text stringlengths 16 3.88k | source stringlengths 60 201 |
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5.8 The Harmonic Oscillator
To illustrate the beauty and efficiency in describing the dynamics of a quan-
tum system using the dirac notation and operator algebra, we reconsider the
one-dimensional harmonic oscillator discussed in section 4.4.2 and described
by the Hamiltonian operator
with
H =
p2
2m
+
1
2
K x2,
[x, p] =... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
+
= ~ω0
¡
a+a+
µ
¢
.
1
2
¶
We introduce the operator
N = a+a,
(5.138)
(5.139)
(5.140)
which is a hermitian operator. Up to an additive constant 1/2 and a scaling
factor ~ω0 equal to the energy of one quantum of the harmonic oscillator it
is equal to the Hamiltonian operator of the harmonic oscillator. Obviously,
N is t... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
constant. This constant follows from the
normalization of this state and being an eigenvector to the number operator.
a+a
n
|
h
n
C
=
|
|
i
|
C = √n.
2 ,
(5.147)
(5.148)
(5.149)
Thus
a
n
i
|
= √n
n
|
,
1
i
−
Clearly, if there is a state with n = 0 application of the annihilation operator
leads to the null-vector in thi... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
155)
(5.156)
(5.157)
5.8.2 Matrix Representation
We can express the normalized position and momentum operators as func-
tions of the creation and annihilation operators
X =
P =
1
√2
j
√2
a+ + a
,
¡
a+
a
−
¢
.
(5.158)
(5.159)
These operators do have the following matrix representations
¡
¢
a
m
h
|
a+a
m
|
h
m
|
h
X
n
i
... | https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/575470ae5ba3cafa4fd36e35b0d29e17_harmonic_oscil.pdf |
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
SLOAN SCHOOL OF MANAGEMENT
15.565 Integrating Information Systems:
Technology, Strategy, and Organizational Factors
15.578 Global Information Systems:
Communications & Connectivity Among Information Systems
Spring 2002
Lecture 9
NETWORK PROTOCOLS
COMPLEXITY OF COMMUNICATION N... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
(E.G., LIBRARY)
5. SESSION: PROCESS-TO-PROCESS (E.G., OS SOFTWARE)
4. TRANSPORT: HOST-TO-HOST (E.G., OS SOFTWARE)
3. NETWORK: ROUTING (E.G., DEVICE DRIVER)
2. DATA: RELIABLE BIT STREAM (E.G., SPECIAL CHIP)
1. PHYSICAL: RAW BIT STREAM (E.G., HARDWARE)
7
6
5
4
3
2
1
APPLICATION LAYER PROTOCOL
PRESENTATION L... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
MULTIPLE ACCESS/COLLISION DETECT)
– SATELLITE
• 5-10 CHANNELS, EACH 50M bps
• UP-LINK & DOWN-LINK = 270 MILLISECONDS
• VSAT
– FIBER-OPTIC
• 100M - 10G bps (Typical)
• INTERNET II (622M -> 2G)
• PROJECT OXYGEN = 1.28T bps (before 2003)
8
2. DATA LINK LAYER
FOCUS: RELIABLE TRANSMISSION: ERROR HANDLING & FLOW ... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
• SIMILAR FOR SHARED LAN (E.G., ETHERNET)
•
IEEE 802 STANDARDS
– MEDIA ACCESS: CSMA/CD AND TOKEN RING
HAWAII
TOKYO
11
3. NETWORK LAYER
ROUTE DETERMINATION (TO BE DISCUSSED MORE LATER)
VIRTUAL CIRCUIT
•
• vs DATAGRAM
• E.G., X.25 NETWORK CCITT 3-LAYER PROTOCOL
•
-- VIRTUAL CIRCUIT ORIGINALLY
• PROCEDURE
... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
AVINGS
(LIKE SOFTWARE MULTIPLEXING)
--
TO USE MULTIPLE “VIRTUAL CIRCUITS”
• FOR INCREASED TRANSMISSION CAPACITY
14
5. SESSION LAYER
• PROVIDE PROCESS-TO-PROCESS COMMUNICATION
(E.G., WEB BROWSER VS. FILE TRANSFER VS. E-MAIL -- SIMULTANEOUS)
6. PRESENTATION LAYER
• TYPICAL ACTIVITIES
--
--
TEXT COMPRESSION & EN... | https://ocw.mit.edu/courses/15-565j-integrating-esystems-global-information-systems-spring-2002/57599d12f997dc9b58d783ce08d5353b_lecture09.pdf |
Lecture 8
8.321 Quantum Theory I, Fall 2017
40
Lecture 8 (Oct. 2, 2017)
8.1 General Time Dependent Hamiltonians
The Schr¨odinger equation dictates that quantum states evolve in time according to
i(cid:126)
d
d
t
| (t)(cid:105) = H(t)|ψ(t)
ψ
(cid:105) .
(8.1)
In the last class, we saw that if the Hamiltonian is independ... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
.6)
(8.7)
2. If [H(t), H(t(cid:48))] = 0 for all t, t(cid:48), then we can simultaneously diagonalize the Hamiltonian at all
times, meaning we can choose a basis of states that are eigenstates of H(t) for all time (the
associated eigenvalues may change as a function of time). We then have
(cid:18)
U (t, t0) = exp
−
ˆ
t... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
2 .
(cid:0)
i
(cid:126)
If we are ignoring terms of order (∆t)2, then we can write
U (t
i+1
, t ) ≈ e−iH(ti)∆t/(cid:126)
i
.
We then build up the finite time-evolution operator as
(8.11)
(8.12)
U (t, t0) = U (t, tN −1)U (tN 1, tN 2) · · · U (ti+1, ti) · · · U (t1, t0) =
−
−
N −1
(cid:89)
i=0
U (ti+1, ti) .
(8.13)
Note t... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
arding
commutation rules when moving the operators around). What guarantees that this limit is
well-defined and exists? This is guaranteed because the operator U (t, t0) is well-defined, and
can be written as a composition of infinitesimal time evolution operators for any partition of
the time interval [t0, t].
An alterna... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
(cid:48)
t0
i
(cid:126)
(cid:0) (cid:1) (cid:0)
dt(cid:48)(cid:48) H t(cid:48)(cid:48) U t(cid:48)(cid:48), t
(cid:1)
0 ,
(8.18)
(cid:48) We can carry this process out an infinite
where the integrand is evaluated at values t(cid:48)(cid:48) ≤ t .
number of times and compose the results to give
(cid:19)2 ˆ
ˆ
(cid:18)
t
t... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
)(cid:48)(cid:1)(cid:3) .
(8.20)
The factor of 1 here deals with overcounting. If we rewrite each term in Eq. (8.19) in a similar
way, there will be a factor of 1 on the nth term to deal with overcounting. We then have
2
n!
ˆ
t
i
U (t, t0) = 1 − (cid:126)
1
n!
+
dt1 H(t1) + · · ·
ˆ
ˆ
(cid:19)
n
t
t0
(cid:18)
−
i
(cid:1... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
I(t)(cid:105) = U −1
0 (t)|ψS(t)(cid:105) ,
(8.24)
with U0(t) the time-evolution operator generated by H0. Contrast this with the expression in the
Heisenberg picture, where the states were defined as
The state |ψI(t)(cid:105) evolves in time according to
|ψH(t)(cid:105) = U −1(t)|ψS(t)(cid:105) ,
i(cid:126)
d
t
d
|ψI(t... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
= i(cid:126)
(cid:18) dU0
dt
UI + U0
(cid:19)
.
dUI
dt
Using Eq. (8.31) on the first term of the right-hand side yields
i(cid:126)
d
dt
U (t) = H0U0UI + i(cid:126)U0
dUI
dt
= H0U + i(cid:126)U0
dUI .
dt
On the other hand, from Eq. (8.30), we have
Thus,
which gives us
i(cid:126) U (t) = H0U + V U .
d
dt
i(cid:126)
U0
dU
... | https://ocw.mit.edu/courses/8-321-quantum-theory-i-fall-2017/579cb61c4eaa494b9adb90830d921099_MIT8_321F17_lec8.pdf |
(cid:0)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
Last
modi(cid:0)ed(cid:1)
September
(cid:2)(cid:3)(cid:4)
(cid:2)(cid:5)(cid:5)(cid:6)
Many(cid:0)body
phenomena
in
condensed
matter
and
atomic
physics
(cid:0)
Lecture
(cid:1)(cid:2)
Vortices(cid:3)
super(cid:4)uidity(cid:2)
Trapped
gases(cid:2)
BEC
at
(cid:5)nite
temperature(... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
ow
v
elocity
(cid:8)h
v
(cid:4)
(cid:4)
(cid:6)(cid:19)(cid:7)
m
r
The
(cid:18)ow
is
irrotational(cid:1)
(cid:4)
(cid:15)
(cid:6)this
is
true
away
from
singularities
in
(cid:4)(cid:7)(cid:1)
and
v
r (cid:1)
(cid:1)
(cid:1)
(cid:1)
h
(cid:8)
m(cid:2)
(cid:4)
(cid:5)
(cid:20)
(cid:10)
m
(cid:6)(cid:9)
(cid:3) (cid:5)
(ci... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
(cid:2)
It
follows
from
the
relation
b
e
t
ween
the
velocity
and
the
phase(cid:1)
Eq(cid:2)
(cid:6)(cid:19)(cid:7)(cid:1)
that
the
circulation
around
any
contour
C
obeys
(cid:22) (cid:4)
d
(cid:4) (cid:9)(cid:7) l
(cid:6)(cid:23)(cid:7)
v
r
(cid:8)h
C
(cid:2)
m
I
with
some
integer
l(cid:2)
We
see
that
The
circulation
i... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
8)(cid:6)
(cid:7)
(cid:1)
Doppler(cid:0)shifted
due
to
the
(cid:18)ow(cid:1)
should
b
e
positive(cid:1)
to
prevent
massive
k
v
k
(cid:0)
(cid:2)
production
of
quasiparticles(cid:2)
This
criterion
de(cid:3)nes
a
critical
velocity
v
(cid:4)
min
(cid:8)(cid:6)
(cid:7)(cid:6)
(cid:6)(cid:26)(cid:7)
k
k
c
k
j
j
above
which
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
are
concentric
circles
parallel
to
the
(cid:6)x(cid:3)
y (cid:7)
plane(cid:2)
Constant
circulation
requires
that
the
velocity
falls
inversely
with
the
distance
(cid:9)
from
the
z
axis(cid:25)
(cid:8)hl
(cid:4)
(cid:5)
v
r
(cid:6)
(cid:7)
(cid:4)
(cid:6)(cid:11)(cid:15)(cid:7)
m
(cid:9)(cid:7)(cid:9)
where
(cid:4)
is
th... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
The
so(cid:0)called
healing
length
(cid:11)
determines
the
size
of
the
vortes
core
where
density
is
depleted
b
e
l
o
w
its
bulk
value(cid:2)
This
qualitative
picture
can
be
con(cid:3)rmed
by
an
analysis
based
on
the
Gross(cid:0)Pitaevskii
equation(cid:2)
One
can
look
for
a
solution
of
the
equation
that
describes
a
vort... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
cid:8)(cid:7)(cid:1)
which
gives
r
(cid:4)
(cid:9)(cid:11) (cid:1)
in
agreement
with
the
estimate
(cid:4)
above(cid:2)
H
Let
us
consider
The
energy
of
the
vortex(cid:1)
that
can
be
estimated
as
the
kinetic
energy
of
the
(cid:18)ow(cid:1)
is
positive(cid:2)
Thus
vortices
do
not
appear
unless
the
system
is
driven(cid:1)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
)(cid:9)(cid:7)
the
density
can
b
e
approximated
by
a
constant
for
(cid:9)
(cid:10)
(cid:11) (cid:1)
while
the
depletion
of
density
in
the
vortex
core(cid:1)
at
(cid:9)
(cid:11) (cid:1)
cuts
the
log
divergence
at
small
(cid:9)(cid:2)(cid:7)
the
contribution
to
the
energy
due
to
the
core(cid:1)
which
can
b
e
estimated
u... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
omes
energetically
favorable
at
Z
Z
Z
(cid:1)
(cid:4)
(cid:4)
(cid:6)(cid:3)(cid:7)
(cid:8)h
b
(cid:27)
(cid:10)
(cid:27)
(cid:4)
E
(cid:6)M
(cid:4)
ln
(cid:6)(cid:11)(cid:17)(cid:7)
c
v
v
(cid:1)
mb
(cid:11)
(cid:0) (cid:1)
Note
the
inverse
square
dependence
of
(cid:27)
on
the
radius
b(cid:1)
which
means
that
it
is
ea... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
1)
d
(cid:4) (cid:27)
(cid:7)b
(cid:1)
which
gives
a
linear
dependence
v
r
(cid:2)
m
H
(cid:1)
N
(cid:6)(cid:27)(cid:7)
(cid:7)b
(cid:27)
(cid:6)(cid:11)(cid:19)(cid:7)
(cid:9)
(cid:8)h
(cid:14)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
gasses
di(cid:28)ers
somewhat
from
BEC
in
a
uniform
system
that
we
discussed
so
far(cid:2)
Most
importantly(cid:1)
t
h
e
BEC
transition
is
accompanied
by
an
abrupt
change
of
density
distribution(cid:2)
This
is
due
to
the
fact
that
the
lowest
energy
quantum
state
in
which
atoms
condense
is
p
e
a
k
ed
at
the
trap
center
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
that
the
density
of
this
state(cid:1)
at
the
peak(cid:1)
(cid:4)
n
N
(cid:6)l
(cid:1)
can
b
e
extremely
high
when
the
numb
e
r
of
atoms
is
large(cid:2)
In
the
presence
of
(cid:5)
(cid:2)
(cid:9)
interactions(cid:1)
on
can
easily
reach
the
limit
when
the
interaction
energy
per
particle
is
much
larger
than
the
level
spac... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
7)
(cid:0)(cid:6)x(cid:7)
(cid:10)
(cid:1)
(cid:0)(cid:6)x(cid:7)
dx
(cid:6)(cid:11)(cid:21)(cid:7)
(cid:1)
h
(cid:8)
(cid:11)
(cid:1)
(cid:1)
(cid:8)
(cid:0)
(cid:9)m
jr
j
(cid:0)
j
j
j
j
(cid:1)
(cid:9)
Z
with
the
particle
numb
e
r
N
(cid:4)
(cid:0)
dx
being
(cid:3)xed
by
a
c
hemical
potential
(cid:5)(cid:2)
(cid:1)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
6)(cid:1)N(cid:6)m(cid:13)
(cid:7)
(cid:2)
At
large
(cid:9)
(cid:3)(cid:3)(cid:9)
(cid:1)
2
2
(cid:9)
;
h
(cid:2)
m(cid:2)
(cid:1)
(cid:1)
N
N
(cid:1)
the
value
R
is
much
larger
than
l
that
satis(cid:3)es
l
(cid:4)
l
(cid:2)
Hence
the
c
(cid:2)
(cid:1)m
(cid:1)
(cid:2)
(cid:2)
(cid:10)
2
2
(cid:2)h
m(cid:2)
(cid:1)
(c... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
each
small
part
of
the
BEC
cloud
as
a
uniform
system(cid:2)
For
the
latter(cid:1)
as
we
already
know(cid:1)
(cid:16)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
)
compensates
the
potential
U
(cid:6)x(cid:7)
variation
in
space(cid:1)
which
gives
n(cid:6)x(cid:7) (cid:4)
(cid:6)(cid:11)(cid:26)(cid:7)
(cid:0)
(cid:6)(cid:5)
U
(cid:6)x(cid:7)(cid:7)
(cid:6)(cid:1)
(cid:3) U (cid:15)
(cid:5)
(cid:15)
(cid:3)
U (cid:10) (cid:5)
(cid:4)
We
note
that
the
argument
used
to
(cid:3)nd
th... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
for
typical
density
n
(cid:4)
N(cid:6)
R
(cid:1)
which
determines
the
scale
of
spatial
nonlo(cid:0)
(cid:5)
(cid:10)
(cid:8)
(cid:4)
(cid:5)
cality
in
BEC
correlations(cid:2)
This
means
that
the
Thomas(cid:0)Fermi
approximation
is
indeed
a
local
density
approximation(cid:2)
The
above
discussion
summarizes
the
situation... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
scattering
(cid:1)
(cid:0) (cid:0)
int
k
k
4
3
k
k
(cid:4)
a
a
a
2
a
1
(cid:6)(cid:9)(cid:14)(cid:7)
H
(cid:9)
1
2
3
4
k
(cid:0)k
(cid:10)k
(cid:0)k
X
(cid:17)
... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
(cid:6)(cid:11)
(cid:10)
f
(cid:7)(cid:6)(cid:11)
(cid:10)
f
(cid:7)(cid:16) (cid:6)(cid:8)
(cid:10)
(cid:8)
(cid:8)
(cid:8)
(cid:7)
(cid:6)(cid:9)(cid:17)(cid:7)
ij(cid:5)mn
i
j
m
n
i
j
m
n
df
(cid:9)(cid:7)
i
(cid:1)
dt
h(cid:8)
(cid:0)
j
j
(cid:0)
(cid:0)
ij(cid:5)mn
X
while
the
rate
of
scattering
in
i
is
j
i
df
(ci... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
T
(cid:6)m
(cid:7)
T
Despite
scattering(cid:1)
quasiparticles
are
well
de(cid:3)ned(cid:25)
(cid:8)(cid:6)
(cid:7)
k
(cid:3)
(cid:8)
q
(cid:3)
(cid:10)
In
the
BEC
state(cid:1)
scattering
is
by
the
presence
of
the
condensate(cid:25)
stimulated
(cid:0)
(cid:0)
p
a
(cid:3)
a
a
(cid:3)
a
(cid:4)
N
(cid:6)(cid:9)(cid:23)(ci... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
6)(cid:9)(cid:7)(cid:8)
h(cid:7)
j
(cid:0)
j
(cid:0)
At
small
velocities(cid:1)
have
(cid:1)
(cid:5)
(cid:1)
(cid:9)
(cid:8)
(cid:5)
p
d
p
(cid:6)(cid:9)(cid:7)
(cid:6)(cid:16)(cid:17)(cid:8) c
h
(cid:7)T
T
T
(cid:9)
(cid:9)
(cid:4)
(cid:6)
(cid:2) f
(cid:6)(cid:2) (cid:8)
(cid:7)
(cid:4)
p
p
n
(cid:5)
(cid:5)(cid:3)(c... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
discuss
the
two(cid:0)(cid:18)uid
hydrodynamics
in
full
generality(cid:2)
Instead(cid:1)
we
describe
a
particular
phenomenon(cid:1)
the
II
sound(cid:1)
a
collective
mode
that
appears
in
the
two(cid:0)(cid:18)uid
regime(cid:2)
In
this
mode(cid:1)
the
relative
fraction
of
the
normal
and
super(cid:18)uid
component
oscilla... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
For
super(cid:18)uid
velocity(cid:1)
o
n
e
can
write
m(cid:2)
(cid:4)
(cid:5)
(cid:6)(cid:14)(cid:17)(cid:7)
v
t
s
(cid:0)r
This
relation(cid:1)
derived
above
from
the
Gross(cid:0)Pitaevskii
equation(cid:1)
is
in
fact
very
general(cid:1)
and
is
true
for
any
super(cid:18)uid(cid:2)
It
follows
from
the
relation
between
t... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
9)C
(cid:2)(cid:9)
(cid:2)T
n
T
(cid:10)
(cid:20)
(cid:20)
The
constants
c
and
c
are
the
isothermal
sound
velocity
and
the
velocity
of
temperature
(cid:3)
(cid:1)
waves
at
constant
density(cid:1)
while
C
(cid:4)
T
(cid:6)(cid:2)(cid:18)(cid:6)(cid:2)T
(cid:7)
is
the
speci(cid:3)c
heat
at
constant
v
olume(cid:1)
(cid:10... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
10)
c
(cid:10)
c
(cid:6)c
(cid:10)
c
(cid:10)
c
(cid:7)
(cid:16)c
c
(cid:6)(cid:16)(cid:19)(cid:7)
I(cid:5)
II
(cid:3)
(cid:1)
(cid:5)
(cid:3)
(cid:1)
(cid:5)
(cid:3)
(cid:1)
(cid:9)
(cid:11)
(cid:9)
(cid:0)
(cid:2)
(cid:3)
q
So
far(cid:1)
the
treatment
w
as
completely
general(cid:1)
applicable
to
any
Bose
system(cid:1... | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
(cid:31)detect
it
people
had
to
use
oscillatory
thermal
sources
and
heat
sensors(cid:2)
(cid:26) | https://ocw.mit.edu/courses/8-514-strongly-correlated-systems-in-condensed-matter-physics-fall-2003/57fb8bd5b6d291ec7eda1bc1c920a403_lec6.pdf |
18.409 An Algorithmist’s Toolkit
October 6, 2009
Lecturer: Jonathan Kelner
Scribe: Alessandro Chiesa (2009)
Lecture 8
1 Administrivia
You should probably know that
• the first problem set (due October 15) is posted on the class website, and
• its hints are also posted there.
Also, today in class there was a maj... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
subset of vertices that defines a cut with low conductance.
However, we want the running time of our algorithm to depend on the cluster size, and not on the size of
the graph. Last time we mentioned that a good example of a problem of this sort is trying to find a cluster
of web pages around mit.edu; we surely do not ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
2.3 Obstacles
We need a bound that says that our general strategy works, and that is why we proved the Lov´asz-Simonovits
theorem. However, the bound we have is global, i.e., it involves the conductance φ(G) and we do not have
the time to compute λ2 for the whole graph to bound the conductance. Moreover, if we exact... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
x, 2m − x
�
1
1 − φ(W )2
2
�t
.
Note that in the last lecture we stated a slightly weaker form of the theorem, where the conductance
ϕ(W ) of the cut (W, W ) was replaced by the conductance φ(G) of the whole graph. Nevertheless, we did
actually prove the stronger version stated above.
The bound above has noth... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
of conductance φ,
otherwise the random walk would mix too quickly.)
8-2
3 PageRank
3.1 Definition
Consider an undirected1 connected graph G = (V, E). Recall that a simple random walk on G is a walk that,
starting at some initial vertex, at each step moves from the current vertex to a randomly chosen neighbor
of ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
α(s) is the unique solution to the
following equation:
prα(s) = αs + (1 − α)W prα(s) ,
(1)
where W is the transition matrix corresponding to a lazy random walk on G.
The point is that one can show that the Lov´asz-Simonovits theorem and its corollary hold for the
PageRank vector prα(s), where s corresponds to the... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
dominant2 because the off-diagonal
elements in each column add up to 1/2, while each diagonal element is 1 − (1 − α)(1/2). By the Gershgorin
circle theorem [2], it must be nonsingular, so that the equation has a unique solution.
Proposition 2 allows us to extend the definition of PageRank: given any vector s ∈ Rn , no... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
≡ prα(W s) satisfies the following equation
Let us verify that x(cid:3) ≡ W prα(s) satisfies the same equation:
x = α(cv + dw) + (1 − α)W x .
α(cv + dw) + (1 − α)W x(cid:3) = αW s + (1 − α)W 2 prα(s)
= W (αs + (1 − α)W prα(s))
= W prα(s)
(cid:3)
= x .
By Proposition 2, the equation has a unique solution, so that ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
Given a fixed (cid:9) > 0, we keep iterating as long as there exists some vertex u such
that r(u) ≥ (cid:9)d(u).
First, we prove that each iteration of the algorithm preserves the invariant p = prα(s − r).
Proposition 5 p(cid:3) = prα(s − r(cid:3)).
8-4
Proof By Proposition 3, it suffices to show that p(cid:3) + prα... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
procedure works.
Theorem 7 Fix (cid:9) > 0. Suppose that in each iteration we pick a vertex u with the property that r(u) ≥ (cid:9)d(u).
� �
Then the process terminates in O 1
(cid:4)α
iterations with vectors p and r that satisfy the following properties:
v
r(
1. maxv d
v
(
)
)
≤ (cid:9).
2. vol(supp(p)) ≤ 1 , ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
note that every vertex in supp(p) must have been picked at least once during the
execution of the algorithm, so that
�
T
i
=1
i=1
(cid:4)α
thus showing (2), and completing the proof of the theorem.
i=1
vol(supp(p)) ≤
di ≤
T �
1
(cid:9)α
,
The theorem we just proved gives the approximation to the PageRank v... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
total running time is almost linear because the running time on each chunk
is almost linear in its volume.
In a random walk scheme, we need to take 1/φ steps in order to get a cut of conductance
Caveat.
√
1/ φ; hence, that takes time that is about (size of chunk) · poly(1√/φ). Similarly, in a PageRank scheme,
we ... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
out a lot of edges from G and still get an approximate
answer, because the running time of the algorithm for the resulting graph will be close to that for a sparse
graph. More precisely, is there any way to “approximate” our graph G with a sparse graph G(cid:3) that has the
property that all of its cuts have more or... | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-409-topics-in-theoretical-computer-science-an-algorithmists-toolkit-fall-2009/5823d396bfe2ae2c5bea9c431601276e_MIT18_409F09_scribe8.pdf |
MIT 2.852
Manufacturing Systems Analysis
Lecture 14-16
Line Optimization
Stanley B. Gershwin
Spring, 2007
Copyright c�2007 Stanley B. Gershwin.
Line Design
• Given a process, find the best set of machines and buffers on
which it can be implemented.
• Best: least capital cost; least operating cost; least average ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
(ie, required accuracy).
• Reducing computation time per iteration is accomplished by
� using analytical models rather than simulations
� using coarser approximations in early iterations and more accurate
evaluations later.
Copyright c
�2007 Stanley B. Gershwin.
5
Problem
Statement
X is a set of possible choices. ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Copyright c
�2007 Stanley B. Gershwin.
9
Continuous
Variables and
Objective
Assume f (t) is decreasing.
• Binary search: Guess t0 and
t1 such that f (t0) > 0 and
f (t1) < 0. Let
t2 = (t0 + t1)/2.
� If f (t2) < 0, then repeat
�
1
=
t2.
with t
= t0 and t
�
0
� If f (t2) > 0, then repeat
=
t1.
t2 and t
�
w... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
99237060547
2.00000381469727
1.99999809265137
2.00000095367432
t1
3
3
2.25
2.25
2.0625
2.0625
2.015625
2.015625
2.00390625
2.00390625
2.0009765625
2.0009765625
2.000244140625
2.000244140625
2.00006103515625
2.00006103515625
2.00001525878906
2.00001525878906
2.00000381469727
2.00000381469727
Copyright �2007 Stanley B... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
f(t1 )
Unconstrained
One-dimensional search
f(t)
t0
t2
t1
t
� Repeat with t�
0 = t1 and
t� = t2 until |f (t� )| is small
1
enough.
0
Copyright �2007 Stanley B. Gershwin.
c
14
Continuous
Variables and
Objective
Unconstrained
One-dimensional search
Example:
f (t) = 4 − t2
t0
0
3
1.33333333333333 ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Constrained
Constrained
Optimum
Equality constrained: solution is
on the constraint surface.
x2
h(x , x ) = 0
1
2
x1
Problems are much easier when
constraint is linear, ie, when the
surface is a plane.
• In that case, replace �J/�x
by its projection onto the
constraint plane.
• But first: find an initial
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
+ y) − (x4 + y4)/4
subject to x − (x − y)2 + 1 ≈ 0
−6
−4
−2
0
2
4
6
−2
0
2
4
6
Solving a nonlinearly-constrained problem is not so easy.
Searching within the boundary is numerically difficult.
Copyright �2007 Stanley B. Gershwin.
c
21
Continuous Nonlinear and Linear Programming
Variables and
Objecti... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
g(x) � 0
f (x) � F
Copyright �2007 Stanley B. Gershwin.
c
25
Buffer Space
Allocation
Problem statement
M
1
B
1
M
2
B
2
M
3
B
3
M
4
B
4
M
5
B
5
M6
Problem: Design the buffer space for a line. The
machines have already been selected. Minimize the
total buffer space needed to achieve a target
pr... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
appears to be a concave function of the
vector (N1, ..., Nk−1).
Copyright �2007 Stanley B. Gershwin.
c
28
Buffer Space
Allocation
Properties of P (N1, ..., Nk−1)
Example — 3-machine line
P
0.91
0.9
0.89
0.88
0.87
0.86
0.85
0.84
0.83
10 20 30 40 50 60
N1
70 80 90
100
30
20
10
Optimal curve
P=0.8800 ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
many iterations.
It would be desirable to transform this problem into one with linear constraints.
Copyright �2007 Stanley B. Gershwin.
c
30
Buffer Space
Allocation
Solution
Dual problem
Maximize P (N1, ..., Nk−1)
subject to
k−1
�
i=1
Ni � N TOTAL specified
Ni � N MIN, i = 1, ..., k − 1.
All the constrain... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
N, . . . , Nk−1) − P (N1, . . . , Ni, . . . , Nk−1)
�N
• Calculate the projected gradient vector (ˆg1, ..., gˆk−1):
gˆi = gi − g¯ where g¯ =
1
k − 1
k−1
�
i=1
gi
Copyright �2007 Stanley B. Gershwin.
c
34
Buffer Space
Allocation
Solution
Dual Algorithm
• The projected gradient gˆ satisfies
k−1
k−1
k−1... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
point N on the constraint plane, the
best improvement is to move in the direction of gˆ; that is,
N + Agˆ.
• To find the best possible improvement, we find A�, the value of
A that maximizes P (N + Agˆ). A is a scalar, so this is a
one-dimensional search.
• N + A�gˆ is the next guess for N , and the process repeats.... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
imal algorithm
e
t
a
r
n
o
i
t
c
u
d
o
r
p
e
g
a
r
e
v
a
m
u
m
x
a
M
i
0.92
0.9
0.88
0.86
0.84
0.82
0.8
0.78
0
50
100
Total buffer space
150
200
P MAX(N TOTAL) as a function of N TOTAL.
Copyright �2007 Stanley B. Gershwin.
c
40
Buffer Space
Allocation
Solution
Primal algorithm
Then, we can... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
i = 1, ..., 19
50
40
30
20
10
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
0
0
2
4
6
8
10
Buffer
12
14
16
18
20
Copyright �2007 Stanley B. Gershwin.
c
43
Buffer Space
Allocation
Example
The “Bowl Phenomena”
• This shows the optimal distribution of buffer space and the resulting
distribution of av... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
a
R
0
0
5
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
46
Buffer Space
Allocation
Example
• Design the buffers for a 20-machine production line.
• The machines have been selected, and the only
decision remaining is the amount of space to
allocate for in-process inventory.
• The goal ... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
minute).
Copyright c
�2007 Stanley B. Gershwin.
51
Example
Buffer Space
Allocation
Are buffers really needed?
Line
Case 1
Case 2
Case 3
Production rate with no buffers,
parts per minute
.487
.475
.475
Yes. How were these numbers calculated?
Copyright c
�2007 Stanley B. Gershwin.
52
Example
Buffer Space
Al... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
0.45
0.4
0.35
e
z
S
i
r
e
f
f
u
B
/
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
=
o
i
t
a
R
0.3
0
5
10
Buffer
15
20
Copyright �2007 Stanley B. Gershwin.
c
55
Buffer Space
Allocation
• Case 4: Same as Case 3 except bottleneck is at Machine 15.
• This shows the optimal distribution of buffer space and t... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
c
57
Buffer Space
Allocation
• Case 5: MTTF bottleneck at Machine 5, MTTR bottleneck at Machine 15.
• This shows the optimal distribution of buffer space and the resulting
Example
distribution of average inventory for Case 5.
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
/
e
z
S
i
r
e
f
f
u
B
55
50
45
40
35
... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
50
45
40
35
30
25
20
15
10
5
0
0
10
20
30
40
50
Buffer
Copyright �2007 Stanley B. Gershwin.
c
60
Buffer Space
Allocation
Example
• This shows the ratio of average inventory to buffer size with optimal buffers
for Case 6.
e
z
S
i
r
e
f
f
u
B
/
l
e
v
e
l
r
e
f
f
u
b
e
g
a
r
e
v
A
=
o
i
t
a
R... | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Spring 2010
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/2-852-manufacturing-systems-analysis-spring-2010/5843df20ba98a98f34469fc47801c1e6_MIT2_852S10_line_opt.pdf |
Lecture 5
8.821/8.871 Holographic duality
Fall 2014
8.821/8.871 Holographic duality
MIT OpenCourseWare Lecture Notes
Hong Liu, Fall 2014
Lecture 5
Reminder from last lecture
The vacuum of Minkowski space can be viewed as an entangled state of left Rindler patch and right Rindler patch
|0(cid:105)M ∝
(cid:88)
n
e−πEn |n... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
:105)M = |0(cid:105)M
This can also be seen geometrically: η translation is a boost in (X, T ), i.e. HRind generates a boost. |0(cid:105)M is
clearly invariant under a boost. Yet boosts act oppositely in the right and left quadrants as indicated in
Fig. 1.
2. If we expand φR (φL) in terms of modes in the right (left) q... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
dE
=
1
T (E)
=
8πGN m
(cid:126)
since for a black hole E = m
ˆ
S(E) =
dE
T (E)
=
4πGN E2
(cid:126)
+ const =
4πr2
s
4(cid:126)GN
=
AH
e(cid:126)GN
2
Lecture 5
8.821/8.871 Holographic duality
Fall 2014
The integral constant can be determined to be 0 since S(E) = 0 for E = 0, AH is the area of black hole horizon.
So we ... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
of thermodynamics with the identification Eq. 1. In particular the 1st law
becomes
dE = T dS + ΩdJ + ΦdQ
Historically, before Hawking’s discovery of black hole radiation, Bekenstein (1972-1974) has found SBH ∝ AH , the
motivation is to save the 2nd law of thermodynamics for a system with black holes. if an ordinary syst... | https://ocw.mit.edu/courses/8-821-string-theory-and-holographic-duality-fall-2014/584b62f85f319bd167efb65047336e0c_MIT8_821S15_Lec5.pdf |
Lecture #2: Background
READINGS AND FIGURES
Readings: Instructor notes on boundary conditions
Frisk, Chapter 3, sections 3.1, 3.2 and 3.3 (up to Eq. 3.38.)
---------------------------------------------------------
DESCRIPTION OF NOTES
Having the basic wave equation for ocean acoustics in hand, we now need to put ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
conditions for some calculations. The discussion in this section of notes loosely follows from Clay and
Medwin’s “Acoustical Oceanography.” (Reference in separate attachment.)
The next small class notes section concerns the “specific acoustic impedance.” In direct analogy
to Ohm’s law, E=I*R, the resistance of the a... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
is “intermediate” to the two above, called a
“mixed” or impedance BC. Frisk shows this type in section 3.2.3. This type of BC is actually capable of
representing reflection from a complex, fully layered medium, and finds wide use in ocean acoustics.
The above three boundary conditions are well known mathematically a... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58540cfbbdd42be15697f7a1523f343b_MIT2_682S12_bglec02.pdf |
Lecture #1: Background
READINGS AND FIGURES
Readings: Frisk, Chapter 1, sections 1.3 and 1.4; Chapter 2, sections 2.1, 2.2, 2.3, 2.4.
Figures used: COA, Figures 1.1, 1.2, 1.3, 1.5
---------------------------------------------------------
DESCRIPTION OF NOTES
The first two pages of the notes are my own answers to ... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
, scattering
theory, and so on. The analogies to quantum mechanical systems are very close in many ways, and many
researchers have exploited this.
The next few pages are my own descriptions of the first few figures from COA (Figs 1.1, 1.2, 1.3,
1.5. The message in all these figures is that, to quote the old Marshal... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
in cylindrical coordinates, which just turns out to be the
familiar Bessel’s equation. The exact and asymptotic forms of the solution are shown, and a little
emphasis made that the spreading law is cylindrical (intensity falls as R, not R^2).
2
MIT OpenCourseWare
http://ocw.mit.edu
2.682 Acoustical Oceanogra... | https://ocw.mit.edu/courses/2-682-acoustical-oceanography-spring-2012/58aeef5d8bd077782924c431bad2f2db_MIT2_682S12_bglec01.pdf |
MIT 3.016 Fall 2005
c
� W.C Carter
Lecture 3
16
Sept. 12 2005: Lecture 3:
Introduction to Mathematica II
Functions and Rules
Besides Mathematica r� ’s large set of builtin mathematical and graphics functions, the
most powerful aspects of Mathematica r� are its ability to recognize and replace patterns
and to ... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
of 10, which is probably
not what was intended. It is much safer to localize variables—in other words, to limit the
scope of their visibility to only those parts of the program that need the variable. Sometimes
this is called a “Context” for the variable in a programming language; Mathematica r� has
contexts as wel... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
5
c
� W.C Carter
Lecture 3
19
Mathematica r� Example: Lecture03
Patterns and Replacement
The real power of patterns and replacement is obtained when defining your own functions.
Mathematica r� Example: Lecture03
Functions with Patterns
It is probably a good idea to define all function with delayed assignment (:... | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
fined so that they can be
reused, either by you or someone else. The conditions for which the function can work should
probably be encoded into the function. In Mathematica r� this can be done with restricted
patterns:
Mathematica r� Example: Lecture03
Functions with Restricted Patterns | https://ocw.mit.edu/courses/3-016-mathematics-for-materials-scientists-and-engineers-fall-2005/58b2b2f2478ab6e14599e1865b1e233e_lecture_03.pdf |
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-1
Lecture 5 - Carrier generation and
recombination (cont.)
February 14, 2007
Contents:
1. G&R rates outside thermal equilibrium (cont.)
2. Dynamics of excess carriers in uniform situations
3. Surface generation and recombination
Reading assig... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
Et
Ev
n>no
ro,ec>ro,ee
ro,hc>ro,he
p>po
thermal equilibrium
with excess carriers
Out of equilibrium, if rate constants are not affected:
rec = cen(Nt − nt)
ree = eent = cenint
rhc = chpnt
rhe = eh(Nt − nt) = chni(Nt − nt)
Recombination: capture of one electron + one hole ⇒
net recombination rate = net elect... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
n-type:
- for p-type:
po (cid:4) n (cid:3) (cid:5) p (cid:3) (cid:4) no
no (cid:4) n (cid:3) (cid:5) p (cid:3) (cid:4) po
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloade... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
(cid:3)
n
τ
1
τ
= Σ
1
τi
The G&R process with the smallest lifetime dominates.
Physical meaning of carrier lifetime:
• U is net recombination rate in unit volume in response to excess
carrier concentration n(cid:3) (linear in n(cid:3)) [cm−3 · s−1]
• n
U
(cid:3) is net recombination rate in unit volume per u... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
∝ Nt
Schroder, D. K. "Carrier Lifetimes in Silicon." IEEE Transactions on Electron Devices 44, no. 1 (1997): 160-170.
Copyright 1997 IEEE. Used with permission.
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusett... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
1E+19 1E+20 1E+21
donor concentration (cm-3)
acceptor concentration (cm-3)
For low doping levels, NA,D < 1017 cm−3 , τtr dominates:
• τ depends on material quality and process → wide data scatter
• Nt correlated with NA,D → τ ∝ N −1
A,D
For high doping levels, NA,D > 1018 cm−3 , τAuger dominates:
• ”intrinsic” ... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
combination,
carrier concentrations change in time:
dn
dt
=
dp
dt
= G − R
• if G > R ⇒ n, p ↑
• if G < R ⇒ n, p ↓
Distinguish between internal and external generation:
G = Gext + Gint
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (htt... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
few τ ’s)
In steady state:
or
Then
generation = recombination
gl =
(cid:3)
n
τ
n (cid:3) = glτ
Cite as: Jesús del Alamo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
amo, course materials for 6.720J Integrated Microelectronic Devices, Spring 2007.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
6.720J/3.43J - Integrated Microelectronic Devices - Spring 2007
Lecture 5-17
Two extreme cases:
n'(t)
glT
τ1>>T
τ2
>>... | https://ocw.mit.edu/courses/6-720j-integrated-microelectronic-devices-spring-2007/58cdd6312da243d9062e56ceee80f955_lecture5.pdf |
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