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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 4
CLOSE TO COLLAPSE
The collapse of the wavefunction
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Travel
• Office hour (this time only):
– This Friday, Sep 21, 4pm
– (instread of Mon, Sep 24, 4pm)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Last time: Wave mechanics
1. The ket Ψ describe the system
2. The evolution is deterministic, but it applies to
stochastic events
3. Classical quantities are replaced by operators
4. The results of measurements are eigenvalues,
and
d th
the ket
k t collapses
ll
iin an eigenvector
i
t
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Commuting Hermitian operators have a set of
common eigenfunctions
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Fifth postulate
• If the measurement of the physical
quantity A gives the result an , the
wavefunction of the system immediately
after the measurement is the eigenvector
ϕn
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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. See Mortimer, R. G. Physical Chemistry. 2nd ed.
San Diego, CA: Elsevier, 2000, page 555, Figure 15.3.
Diagram showing the probability densities of
the first 3 energy states in a 1D quantum well of width L.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
“Collapse” of the wavefunction
x
x
a
(a)
x
(b)
a
a
(c)
The wave function of a particle in a box. (a) Before a position measurement (schematic). The probability
density is nonzero over the entire box (except for the endpoints). (b) Immediately after the position
measurement (schematic). In a very short time, the particle cannot have moved far from the position given
by the measurement, and the probability density must be a sharply peaked function. (c) Shortly after a
position measurement (schematic). After a short time, the probability density can be nonzero over a larger
region.
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Quantum double-slit
Image removed due to copyright restrictions.
Please see any experimental
verification of the double-slit experiment, such as
http://commons.wikimedia.org/wiki/Image:Doubleslitexperiment_results_Tanamura_1.gif
Image of a double-slit experiment simulation removed
due to copyright restrictions. Please see "Double Slit Experiment."
in Visual Quantum Mechanics.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
4
Deterministic vs. stochastic
• Classical,
•
Classical macroscopic objects: we have wellwell
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.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
When scientists turn bad…
Image from Wikimedia Commons, http://commons.wikimedia.org.
5
Cat wavefunction
1
2
⎛
⎛ t
⎞ ⎞
Ψcat (t ) = Ψalive ⎜⎜ exp⎜ − ⎟ ⎟⎟ + Ψdead
⎝
⎝ τ
⎠
⎠
1
2
⎛
⎛ t
⎞ ⎞
⎜⎜1−
exp⎜ − ⎟ ⎟⎟
⎝ τ
⎠
⎠
⎝
• There is not a value of the observable until it’s
measured (a conceptually different “statistics”
from thermodynamics)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Uncertainties, and Heisenberg’s Indetermination Principle
(∆A)2 = (A −
∆A∆B ≥
1
2
A
)
2
= A2 − A
2
[A, B ]
d ⎤
⎡
⎢⎣
x
,−
ih
dx
⎥⎦ =
i
h
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
6
Linewidth Broadening
Image removed due to copyright restrictions.
Please see: Fig. 2 in Uhlenberg, G., et al. "Magneto-optical Trapping
of Silver Ions." Physical Review A 62 (November 2000): 063404.
∆E∆t ≥
h
2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Top Three List
• Albert Einstein: “Gott
•
Gott wurfelt nicht!”
nicht!
[God does
not play dice!]
• Werner Heisenberg “I myself . . . only came to believe in the uncertainty relations after many pangs of conscience. . .”
• Erwin Schrödinger: “Had
•
Had I known that
that
we were
not going to get rid of this damned quantum jumping, I never would have involved myself in this business!”
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
7
Spherical Coordinates
z
P
r=r
θ
0
y
φ
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ
x
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Angular Momentum
Classical
Quantum
r r r
L=r×p
⎛ ∂
∂ ⎞
Lˆx = yˆ p̂z − ẑpˆ y = −ih ⎜ y − z ⎟
∂y ⎠
⎝ ∂z
∂ ⎞
⎛ ∂
ˆˆ x − xˆˆpz = −ih ⎜ z − x ⎟
L̂y = zp
∂z ⎠
⎝ ∂x
⎛ ∂
∂ ⎞
Lˆz = x̂p̂ y − ŷpˆ x = −ih ⎜ x − y ⎟
∂x ⎠
⎝ ∂y
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
8
Commutation Relation
Lˆ2 = Lˆ2x + Lˆ2y + Lˆ2z
⎡ Lˆ2 , Lˆx ⎤ = ⎡ Lˆ2 , Lˆ y ⎤ = ⎡ Lˆ2 , Lˆz ⎤ = 0
⎣
⎦ ⎣
⎦ ⎣
⎦
ˆ ≠0
⎡ Lˆx , Lˆ y ⎤ = ihL
i
h
L
z
⎣
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Angular Momentum in
Spherical Coordinates
L̂z = −ih
∂
∂ϕ
⎛ 1 ∂ ⎛
∂ ⎞
1
∂2 ⎞
L̂ = −h
h ⎜
⎜ siin θ
⎟+ 2
2 ⎟
θ
∂
θ
∂
θ
θ
∂
ϕ
sin
sin
⎝
⎠
⎝
⎠
2
2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
9
Eigenfunctions of Lz , L2
∂ m
Lˆ zYl m (θ , ϕ ) = −ih
Yl (θ , ϕ ) = mhYl m (θ , ϕ )
∂ϕ
⎛ 1 ∂ ⎛
∂
−h 2 ⎜
⎜ sin θ
i θ ∂θ ⎝
∂θ
⎝ sin
1
∂2 ⎞ m
⎞
Y θ , ϕ ) = h 2l ( l +1) Yl m (θ , ϕ )
+
⎟
2
2 ⎟ l (
i θ ∂ϕ ⎠
⎠ sin
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Simultaneous eigenfunctions of L2, Lz
LˆzYl m (θ , ϕ ) = mhYl m (θ , ϕ )
Lˆ2Yl m (θ , ϕ ) = h 2l ( l + 1) Yl m (θ , ϕ )
Yl m (θ , ϕ ) = Θlm (θ ) Φ m (ϕ )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
10
Spherical Harmonics in Real Form
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Same as a beating drum…
01
11
21
Copyright © 1999, 2000 David M. Harrison. This material may be distributed only
subject to the terms and conditions set forth in the Open Publication License,
vX.Y or later (the latest version is presently available at http://www.opencontent.org/openpub/).
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
11
…for the career helioseismologist
Image courtesy of NSO/AURA/NSF. Used with permission.
Normal modes (i.e. sound, or seismic waves) for the Sun (basically jello in a 3d spherical box)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Angular Momentum, then…
m=2
m=1
y
m=0
x
m = -1
m = -2
L2 = l (l +
1)h 2 = 0, 2h 2 , 6h 2
...
Lz = 0, ± h, ± 2h, ± 3h
...
Cones of possible angular momentum directions for
l = 2. These cones are similar to the cones of
precession of a gyroscope, and represent possible
directions for the angular momentum vector . The z
component is arbitrarily chosen as the one component
that can have a definite value.
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
12
An electron in a central potential (I)
h2 2
r
Hˆ = −
∇ + V (r*)
2µ
∇ 2 needs
d to bbe iin spherical
h i l coordinates
di
1
1
h2 ⎡ 1 ∂ ⎛ 2 ∂ ⎞
∂ ⎛
∂ ⎞
∂2 ⎤
ϑ
Hˆ = −
r
sin
+
+
⎢
⎜
⎟
⎜
⎟
⎥ + V (r )
∂ϑ ⎠ r 2 sin 2 ϑ ∂ϕ 2 ⎦
2 µ ⎣ r 2 ∂r ⎝ ∂r ⎠ r 2 sin ϑ ∂ϑ ⎝
h 2 ⎡ 1 ∂ ⎛ 2 ∂ ⎞ L2 ⎤
Hˆ = −
⎢
⎜r
⎟−
⎥ + V (r )
2 µ ⎣ r 2 ∂r ⎝ ∂r ⎠ h 2 r 2 ⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central potential (II)
h2 1 d ⎛ 2 d ⎞
L2
ˆ
Ĥ = −
+ V (r )
⎜r
⎟+
2µ r 2 dr ⎝ dr ⎠ 2µ r 2
r
ψ nlm (r ) = Rnlm (r )Ylm (ϑ , ϕ )
⎡ h 2 1 d ⎛ 2 d ⎞ h 2 l (l +1)
⎤
r
V
(
r
)
−
+
+
⎢
⎜
⎟
⎥ Rnl (r ) = Enl Rnl (r )
2
2
2
r
d
r
dr
2
r
µ
µ
⎝
⎠
⎣
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
13
An electron in a central potential (III)
h 2 l (l + 1) Ze 2
Veff (r ) =
−
2
2µ r
4πε 0 r
unl (r ) = r Rnl (r )
⎡ h2 d 2
⎤
−
+
V
(
r
)
eff
⎢
⎥ unl (r ) = Enl unl (r )
2
⎣ 2µ dr
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
What is the Veff(r) potential ?
2
Vcentripetal(r)
1
1018v(r) (J)
1010r(m)
1
Veff(r)
-1
2
3
4
5
6
VCoulomb(r)
-2
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
14
R10
r2R210
2
0.4
1
The Radial Wavefunctions
for Coulomb V(r)
0
0.2
0
1
2
3
4
0
r
R20
0.6
0.4
0.2
0
-0.2
Z
2
6
0.15
0.1
0.05
0
10
r
R21
R30
0
2
6
10
0.15
0.1
0.05
0
r
4
0
-0.1
8
12
16
0
r
0
2
6
10
r
0
2
6
10
r
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
4
0
-0.04
8
12
16
0.4
r
0
rR
2
R32
0.04
0
2
32
0.8
0.02
Figures by MIT OpenCourseWare.
4
0.04
r
R31
X
3
0.08
0.2
Y
2
r2R230
0.4
r
1
2
20
r2R221
0.12
0.08
0.04
0
Thickness dr
0
rR
2
0.4
0
4
8
12
16
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.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
The Grand Table
Shell
Quantum numbers
n
l
m
Spectroscopic
notation
K
1
0
0
1s
L
2
0
0
2s
2
1
0
2p0
2
1
_0
+
3
0
0
3s
3
1
0
3p0
3
1
_1
+
3
2
0
3
2
_1
+
3d+1
_
3
2
_2
+
3d_+2
M
2p+1
_
3p+1
_
3d0
Wave function Ψnlm (r, θ, φ)
1
3/2
π (Z/a0) exp (-Zr/a0)
1
(Z/a0)3/2 (1-Zr/2a0) exp (-Zr/a0)
2 2π
1
(Z/a0)3/2 (Zr/a0) exp (-Zr/2a0) cos θ
4 2π
_ 1
_
+ 8 (Z/a0)3/2 (Zr/a0) exp (-Zr/2a0) sin θ exp (+iφ)
π
1
(Z/a0)3/2 (1-2Zr/3a0 + 2Z2r2/27a ) exp (-Zr/3a0)
3 3π
2 2
27 π
_
+
(Z/a0)3/2 (1-Zr/6a0) (Zr/a0) exp (-Zr/3a0) cos θ
2
27 π
1
(Z/a0)3/2 (Z r /a ) exp (-Zr/3a0) (3 cos θ − 1)
1
_
(Z/a0)3/2 (Z r /a ) exp (-Zr/3a0 ) sin θ cos θ exp (+iφ)
81 6π
_
+
(Z/a0)3/2 (1-Zr/6a0) (Zr/a0) exp (-Zr/3a0) sin θ exp (+iφ)
_
81 π
1
162 π
2 2
2
2 2
(Z/a0)3/2 (Z2r2/a ) exp (-Zr/3a0 ) sin2 θ exp (+2iφ)
_
The complete normalised hydrogenic wave functions corresponding to the first three shells, for an 'infinitely heavy'
nucleus. The quantity a0 = 4πε0h2/me2 is the first Bohr radius. In order to take into account the reduced mass effect one
should replace a0 by aµ = a0 (m/µ)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Figure by MIT OpenCourseWare.
15
Solutions in the central Coulomb Potential: the Alphabet Soup
http://www.orbitals.com/orb/orbtable.htm
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Courtesy of David Manthey. Used with permission.
Source: http://www.orbitals.com/orb/orbtable.htm.
16