3.012 fund of mat sci: bonding – lecture 5/6 the hydrogen … · 2020-07-09 · 3.012 fund of mat...
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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
Graph of the probability density for positions of a particle in a one-dimensional hard box removed for copyright reasons.
Graphs of the probability density for positions of a particle in a one-dimensional hard box according to classical mechanics 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 polar coordinates 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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Radial Wavefunctions
for Coulomb V(r)
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 Density2
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 atoms
Figure by MIT OCW.
Orb
ital E
nerg
y (k
J / m
ol)
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
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
Y
ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI
Figure by MIT OCW.