbitals removed for copyright reasons.See "n and l versus m" at http://www.orbitals.com/orb/orbtable.htm.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Orbital levels in multi-electron atomsO
rbita
l Ene
rgy
(kJ /
mol
)0
-82
-146
-328
-1313
0-82
-146
-328
-13131s 1s
4s4s
2s
2s
3s3s
4p 4p
3p 3p
2p 2p
4d 4d
3d3d
4f4f
Hydrogen Multielectron Atoms
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Screening
Auf-bau
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
6s
5s
4s
4p
5p
6p5d
4d
4f
3d
3s
3p
2s
2p
1s
LOW
EN
ERG
YH
IGH
EN
ERG
YENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI
Figure by MIT OCW.