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Chapter 2
Fine Structure
B. Zwiebach
c
(cid:13)
2.1 Review of hydrogen atom
The hydrogen atom Hamiltonian is by now familiar to you. You have found the bound state
spectrum in more than one way and learned about the large degeneracy that exists for all
states except the ground state. We will call the hydrogen atom Hami... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
2. Then,
e2
~c ≃
1
137
,
α
≡
e2
a0
=
me4
~2 =
mα2~2c2
~2
= α2 mc2 .
(2.1.4)
(2.1.5)
This states that the energy scale of hydrogen bound states is a factor of α2 smaller than
the rest energy of the electron, that is, about 19000 times smaller. We can thus rewrite the
possible energies as:
The typical momentum in the hyd... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
1, . . . , n
1
−
m =
−
ℓ, . . . , ℓ
# of states with energy En =
n
−
1
(2ℓ + 1) = n2
Xℓ=0
The states of hydrogen are shown in this energy diagram, which is not drawn to scale,
2.1. REVIEW OF HYDROGEN ATOM
27
...
n = 4
n = 3
n = 2
n = 1
S
ℓ = 0
...
P
ℓ = 1
...
D
ℓ = 2
...
F
ℓ = 3
...
N = 3 N = 2 N = 1 N = 0
N = 2 N = 1... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
of n can be read from the exponential factor. The value of ℓ can be read from
the radial prefactor, or from the spherical harmonic. The value of m can be read from
the spherical harmonic. For the ground state n = 1, ℓ = 0 and m = 0. The normalized
wavefunction is
na0 Yℓ,m(θ, φ) ,
(2.1.11)
e−
−
(cid:19)
·
r
(2.1.12)
Com... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
following analysis. It will determine the fine-structure of the hydrogen atom. The
corrections will break much of the degeneracy of the spectrum.
5. In order to understand better the spectrum and the properties of the Hydrogen atom
one can apply an electric field, leading to the Stark effect or a magnetic field, leading
to... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
states are eigenstates of ˆJ2 and ˆJz,
where the total angular momentum ˆJ is obtained by adding the orbital angular momentum
ˆL to the spin angular momentum ˆS:
ˆJ = ˆL + ˆS .
(2.1.15)
2.1. REVIEW OF HYDROGEN ATOM
29
s we are tensoring a a full ℓ multiplet to an s multiplet (here, of course,
When we form ℓ
s are eige... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
j = 1/2 multiplet. We use the notation Lj for the coupled
multiplets, with L = S, P, D, F for ℓ = 0, 1, 2, and 3 (see (2.1.10). The change of basis is
summarized by the replacements
ℓ
⊗
1
2 →
L(ℓ)
j=ℓ+
1
2 ⊕
L(ℓ)
j=ℓ
1
2
−
(2.1.18)
or more explicitly,
1
0
⊗
⊗
P 3
S 1
2
1
2 →
1
2 →
1
2 →
1
2 →
Thus, by the time we combi... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
n = 1
S
ℓ = 0
...
4S 1
2
3S 1
2
(2)
2S 1
2
(2)
1S 1
2
(2)
The number of states is indicated in parenthesis.
2.2 The Pauli equation
µ
In the hydrogen atom the spin-orbit coupling arises because the electron is moving in the
electric field of the proton. Since the electron is moving relative to the frame where we have
a s... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
ec
µ = 2 −
ˆS
~ =
1
2 σ =
ˆS =
σ .
−
−
−
2
2
µB =
e~
2mec ≃
9.274
10−
×
21 erg
gauss
= 5.79
9
10−
×
eV
gauss
.
31
(2.2.3)
(2.2.4)
(for SI values use Tesla = 104 gauss). The coupling of an electron to an external magnetic
field is therefore represented by a Hamiltonian HB given by
HB =
µ
−
·
B =
e~
2mec
B .
σ
·
(2.2.5)
O... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
that the Hamiltonian (2.2.8) can be rewritten as
(σ
ˆp)
·
·
(σ
·
ˆp) = ˆp2 12
×
2 .
H =
1
2m
(σ
·
ˆp)(σ
ˆp) .
·
(2.2.10)
(2.2.11)
(2.2.8)
(2.2.9)
32
CHAPTER 2. HYDROGEN ATOM FINE STRUCTURE
So far, this is all just rewriting with no change in physics. But new things happen when we
the particle is charged and we couple ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
π
= 0 because the various πi do not commute. Note that the replacement
ˆπ term.
×
ˆπ
×
To evaluate that term we use
( ˆπ
×
ˆπ)k = ǫijk ˆπiˆπj = 1
2 ǫijk[ˆπi, ˆπj] .
(2.2.15)
The commutator here is
[ πi , πj ] =
pi −
q
c
Ai , pj −
q
c
h
i
Aj
.
(2.2.16)
As usual, the ˆp components can be thought of as derivatives acting ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
e
c
i~q
c
i
2m
e~
2mc
σ
B
·
−
eΦ(ˆx) .
B
σ
·
−
eΦ(ˆx)
(2.2.20)
(cid:17)
The second term in this expanded Pauli Hamiltonian gives the coupling of the electron spin
to the magnetic field and agrees precisely with the expected coupling (2.2.5). We thus see
that the Pauli equation predicts the g = 2 value in the electron ma... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
Hamiltonian,
and the next term is the first nontrivial relativistic correction. For small momenta we will
treat that term as a perturbation.
More elegantly, Dirac wanted to find a Hamiltonian linear in momenta and without
square roots. This would be possible if one could write the relativistic energy as the square
of a l... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
is
·
(cid:1)
where Ψ is a Dirac spinor, a four-component column vector that can be thought to be
composed by two two-component Pauli spinors χ and η:
(cid:0)
(2.3.9)
i~ ∂Ψ
∂t
=
cα
p + βmc2
Ψ ,
Ψ =
χ
η
(cid:18)
(cid:19)
,
χ =
χ1
χ2(cid:19)
(cid:18)
,
η =
η1
η2(cid:19)
(cid:18)
.
(2.3.10)
The coupling to electromagnetic ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
)
(2.3.14)
6
2.4. FINE STRUCTURE OF HYDROGEN
35
The first correction is the relativistic energy correction anticipated earlier. The second is
the spin-orbit coupling, and the third is the Darwin correction, that as we shall see affects
only ℓ = 0 states.
Recall that the energy scale for H (0) eigenstates is α2mc2. We wi... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
δHDarwin =
e2~2
8m2c2 ∇
−
e2~2
8m2c2 (
−
To estimate this correction note that, due to the δ function the the integral in the expecta-
3
0 . We will therefore have
a−
tion value will introduce a factor
e2~2
m2c2a3
e2~2
m2c2 δ(r) .
δHDarwin ∼
2
ψ(0)
|
|
4πδ(r)) =
α4 mc2 ,
(2.3.19)
(2.3.20)
π
2
∼
=
(cid:19)
(cid:18)
0 ∼
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
function
at the origin, the first order correction to the energy vanishes unless the wavefunction is
non-zero at the origin. This can only happen for nS states. There is no need to use the
apparatus of degenerate perturbation theory. Indeed, for fixed n there are two orthogonal
ℓ = 0 states, one with electron spin up and... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
electron is no longer a point particle. Still the fact remains
that in a relativistic treatment of an electron, its Compton wavelength is relevant and is
physically the shortest distance an electron can be localized.
The potential energy V (r) of the electron, as a point particle, is the product of the
e) times the ele... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
−
V (r) =
d3u ρ0(u)V (r + u) .
(2.4.28)
Zelectron
e
This equation has a clear interpretation: the potential energy is obtained as a weighted
integral of potential due to the proton over the extended electron. If the electron charge
would be perfectly localized, ρ0(u) = δ(u) and ˜V (r) would just be equal to V (r). We w... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
u ρ0 (u) ρi + 1
2
∂i∂jV
Z
Xi,j
d3u ρ0 (u) uiuj +. . . (2.4.32)
r
Z
Due to spherical symmetry the first integral vanishes and the second takes the form
r
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
d3u ρ0 (u) uiuj = 1
3 δij
d3u ρ0 (u) u2 .
Z
Z
(2.4.33)
Indeed the integral must vanish for i
Since u2 = u2
1 + u2
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
u2 =
Z
u0
4πu2du u2
4π
3 u3
0
=
u0
3
u3
0 Z
0
0
Z
u4du = 3
5 u2
0 .
(2.4.37)
6
2.4. FINE STRUCTURE OF HYDROGEN
Therefore,
2V .
39
(2.4.38)
If we choose the radius u0 of the charge distribution to be the Compton wavelength ~
the electron we get,
mc of
δV = 1
10 u2
0 ∇
δV =
~2
10 m2c2 ∇
2V .
(2.4.39)
Comparing with (2.3... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
the matrix element we use the Hermiticity of p2 to move one of the factors
into the bra
1
8m3c2 h
where in the right-hand side we evaluated the trivial expectation value for the spin degrees
of freedom. To simplify the evaluation we use the Schr¨odinger equation, which tells us that
p2ψnℓmi
n,ℓmℓms;rel =
p2ψnℓm|
(2.4.4... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
computation of the expectation value of V (r) and
V 2(r) in the ψnℓm state. The expectation value of V (r) is obtained from the virial theorem
that states that
n . For V 2(r) we have
= 2E(0)
V
h
i
V 2
h
i
= e4
1
r2
D
E
= e4
a2
0n3
1
ℓ + 1
2
Back into (2.4.44) we find
(cid:0)
(cid:1)
=
e2
2a0
1
n2
2 4n
ℓ + 1
2
(cid:19)
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
of fixed n and ℓ, δHrel is in fact a multiple of the identity matrix,
since the matrix elements are independent of m and ms (the Lz and Sz) eigenvalues. A
matrix equal to a multiple of the identity is invariant under any orthonormal change of
1
2 multiplet, the resulting j multiplets provide an alternative orthonormal
b... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
si
(2.4.50)
2f (n, ℓ) = f (n, ℓ) .
ci|
|
2.4.3 Spin orbit coupling
The spin-orbit contribution to the Hamiltonian is
δHspin-orbit =
e2
2m2c2
1
r3
L .
S
·
(2.4.51)
Note that δHspin-orbit commutes with L2 because L2 commutes with any ˆLi and any ˆSi.
Moreover, δHspin-orbit commutes with J2 and with Jz since, in fact, [ ˆ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
1/r3 in these states. It is known that
nℓmℓ
.
(2.4.53)
1
ℓ + 1
2
n3a3
0ℓ
(ℓ + 1)
D
(cid:12)
(cid:12)
(cid:12)
Because of the mℓ independence of this expectation value (and its obvious ms independence)
1
the operator 1/r3 is a multiple of the identity matrix in each ℓ
2 multiplet. It follows that
it is the same multiple... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
identically acting on any ℓ = 0 state, it is physically reasonable, as we will
do, to assume that the spin-orbit correction vanishes for ℓ = 0 states. On the other hand
0, while somewhat ambiguous, is nonzero. We set
the limit of the above formula as ℓ
1
j = ℓ + 1
2 (the other possibility j = ℓ
2 does not apply for ℓ =... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
. The result,
therefore will give the shifts of the coupled states. Collecting our results (2.4.46) and (2.4.56)
we have
nℓjmj
D
δHrel + δHspin-orbit
(cid:12)
(cid:12)
(cid:12)
(E(0)
n )2
2mc2
=
−
3
(
nℓjmj
(cid:12)
4n
(cid:12)
(cid:12)
ℓ + 1
2
E
+
2n
j(j + 1)
−
ℓ + 1
2
ℓ
(cid:2)
ℓ(ℓ + 1)
(ℓ + 1)
3
4
−
)
(cid:3)
(2.4.5... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
brackets above
2 or the multiplet can have ℓ = j + 1
−
1
f (j, ℓ)
≡
j(j + 1)
ℓ
−
ℓ + 1
2
3ℓ(ℓ + 1)
(ℓ + 1)
3
4
.
−
(2.4.60)
The evaluation of this expression in both cases gives the same result:
(cid:0)
(cid:1)
=
=
ℓ=j
1
2
−
ℓ=j+ 1
2
f (j, ℓ)
(cid:12)
(cid:12)
(cid:12)
f (j, ℓ)
(cid:12)
(cid:12)
(cid:12)
1
−
j
3(j
1
2
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
(fs) shifts:
E(1)
nℓj,mj;fs =
(E(0)
n )2
2mc2
−
4n
j + 1
2 −
h
3
=
i
α4(mc2)
−
1
2n4
h
n
j + 1
2 −
3
4
.
i
More briefly we can write
E(1)
nℓj,mj;fine =
α4mc2
−
Sn,j , with Sn,j ≡
·
1
2n4
h
n
j + 1
2 −
3
4
.
i
Let us consider a few remarks:
(2.4.62)
(2.4.63)
1. The dependence on j and absence of dependence on ℓ in the ene... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
n
j + 1
2 ≥
n
jmax + 1
2
=
n
ℓmax + 1
2 + 1
2
=
n
n
= 1
→
n
j + 1
2 −
3
4 ≥
1
4 .
(2.4.64)
5. For a given fixed n, states with lower values of j get pushed further down. As n
increases splittings fall off like n−
3.
A table of values of Sn,j is given here below
1
n j Sn,j
1
2
1
2
1
8
5
128
2
3
3
2
1
2
3
2
5
2
1
128
1
72
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
2P1/2 and is of
order α5. There is also hyperfine splitting, which arises from the coupling of the magnetic
moment of the proton to the magnetic moment of the electron. Such coupling leads to a
splitting that is a factor me/mp smaller than fine structure.
2.5 Zeeman effect
In remarkable experiment done in 1896, the Dutch ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
+ 2 ˆSz) .
(2.5.3)
When we consider the Zeeman effect on Hydrogen we must not forget fine structure.
The full Hamiltonian to be considered is
H = H (0) + δHfs + δHZeeman .
(2.5.4)
(2.5.2)
46
CHAPTER 2. HYDROGEN ATOM FINE STRUCTURE
Recall that in fine structure, there is an internal magnetic field Bint associated with spin... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
eman magnetic field is neither weak nor strong, we must take the sum
of the Zeeman and fine structure Hamiltonians as the perturbation. No simplification is
possible and one must diagonalize the perturbation.
Weak-field Zeeman effect. The approximate eigenstates of ˜H (0) are the coupled states
that exhibit fine structure co... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
atisfies
(2.5.9)
( ˆLz + 2 ˆSz)
nℓjmji
.
|
It follows that we only need to concern ourselves with ˆSz matrix elements.
nℓjmj|
h
nℓjmj|
h
nℓjmji
= ~m +
ˆSz|
Let’s talk about vector operators. The operator ˆV is said to be a vector operator
under an angular momentum operator ˆJ if the following commutator holds for all va... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
side of (2.5.12) on
such eigenstates is necessarily zero:
(cid:0)
(cid:1)
k′; jm′j|
h
h
ˆJ2 , [ˆJ2 , ˆV]
i
k; jmji
|
= 0 ,
(2.5.13)
as can be seen by expanding the outer commutator and noticing that ˆJ2 gives the same
eigenvalue when acting on the bra and on the ket. Therefore the matrix elements of the
right-hand side... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
izing
to the z-component
·
nℓ jmj|
h
ˆSz|
nℓ jmji
=
~mj h
ˆS
ˆJ
nℓ jmji
nℓ jmj|
|
~2 j(j + 1)
·
.
(2.5.17)
We already see the appearance of the predicted ~mj factor. The matrix element in the
numerator is still to be calculated but it will introduce no mj dependence. In fact ˆS
ˆJ is a
scalar operator (it commutes with... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
finally have for the Zeeman energy shifts in (2.5.8)
E(1)
nℓjmj
=
e~
2mc
B gJ (ℓ) mj .
(2.5.21)
(2.5.22)
Here the Bohr magneton e~
9eV/gauss. This is our final result for the weak-
field Zeeman energy corrections to the fine structure energy levels. Since all degeneracies
within j multiplets are broken and j multiplets wit... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
ˇH (0) and have
energies
Enℓmℓms = E(0)
n +
B(mℓ + 2ms) .
(2.5.26)
e~
2mc
Some of the degeneracy of H (0) has been removed, but some remains. For a fixed principal
1
quantum number n there are degeneracies among ℓ
2 states and degeneracies among such
multiplets with ℓ and ℓ′ different. This is illustrated in Figure 2.2.
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
Ignoring ℓ = 0 states, we have to re-examine the relativistic
correction and spin orbit. The relativistic correction was computed in the uncoupled basis
and one can use the result because the states are unchanged and the perturbation was
shown to be diagonal in this basis. For spin-orbit the calculation was done in the... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2018/439fc15f7ed10dd76c69dc3f7fea600e_MIT8_06S18ch2.pdf |
8.04: Quantum Mechanics
Massachusetts Institute of Technology
Professor Allan Adams
2013 February 5
Lecture 1
Introduction to Superposition
Assigned Reading:
E&R 16,7, 21,2,3,4,5, 3all NOT 4all!!!
1all, 23,5,6 NOT 2-4!!!
Li.
12,3,4 NOT 1-5!!!
Ga.
3
Sh.
I want to start by describing to you, the readers, ... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
though, here is what is
inside the boxes: a magnetic field B = B0ez splits electrons by color, such that
electrons of one color are sent in one direction toward a screen, and electrons of
the other color are sent in another direction toward the same large screen. The
electrons of each color always land in the same p... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
being male and being a bachelor are
correlated. So are color and hardness correlated? Well, this is easy to test with boxes!
Empirically, it is found that if one property is measured and all the electrons with a single
value of that property have the other property measured, the value of the other property is
found... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
exit as black.
This is completely and utterly wrong! In fact, half of the electrons that were previously
measured to only be white as white, and half now exit as black! The same goes for any
other pair of results from the first two boxes, if hardness and color were interchanged, et
cetera. Apparently, the presence o... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
able. For instance, in the displayed apparatus, if the hardness and color are measured of the
output that should be hard and black, all are hard, but only half are black.
Thus, simultaneously measuring hardness and color is fundamentally disallowed! The general
statement of this is the uncertainty principle, which i... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
electrons! Half take the hard output path,
and half take the soft output path. Those that take the hard path exit as hard, while those
that take the soft path exit as soft, so the final output should have a measurement of half
of the electrons as hard and half as soft. Indeed, this is empirically correct!
Second, le... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
a given wall “out” means the system is unchanged from before. Having a given wall
“in” means that path is blocked.
Figure 11: Block soft particle path
Fourth, let us send in white electrons and measure the color at the end while having a wall
in the soft path.
We expect that the overall output will decrease by hal... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
they take is not an individual path, not both paths, and not no path at all. There
do not appear to be any other logical possibilities, so what are they doing anyway?
If the experiments are accurate and the arguments correct, the electrons are in fact doing
8.04: Lecture 1
9
something we have never dreamed of bef... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
in a superposition of the two
values, with the probability that we subsequently measure it to have one value or the other
depending on the details of the superposition. For instance, an electron that is white is in
an equal superposition of being hard and soft, as the probability of measuring each value of
hardness... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
Ware
http://ocw.mit.edu
8.04 Quantum Physics I
Spring 2013
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/43a8712da99ace660cf042c1f1371b46_MIT8_04S13_Lec01.pdf |
Extrinsic Semiconductors
• Adding ‘correct’ impurities can lead to controlled domination of one carrier type
– n-type is dominated by electrons
– p-type if dominated by holes
• Adding other impurities can degrade electrical properties
Impurities with close electronic
structure to host
isoelectronic
hydrogenic
... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
99 E.A. Fitzgerald
2
The Power of Doping
• Can make the material n-type or p-type: Hydrogenic impurities are nearly fully
ionized at room temperature
2 for Si: ~1020cm-3
– ni
– Add 1018cm-3 donors to Si: n~Nd
– n~1018cm-3, p~102 (ni
2/Nd)
• Can change conductivity drastically
– 1 part in 107 impurity in a crystal ... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
+
1
+ ...
1
vthσi Ni
τi =
li =
vth
liσi Ni = 1
where σ is the cross-section of the scatterer, N is the
number of scatterers per volume, and l is the average
distance before collisions
The mechanism that will tend to dominate the scattering will be the mechanism with the
shortest l (most numerous), unless there... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
•Barrier to capture carrier is Eg/2
•Since the probability of the carrier transition is ~e-ΔE/kT, trapping a carrier with a deep state is
very probable
•A trapped carrier can then help attract another carrier, increasing recombination through the
deep state
1
1
=
ττ
int rinsic
+
1
τ
deep
©1999 E.A. Fitzgera... | https://ocw.mit.edu/courses/3-225-electronic-and-mechanical-properties-of-materials-fall-2007/43fea384070aa4c2f247637091f10d55_lecture_8.pdf |
6.S096 Lecture 1 – Introduction to C
Welcome to the Memory Jungle
Andre Kessler
Andre Kessler
6.S096 Lecture 1 – Introduction to C
1 / 30
Outline
1 Motivation
1
2 Class Logistics
2
3 Memory Model
3
4 Compiling
4
5 Wrap-up
5
Andre Kessler
6.S096 Lecture 1 – Introduction to C
2 /
30
First Example (Pyth... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
6.S096 Lecture 1 – Introduction to C
4 /
30
Motivation
Why C or C++?
Speed
Graph of prog
Source: http://benchmarksgame.alioth.debian.org/u64q/which-programs-are-fastest.php.
ram s peed acro ss language im plementations remove d due to copy right r estrictions.
Andre Kessler
6.S096 Lecture 1 – Introduction to C... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
.
Andre Kessler
6.S096 Lecture 1 – Introduction to C
/
9
30
Class Logistics
Course Syllabus
Day
1
2
3
4
5
6
7
8
9
10
Topic
Introduction to C: memory and the compiler
Subtleties of C: memory, floating point
Guest lectures: Assembly and Secure C
Transition from C to C++
Object-oriente... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
Andre Kessler
6.S096 Lecture 1 – Introduction to C
12
/
30
Class Logistics
The Minimal C Program
nothing.c: takes no arguments, does nothing, returns 0 (“exit success”)
int main(void) {
return 0;
}
1
1
2
2
3
3
4
4
To compile: make nothing
Previous step produced an executable named nothing
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
6
/
30
Class Logistics
Hello, world!
hello.c: takes no arguments, prints “Hello, world!”, returns 0
#include <stdio.h>
int main(void) {
printf( "Hello, world!\n" );
return 0;
}
1
1
2
2
3
3
4
4
To compile: make hello
Previous step produced an executable named hello
To run: ./hello
Hello... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
Kessler
6.S096 Lecture 1 – Introduction to C
19
/
30
Memory Model
It’s all about the memory
int a = 5;
int *a ptr = &a;
&a
&a ptr
Memory Address
0x7fff6f641914 0x000000000005 a
0x7fff6f641918 0x???????????? a ptr
Value
Identifier
Note: definitely a 64-bit machine, since the addresses are ... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
long double ( 80)
Andre Kessler
6.S096 Lecture 1 – Introduction to C
22
/
30
Memory Model
C Data Types
Table of C data types removed due to copyright restrictions.
Courtesy of http://en.cppreference.com/w/cpp/language/types
Andre Kessler
6.S096 Lecture 1 – Introduction to C
23
/
30
... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
096 Lecture 1 – Introduction to C
26
/
30
Examples
Compiling
Time for some examples!
Andre Kessler
6.S096 Lecture 1 – Introduction to C
27
/
30
With great power comes great responsibility
Compiling
C is focused on speed; always checki
would slow you down.
simple typo for( int i = 0;... | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-s096-effective-programming-in-c-and-c-january-iap-2014/443693ecfb1cc3e6fe78034a2ce179d0_MIT6_S096IAP14_Lecture1.pdf |
Imperative Programming in
Haskell?
Armando Solar-Lezama
Computer Science and Artificial Intelligence Laboratory
MIT
With content from Nirav Dave (used with permission) and
examples from Dan Piponi’s great blog Post
“You could have invented Monads! And Maybe You Already Have”
http://blog.sigfpe.com/2006/08/you-c... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
f x)
– (*) f g = (bind f) . (g) = \x ((bind f) (g x))
• Some useful identities
– unit * f = f * unit = f
– lift f * lift g = lift (f . g)
October 7, 2015
L09-4
Random Numbers
• Consider the “function” rand()
– Not really a function, but you can make it a
function
• rand: StdGen->(int, StdGen)
– think of... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
rand) (unit 5) )
October 7, 2015
L09-7
Lift and composition
• We can again define
– lift f x = unit (f x)
– (*) f g = (bind f) . g
• And again it is true that
– unit * f = f * unit = f
– lift f * lift g = lift (f . g)
October 7, 2015
L09-8
Monads as a type class
• Monad is a typeclass that ... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
� let p=e in do dostmts
return 5 >>= \x (timesrand x >>= \y do plusrand y)
return 5 >>= \x (timesrand x >>= \y plusrand y)
Int->MyRand Int
October 7, 2015
L09-15
do syntactic sugar for Monads
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
fail... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
wc :: String -> (Int,Int,Int)
wcs :: String -> Bool -> (Int,Int,Int)
-> (Int,Int,Int)
wc cs = wcs cs False (0,0,0)
wcs [] inWord (nc,nw,nl) = (nc,nw,nl)
wcs (c:cs) inWord (nc,nw,nl) =
if (isNewLine c) then
Suppose we want to read the
wcs cs False ((nc+1),nw,(nl+1))
string from an input file
else if (isSpace... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
September 29, 2010
L09-21
Ugly, yes, but may still be okay
• Issue: If we rely on strict
execution, this cannot be simplified
let unused = bigComputation input
in 2
• To this…
2
September 29, 2010
L09-22
Monads: A Review
Monad is a type class with the following operations
class (Monad m... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
readFileContents :: Handle -> String -> IO String
readFileContents h rcs =
do b <- hIsEOF h
if (not b) then return []
else do c <- hGetChar h
readFileContents h (c:rcs)
reading
a file
September 29, 2010
L09-25
Monadic vs bogus code
getFileContents :: String -> IO String
getFileContents filename =
do... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
-28
Monadic Word Count Program
version 2
file name
wc :: String -> IO (Int,Int,Int)
wc filename =
do
h <- openFile filename ReadMode
(nc,nw,nl) <- wch h False (0,0,0)
hClose h
return (nc,nw,nl)
wch :: Handle -> Bool -> (Int,Int,Int)
wcs :: String -> Bool -> (Int,Int,Int)
-> (Int,Int,Int)
-> IO (Int,I... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
fib n =
if (n<=1) then n
else
let
n1 = n - 1
n2 = n - 2
f1 = fib n1
f2 = fib n2
in f1 + f2
fib :: (Monad m) => Int -> m Int
fib n =
if (n<=1) then return n
else
do
n1 <- return (n-1)
n2 <- return (n-2)
f1 <- fib n1
f2 <- fib n2
return (f1+f2)
- monadic fib will work inside any other m... | https://ocw.mit.edu/courses/6-820-fundamentals-of-program-analysis-fall-2015/444a3c5a42232c471d9386191f58fc9f_MIT6_820F15_L09.pdf |
18.417 Introduction to Computational Molecular Biology
Lecture 7: September 30, 2004
Lecturer: Ross Lippert
Scribe: Mark Halsey
Editor: Rob Beverly
Divide and Conquer: More Efficient Dynamic Programming
Introduction
We have seen both global and local alignment problems in previous lectures. Briefly:
• Global Align... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
Four-Russians Speedup: computing alignment in
2 nO( logn ) time.
7-1
7-2
Lecture 7: September 30, 2004
Space-Efficient Sequence Alignment
The space complexity of the algorithms we have seen previously is proportional to
the number of vertices in the edit graph, i.e. O(nm). Observe however that the only
values ne... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
.
m/2
m
Prefix(i)
(i, m/2)
Suffix(i)
m
Adapted from Figure 7.1: Linear-Space Sequence Alignment
Note that the optimal alignment is simply pref ix(i) + suf f ix(i). pref ix(i) can be
Lecture 7: September 30, 2004
7-3
computed by finding the score si,
, i.e. we compute the score in linear space as shown
earlier for ... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
area
2
4+ area + · · · . Thus, the total time complexity is O(nm).
0 m/8
m/4
3m/8 m/2
5m/8
3m/4 7m/8 m
Adapted from Figure 7.2: Iteratively Computing the Optimal Alignment Midpoints
7-4
Lecture 7: September 30, 2004
Block Alignment and the Four-Russians Speedup
The time complexity of the dynamic programming globa... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
(from the top left to the bottom right) or
along the edges of a block. Thus we are restricting entry and exit to the corners of
blocks.
The block alignment problem is:
• Given: Two strings v and w partitioned into blocks of size t
Lecture 7: September 30, 2004
7-5
• Output: The block alignment of v and w with t... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
Looking up an
element in the lookup table takes O(t). Therefore, given a lookup table, the block
2 n
alignment algorithm takes O( n ) = O( logn ). We then add the time to compute the
lookup table, but see that the overall time is dominated by the n2 term. Therefore,
the overall running time is: O( logn ).
2 n
t
t... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
the fact that the scores in the first row and column are not arbitrary. The scores must
be both monotonically increasing and adjacent elements cannot differ by more than
1. Thus, the possible scores can be encoded as a vector of differences. This table is
depicted in Figure 7.4.
Figure 7.4: Mini-Alignment Lookup Table... | https://ocw.mit.edu/courses/18-417-introduction-to-computational-molecular-biology-fall-2004/446cb4bf40bde157d9dd83d7f1c6c650_lecture_07.pdf |
16
RICHARD B. MELROSE
3. Measureability of functions
Suppose that M is a �-algebra on a set X 4 and N is a �-algebra on
another set Y. A map f : X � Y is said to be measurable with respect
to these given �-algebras on X and Y if
(3.1)
f −1(E) ≤ M � E ≤ N .
Notice how similar this is to one of the characterizati... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
sees that if M is any �-algebra on
X then
(3.4)
f�(M) = E ∀ Y ; f −1(E) ≤ M
is always a �-algebra on Y.
�
In particular if f −1(A) ≤ M for all A ≤ G ∀ N then f�(M) is a �-
algebra containing G, hence containing N by the generating condition.
Thus f −1(E) ≤ M for all E ≤ N so f is measurable.
�
Proposition 3.2. ... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
X. More generally, for an
extended function f : X � [−⊂, ⊂] we take as the ‘Borel’ �-algebra
in [−⊂, ⊂] the smallest �-algebra containing all open subsets of R and
all sets (a, ⊂] and [−⊂, b); in fact it is generated by the sets (a, ⊂].
(See Problem 6.)
Our main task is to define the integral of a measurable functi... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
and this limit is uniform on any measurable set on which f is finite.
Proof. Folland [1] page 45 has a nice proof. For each integer n > 0 and
0 ⊃ k ⊃ 22n − 1, set
En,k = {x ≤ X; 2−nk ⊃ f (x) < 2−n(k + 1)},
∗E = {x ≤ X; f (x) → 2n}.
n
18
RICHARD B. MELROSE
These are measurable sets. On increasing n by one, th... | https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/447c5870100c43f0ded7978fc946e4f4_section3.pdf |
BUILD IT
Teaching Notes
Charcoal Press
This project is a very low-cost device for forming charcoal briquettes. It is composed of three welded
metal parts and a wooden block. To make it, you will learn to use a band saw, a sander, a drill press,
the OMAX water jet cutter and several hand tools. You will also learn to ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
used to starting from the material in it’s raw
form.
Teaching Notes
Build-It Background
The Build-It modules were designed to give students experience
with a variety of tools and manufacturing techniques while at the same time
exposing them to some simple, appropriate technologies. These modules
show them both the ... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
to use the tools
and equipment.
Each of these projects takes three 1.5-hour sessions; it is best that these
sessions are a full week apart, to give the students time to finish the
necessary work before the next session. It is intended that the projects
are done as part of a hands-on tutorial, with plenty of guidanc... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
part to spin (with a surprising amount of force). You can use a vise to hold
the part so that it won’t spin.
To cut steel, the band saw should be set at a cutting speed of about
100 ft/sec. Be sure that the guard is set to the right height before beginning
to cut.
Once the initial cut is made, turning the pipe slo... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
hole, that is start by drilling a smaller hole, then move
up to the larger bit size.
Drill the hole at least 2.6 inches deep, so that the ejector can easily fit into
the hole when the briquettes are being formed.
Plunger
Cup
Ejector
Block
Fig 1 The four components of the $2
charcoal press
Additional Teaching Note... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
as shown in Figure 2a. For this, you will need a more precise
measurement of the inner diameter: allow for a 0.002” interference fit,
meaning that the inner part is slightly larger than the outer part, and will
therefore stay in place when forced into position. The larger the interference,
the tighter the fit; if t... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
ave a 0.8” hole in the center so that the ejector
can fit through it. We will use the water jet cutter to make these parts. If
you are unfamiliar with the OMAX software and other CAD programs, you
may want to run through the tutorial before starting. Use the OMAX Layout
program to generate the tool path for the thr... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
start welding, so that adjacent material will melt together and
form a good joint and so the pieces do not move during the welding process.
You will use the welding fixtures in the shop to hold the pieces together
while you weld them.
Be sure to use protective clothing and goggles, as the radiation and extreme
lig... | https://ocw.mit.edu/courses/ec-720j-d-lab-ii-design-spring-2010/449159c23d3552f73d7d59c5808c7f55_MITEC_720JS10_bldit_chrc.pdf |
MIT OpenCourseWare
http://ocw.mit.edu
8.512 Theory of Solids II
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
Lecture 3: Properties of t h e Response Function
In this lecture we will discuss some general properties of the response functions X, and som... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
closing the contour in the upper 1/2 plane ensures that leJiwtl + 0 on the curved portion of the
KramersKronig
2
contour. Since we have ensured that χ(ω) is analytic in the upper half plane, Cauchy’s residue
theorem guarantees that the integral over the entire contour is 0. As a result, the piece we need,
i.e. t... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
q, ω) ensures that χ(�
Assuming that χ(�q, ω) → 0 as |ω| → ∞,
0 =
�
∞
−∞
dω�χ(q, ω�
�) Pr
�
�
1
ω − ω�
+ iπχ(�q, ω)
(3.9)
where the additional term iπχ(�
we picked up by making a hump over the pole. Thus for fixed �q,
q, ω) is one half of the contribution from the pole at ω�
= ω that
χ�(�
q, ω) = −
χ��(... | https://ocw.mit.edu/courses/8-512-theory-of-solids-ii-spring-2009/44bb605cb656287b1aaff1a93bcfa8fc_MIT8_512s09_lec03_rev.pdf |
Application Specific Integrated Circuit Design
Lecture 5
Vladimir Stojanoviü
6.973 Communication System Design – Spring 2006
Massachusetts Institute of Technology
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts ... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
gate arrays)
Field Programmable Gate Arrays (FPGA)
On-the fly reconfigurable interconnect
Flexibility vs. cost
Tighter control over transistors increases design cost
Can make faster designs but harder to verify and more expensive
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication Sys... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
size
Example- chip-in-a-day flow (B. Brodersen, UC Berkely)
A bunch of macros pre-generated (multipliers, adders, memories)
Easy to do COmm system design
Courtesy of Anantha Chandrakasan. Used with permission.
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT O... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
(6 transistors in basic cell)
Logic slow and big due t o fixed transistors and
wiring overhead
- Advanced cell -based arrays hardwire logic functions
(NANDs/NORs/LUTs) which are personalized with
metal
Courtesy of Arvind and Krste Asanovic. Used with permission.
Cite as: Vladimir Stojanovic, course materials for... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communication System Design
9
Standard cell ASICs
Also called Cell-Based ICs (CBICs)
Fixed library of cells
Memory gene... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
[DD Month WYY].
6.973 Communication System Design
ASIC flow subset – 6.375 and 6.973
Cite as: Vladimir Stojanovic, course materials for 6.973 Communication System Design, Spring 2006.
MIT OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].
6.973 Communicatio... | https://ocw.mit.edu/courses/6-973-communication-system-design-spring-2006/44d1cff68d0f994cca25442de3e1ac78_lecture_5.pdf |
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