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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)
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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 the ket collapses in an eigenvector
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)
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)
“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)
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.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
When scientists turn bad…
Image from Wikimedia Commons, http://commons.wikimedia.org.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Cat wavefunction
Ψcat (t ) = Ψalive
⎛
⎛ t
⎜⎜ exp⎜ −
⎝ τ
⎝
1
2
⎞⎞
⎟ ⎟⎟ + Ψdead
⎠⎠
⎛
⎛ 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)
1
2
Uncertainties, and Heisenberg’s
Indetermination Principle
(ΔA)
2
= (A − A
1
ΔAΔB ≥
2
)
2
= A − A
2
[A, B]
d⎤
⎡
⎢ x,−i= dx ⎥ = i=
⎣
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Linewidth Broadening
=
ΔEΔt ≥
2
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.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Top Three List
• Albert Einstein: “Gott wurfelt 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 I known 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)
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
G G G
L=r×p
⎛ ∂
∂ ⎞
ˆ
ˆ ˆ z − zp
ˆ ˆ y = −i = ⎜ y − z ⎟
Lx = yp
∂y ⎠
⎝ ∂z
∂ ⎞
⎛ ∂
ˆ
ˆˆ z = −i= ⎜ z − x ⎟
ˆ ˆ x − xp
Ly = zp
∂z ⎠
⎝ ∂x
⎛ ∂
∂ ⎞
ˆ
ˆˆ y − yp
ˆ ˆ x = −i = ⎜ x − y ⎟
Lz = xp
∂x ⎠
⎝ ∂y
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Commutation Relation
2
2
2
2
ˆ
ˆ
ˆ
ˆ
L = Lx + Ly + Lz
⎡ Lˆ2 , Lˆx ⎤ = ⎡ Lˆ2 , Lˆ y ⎤ = ⎡ Lˆ2 , Lˆz ⎤ = 0
⎣
⎦ ⎣
⎦ ⎣
⎦
⎡ Lˆx , Lˆ y ⎤ = i=Lˆz ≠ 0
⎣
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Angular Momentum in
Spherical Coordinates
∂
ˆ
Lz = −i=
∂ϕ
2
⎛
⎞
1
1
∂
∂
∂
⎛
⎞
2
2
Lˆ = − = ⎜
⎜ sin θ
⎟+ 2
2 ⎟
∂θ ⎠ sin θ ∂ϕ ⎠
⎝ sin θ ∂θ ⎝
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Eigenfunctions of Lz , L2
∂ m
m
ˆ
LzYl (θ , ϕ ) = −i=
Yl (θ , ϕ ) = m=Yl m (θ , ϕ )
∂ϕ
2
⎛
⎞ m
∂
∂
∂
1
1
⎛
⎞
2
2
m
−= ⎜
+
=
+
θ
Y
θ
ϕ
l
l
Y
sin
,
1
=
(
)
(
)
l (θ , ϕ )
⎜
⎟
2
2 ⎟ l
∂θ ⎠ sin θ ∂ϕ ⎠
⎝ sin θ ∂θ ⎝
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Simultaneous eigenfunctions of
L2 , L z
m
m
ˆ
LzYl (θ , ϕ ) = m=Yl (θ , ϕ )
2 m
2
m
ˆ
L Yl (θ , ϕ ) = = l ( l + 1) Yl (θ , ϕ )
Yl
m
(θ , ϕ ) = Θ (θ ) Φ m (ϕ )
m
l
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
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)
…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
m=0
x
y
m = -1
m = -2
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.
L2 = l (l + 1)= 2 = 0, 2= 2 , 6= 2 ...
Lz = 0, ± =, ± 2=, ± 3=...
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central
potential (I)
2
=
G
∇ 2 + V (r*)
Hˆ = −
2μ
∇ 2 needs to be in spherical coordinates
2
=
Hˆ = −
2μ
⎡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
=
Hˆ = −
2μ
⎡ 1 ∂ ⎛ 2 ∂ ⎞ L2 ⎤
⎢ 2
⎜r
⎟ − 2 2 ⎥ + V (r )
⎣ r ∂r ⎝ ∂r ⎠ = r ⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central
potential (II)
2
2
= 1 d ⎛ 2 d ⎞
L
ˆ
+ V (r )
H =−
⎜r
⎟+
2
2
2μ r dr ⎝ dr ⎠ 2μ r
G
ψ nlm (r ) = Rnlm (r )Ylm (ϑ , ϕ )
⎡ = 2 1 d ⎛ 2 d ⎞ = 2 l (l + 1)
⎤
+ V (r ) ⎥ Rnl (r ) = Enl Rnl (r )
⎢−
⎜r
⎟+
2
2
⎣ 2μ r dr ⎝ dr ⎠ 2μ r
⎦
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
An electron in a central
potential (III)
unl (r ) = r Rnl (r )
= l (l + 1) Ze
Veff (r ) =
−
2
2μ r
4πε 0 r
2
⎡ = d
⎤
(
)
(
)
(
)
V
r
u
r
E
u
r
+
=
eff
nl
nl
nl
⎢−
⎥
2
⎣ 2μ dr
⎦
2
2
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
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)
The Radial
Wavefunctions
for Coulomb V(r)
Z
R10
r2R210
2
0.4
1
0
0.2
0
1
2
3
4
R20
0.6
0.4
0.2
0
-0.2
0.12
0.08
0.04
0
R30
2
6
r
r2R221
0
2
6
10
r
3
4
r
0
2
6
10
r
0
2
6
10
r
2
30
0.08
4
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
0.8
0.04
4
0
-0.04
8
12
16
r
0.4
0
r2R232
R32
0.04
Figures by MIT OpenCourseWare.
2
rR
0
-0.1
X
0.15
0.1
0.05
0
2
0.2
Y
1
2
20
0.15
0.1
0.05
0
10
0.4
Thickness dr
0
rR
2
R21
r
0
r
0.8
0.02
0
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.
3.23 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)
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Solutions in the central
Coulomb Potential: the Alphabet
Soup
http://www.orbitals.com/orb/orbtable.htm
Courtesy of David Manthey. Used with permission.
Source: http://www.orbitals.com/orb/orbtable.htm
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)