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 Operators
1.
The eigenvalues of a Hermitian operator are real
2.
Two eigenfunctions corresponding to different eigenvalues
are 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 Hermitian
operator 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
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."
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
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
Ψ Aˆ Ψ
(“expectation”) value is A =
Ψ Ψ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Deterministic vs. stochastic
• Classical, macroscopic objects: we have welldefined 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
z
P
r=r
θ
0
y
φ
x
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ
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.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Angular Momentum
Classical
Quantum
r r r
L=r×p
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Commutation Relation
2
2
2
2
ˆ
ˆ
ˆ
ˆ
L = Lx + Ly + Lz
⎡ Lˆ2 , Lˆx ⎤ = ⎡ Lˆ2 , Lˆ y ⎤ = ⎡ Lˆ2 , Lˆ z ⎤ = 0
⎣
⎦ ⎣
⎦ ⎣
⎦
⎡ Lˆx , Lˆ y ⎤ = ihLˆ z ≠ 0
⎣
⎦
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Angular Momentum in
Spherical Coordinates
∂
ˆ
Lz = −ih
∂ϕ
2
⎛
⎞
∂
∂
∂
1
1
⎛
⎞
2
2
ˆ
L = −h ⎜
⎜ sin θ
⎟+ 2
2 ⎟
∂θ ⎠ sin θ ∂ϕ ⎠
⎝ sin θ ∂θ ⎝
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Simultaneous eigenfunctions of L2, Lz
m
m
ˆ
LzYl (θ , ϕ ) = mhYl (θ , ϕ )
2 m
2
m
ˆ
L Yl (θ , ϕ ) = h l ( l + 1) Yl (θ , ϕ )
Yl
m
(θ , ϕ ) = Θ (θ ) Φ m (ϕ )
m
l
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
h
r
Hˆ = −
r)
∇ 2 + V (*
2me
∇ 2 needs to be in spherical coordinates
2
h
Hˆ = −
2me
⎡1 ∂ ⎛ 2 ∂ ⎞
∂ ⎛
∂ ⎞
∂2 ⎤
1
1
+ V (r )
⎢ 2
⎜r
⎟+ 2
⎜ sin ϑ
⎟+ 2 2
2⎥
∂ϑ ⎠ r sin ϑ ∂ϕ ⎦
⎣ r ∂r ⎝ ∂r ⎠ r sin ϑ ∂ϑ ⎝
2
h
Hˆ = −
2me
⎡ 1 ∂ ⎛ 2 ∂ ⎞ Lˆ2 ⎤
⎢ 2
⎜r
⎟ − 2 2 ⎥ + V (r )
⎣ r ∂r ⎝ ∂r ⎠ h r ⎦
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
An electron in a central potential (II)
2
2
ˆ
1
d
d
L
h
⎛ 2 ⎞
ˆ
H =−
+ V (r )
⎜r
⎟+
2
2
2me r dr ⎝ dr ⎠ 2me r
r
ψ (r ) = R(r )Y (ϑ , ϕ )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
An electron in a central potential (III)
⎡ h 2 1 d ⎛ 2 d ⎞ h 2 l (l + 1)
⎤
+ V (r ) ⎥ Rnl (r ) = Enl Rnl (r )
⎢−
⎜r
⎟+
2
2
⎣ 2me r dr ⎝ dr ⎠ 2me r
⎦
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
What is the V(r) potential ?
2
Vcentripetal(r)
1
1018v(r) (J)
1010r(m)
1
-1
2
3
Veff(r)
-2
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
4
VCoulomb(r)
5
6
The Radial
Wavefunctions
for Coulomb V(r)
R10
2
1
0
0
1
2
3
4
r
R20
0.6
0.4
0.2
0
-0.2
2
6
10
r
R21
0.12
0.08
0.04
0
R30
0
2
6
10
r
0.4
0.2
4
0
-0.1
8
12
16
r
R31
0.08
0.04
0
-0.04
4
8
12
16
4
8
12
16
r
R32
0.04
0.02
0
0
r
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 the
first Bohr radius.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
The Radial Density
R10
0.4
1
0
Z
r2R210
2
0.2
0
1
2
3
4
R20
6
2
r
y
R30
r
r2R221
0
2
6
10
0.15
0.1
0.05
0
r
rR
2
0.4
2
3
4
r
0
2
6
10
r
0
2
6
10
r
2
30
4
0
-0.1
8
12
16
0.04
r
R31
0
0
4
8
12
16
r
0
4
8
12
16
r
0
4
8
12
16
r
r2R231
0.08
Figure by MIT OCW.
1
0.08
0.2
X
0.15
0.1
0.05
0
10
R21
0.12
0.08
0.04
0
0
r2R220
0.6
0.4
0.2
0
-0.2
Thickness dr
0
r
0.8
0.04
4
0
-0.04
8
12
16
0.4
r
0
rR
2
R32
0.04
0.8
0.02
0
2
32
0.4
0
4
8
12
16
r
0
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 the
first Bohr radius.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
Three Quantum Numbers
• Principal quantum number n (energy,
accidental degeneracy)
2
2
2
2
e
Z
Z
Z
= − (13.6058 eV ) 2 = − (1 Ry ) 2
En = −
2
8πε 0 a0 n
n
n
• 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
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
0
-82
-146
4s
4p
4d
3s
3p
3d
4f
0
-82
-146
4p
4s
3p
4d
3d
Orbital Energy (kJ / mol)
3s
-328
2s
2p
-328
2p
2s
-1313
Figure by MIT OCW.
1s
-1313
Hydrogen
1s
Multielectron Atoms
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
4f
Screening
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI
Auf-bau
6p
5d
HIGH ENERGY
6s
5p
4d
5s
4p
4s
LOW ENERGY
4f
3d
3p
3s
2p
2s
1s
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)