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.
See Mortimer, R. G. Physical Chemistry. 2nd ed.
San Diego, CA: Elsevier, 2000, p. 554, figure 15.2.
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. 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
Veff(r)
2
3
4
5
6
VCoulomb(r)
-2
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
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.
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The Radial Density
R10
r2R210
2
0.4
1
0
0.2
0
1
2
3
4
R20
Z
2
6
Y
r
r2R221
0.12
0.08
0.04
0
R30
0
2
6
10
0.15
0.1
0.05
0
r
0.4
3
4
r
0
2
6
10
r
0
2
6
10
r
0.08
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
r R 31
2
2
0.08
0.8
0.04
Figure by MIT OCW.
2
r2R230
0.2
X
1
2
20
0.15
0.1
0.05
0
10
R21
Thickness dr
0
rR
2
0.6
0.4
0.2
0
-0.2
r
0
r
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
4f
3d
Orbital Energy (kJ / mol)
3s
-328
2s
2p
-328
2p
2s
-1313
1s
-1313
Hydrogen
1s
Multielectron Atoms
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Figure by MIT OCW.
Screening
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
ENERGY LEVELS OF THE ELECTRONS ABOUT THEIR NUCLEI
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)
Auf-bau