text stringlengths 16 3.88k | source stringlengths 60 201 |
|---|---|
G
Tk
B
2
∝
e
=
p
i
n
i
n
i
> n is the concentration [#/cm3] of electrons
> p is the concentration [#/cm3] of holes
Every thermally
generated electron
leaves behind a hole
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseW... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
ceptor dopant concentrations ar
e
NA [#/cm3] (cid:198) p-type material
• Boron (3 valence electrons)
> Donor dopant concentrations are N
D
[#/cm3] (cid:198) n-type material
• Phosphorous (5 valence electrons)
> These introduce new energy levels
close to valence or conduction
bands (~0.05 eV)
> Dopant concentrations... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
/6.777J Spring 2007, Lecture 7E - 10
Main results
> Donors or acceptors are fully ionized
> Define n0 and p0 as the electron and hole
N
D
N
→ +
+
D
−
e
concentrations at equilibrium
> n0 and p0 follow a mass-action law
N
N
→ +
+
h
A
N
≈
D
N
A
≈
N
−
A
N
+
D
−
A
n p
0
0
2
n=
i
-
+
-
+
-
+
− =
AN
p
+ =
DN
n
n
p=
Cite as... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 7E - 12
Main results
> Therefore, the equilibrium
majority carrier concentration is
determined by the net doping
and the minority carrier
concentration is determined b... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
′
n
τ
m
Electron-hole
pair (EHP)
′
n
n t
( ) ~ (0)
′
t
−
e τ
m
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
obey a drift/diffusion flux
relation
> Drift: carriers move in an
electric field
> Diffusion: carriers move in a
concentration gradient
> For p-type material
• n is small (cid:206) drift current is
small
• ∇n can be big (cid:206) diffusion
current dominates
Electric field [V/cm]
drift
E
n
(
q
μ=
e
n
diffusion
)
D ... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
> At equilibrium diffusive driving force =
electric field in the vicinity of the
junction
> In order to set up this electric field, the
ionized donors and acceptors are
relatively depleted of mobile carriers
near the junction – the space-charge
layer (SCL) or depletion region
Oxide... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
.2 in: Senturia, Stephen D. M
Boston, MA: Kluwer Academic Publishers, 2001, p. 359. ISBN: 9780792372462.
icrosystem Design.
Image by MIT OpenCourseWare.
p
n
Equilibrium
IN
Ec
Ev
Ec
Ev
IP
Forward bias
Image by MIT OpenCourseWare.
Adapted from Figu
re 6.1 in: Pierret, Robert F. Semiconductor
Device Fundamentals. Reading,... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 7E - 20
The Junction-Isolated Diffused Resistor
> The structure is a diode, but with two
contacts
> The diode action prevents currents
into the substrate pr... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
the channel region of the
substrate can either increase the surface concentration
(accumulation) or deplete the surface and eventually invert the
surface
Source
Gate
Drain
Oxide
Channels
p-type substrate
n+ regions
G
B
D
S
Body
Adapted from Figure 14.10 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
A
m
(
D
I
1000
800
600
400
200
0
VGS = 4 V
triode
ID,sat
VGS = 3 V
saturation
VGS = 2 V
0
Image by MIT OpenCourseWare.
1
2
3
4
5
Adapted from Figure 14.13 in: Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 369. ISBN: 9780792372462.
VDS
G
S
e-
D
e-
G
S
e-
VDS>0
V VDS
>
-V
T
GS
... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
circuits involving MOSFETS
> Can use either full nonlinear
characteristics
> Or linearized small-signal
model
• Different models include
different components
G
G
2
K/2(V -V )
Tn
GS
D
S
Simple large-signal
model, in saturation
D
CGS
g vm gs
r0
+
-
S
Simple small-signal
model, in saturation
Cite as: Joel Voldman, co... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
B
_
Top view
Adapted from Figures 14.23 and 14.24 in Senturia, Stephen D. Microsystem Design. Boston, MA: Kluwer Academic Publishers,
2001, p. 382. ISBN: 9780792372462.
LM158
Image by MIT OpenCourseWare.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
v
−
s
−
R
1
A
(0
=
=
v V
−
−
0
R
2
−
v
−
)
⇓
1
⎛
⎜
⎝
1
+
1
A
+
R
2
R
1
⎞
⎟
⎠
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎣
A
1
⇓ → ∞
V
o
V
s
R
2
R
1
= −
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachuset... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
2
R
1
R
2
R
1
Vs
+
_
V0
R1
R2
Non-inverting
Image by MIT OpenCourseWare.
Adapted from Figure 14.28 in Senturia, Stephen D. Microsystem Design.
Boston, MA: Kluwer Academic Publishers, 2001, p. 388. ISBN: 9780792372462.
Cite as: Joel Voldman, course materials for 6.777J / 2.372J Design and Fabrication of Microelectromech... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
J / 2.372J Design and Fabrication of Microelectromechanical Devices, Spring 2007. MIT
OpenCourseWare (http://ocw.mit.edu/), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
JV: 2.372J/6.777J Spring 2007, Lecture 7E - 35
Differentiator
C
Vs
+
_
R1
_
+
V0
Adapted from Figure 14.31 in Senturia, Ste... | https://ocw.mit.edu/courses/6-777j-design-and-fabrication-of-microelectromechanical-devices-spring-2007/6de08ad7d4ee670c283ce5ec0e6a653b_07lecture07e.pdf |
Turbulent Flow and Transport
3
Concepts in Turbulence
3.1
Comments on laminar flow, its stability, and the transition to turbulent flow.
3.2
Features of turbulent flows (high Reynolds number, "randomness", three−
dimensionality of fluctuations, intermittency near free boundaries, etc.
3.3 The energy cascade and... | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/6dea1c01310528dc771c046e9352c01e_Concepts.pdf |
, read for example Schlichting. 7th ed. 578−586,
or White: 436−440. | https://ocw.mit.edu/courses/2-27-turbulent-flow-and-transport-spring-2002/6dea1c01310528dc771c046e9352c01e_Concepts.pdf |
StokesI Theorem
Our text states and proves Stokes' Theorem in 12.11, but ituses
the scalar form for writing both the line integral and the surface integral
involved. In the applications, it is the vector form of the theorem that is
most likely to be quoted, since the notations dxAdy =d the like are not
in common u... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
( F a T) ds --
JJs ((curl T)-$] d~ .
Jc
Here the orientation of C is that derived from the counterclockwise
orientation of D; and the normal fi to the surface S points in the same
direction as 3rJu Y 1gJv .
Remark 1. The relation between T and 2 is often described
4
informally as follows: "If youralkaround C in ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
Proof of the theorem. The proof consists of verifying the
following three equations:
m e theorem follo& by adding these equations together.
We shall in fact verify only the first equation. The others are
proved similarly. Alternatively, if one makes the substitutions
>
1
j and j + k m.d k -+i m d x--sy and y + z... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
/ ~ v ) du dv .
We use the chain rule to compute the integrand. We have
where P and its partials are evaluated at ~(u,v), of course.
Subtracting, we see that the first and last terms cancel each other.
The double integral ( * ) then takes the form
Nclw we compute the surfac~ integral of our theorem. Since
curl F ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
t o S,
x
2
-
1 z = 0
Let
= x t -
and let 3,
b e
t h e unit normal to S 2 ,
both with positive d
component. Evaluate
y o d l d S and Jj 1 * a 2 d s .
S1
s2
-'Grad 'Curl - - -
Div and all that.
We study two questions about these operations:
I. DO they have natural (i.e., coordinate-free) physical
or geo... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
a physical interpretation of curl F, let us
-+
,
imagine F to be the velocity vector field of a moving fluid.
+
Let us place a small paddle wheel of radius P
in the fluid,
with its axis along n. Eventually, the paddle wheel settles
-+
down to rotating steadily with angular speed w (considered as
positive if cou... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
s i d e r t h e n e x t t w o o p e r a t i o n s , c u r l and
d i v . Again, w e c o n s i d e r o n l y f i e l d s t h a t a r e c o n t i n u o u s l y
d i f f e r e n t i a b l e i n a r e g i o n U o f R . There a r e analogues of
3
a l l t h e e a r l i e r theorems:
I
Theorem 5 . - I f F - - - -
i s a ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
h y p o t h e s i s , G = c u r l ?
d e f i n e d i n U. W e corn-
f o r some
+-
p u t e :
G n d s =
S2
11 c u r l F mii d~ = -
s 2
.
1
C
H mda.
-
Adding, w e see t h a t ,
ema ark. The convers'e of Theorem 5 h o l d s a l s o , b u t w e
s h a l l n o t a t t e m p t t o prove it.
Theorem 6 . - I f 2 = c ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
to construct a specific such function
in the given cases.
if 2
Note that
-+
= curl ?$ and G = curl H, then
-+
+ +
curl(H-F) = 8 . Hence by Theorem 3, H - 5 = grad 4 in U,
for some 4.
Theorem 8. -The condition
-t
d i v G = O
&I
U
does not in general imply - that if - - - -
---
is a curl in U.
Proof. L ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
a of s p h e r e ) # 0.
0
erna ark. Suppose w e s a y t h a t a r e g i o n U
i n R3
i s "two-
simply c o n n e c t e d w i f e v e r y c l o s e d s u r f a c e i n U bounds a
s o l i d r e g i o n l y i n g i n U.* The r e g i o n U = R - ( o r i g i n )
3
i s nott'two-
simply connected", f o r example, b u t... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
i s much more one can s a y a b o u t t h e s e matters, b u t
one needs t o i n t r o d u c e a b i t of a l g e b r a i c topology i n o r d e r t o do
s o .
I t i s a b i t l a t e i n t h e s e m e s t e r f o r t h a t !
*The proper mathematical term for t h i s is ffharalogically t r i v i a l in dimension tw... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/6e01501b3081488401fafd1936cf643b_MIT18_024s11_ChFnotes.pdf |
8.06 Spring 2016 Lecture Notes
4. Identical particles
Aram Harrow
Last updated: May 19, 2016
Contents
1 Fermions and Bosons
Introduction and two-particle systems
. . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
1.2 N particles
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
Charged particles in a magnetic field
21
3.1 The Pauli Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 The de H... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
·) denotes an ordered list, in which different posi
tions have different meanings; e.g. in general (rr1, pr1, rr2, pr2) = (rr2, pr2, rr1, pr1).
1
To describe indistinguishable particles, we can use set notation. For example, the sets {a, b, c}
and {c, a, b} are equal. We can thus denote the state of N indistinguish... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
, since it may be that
swapping two positions could result in an unobservable change, such as multiplying by an overall
phase.
To be more concrete, consider the case of two indistinguishable particles. Then we should have
|ψ((cid:126)r1, (cid:126)r2)| = |ψ((cid:126)r2, (cid:126)r1)|, or equivalently
ψ((cid:126)r1, (cid... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
3/2, etc.) are fermions and that particles with integer spin (0, 1, etc.) are bosons. The proof
of this involves field theory (or at least the existence of antiparticles) and is beyond the scope of
8.06 (but could conceivable be a term-paper topic).
To find a basis for the symmetric and antisymmetric subspaces, we can co... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
105) and |β(cid:105),
in which case we get back either the same state (symmetric subspace) or the same state multiplied
by -1 (antisymmetric subspace).
after normalizing if
√
α
β
β
2
If V is d-dimensional and has basis {|1(cid:105), . . . , |d(cid:105)} then V ⊗ V is d2-dimensional and has basis
{|1(cid:105) ⊗ |1(cid:1... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
eyl duality” is the
phrase to google to learn more.)
(cid:1) = d(d+1)
2
2
In this case, we use spin
Example: spin-1/2 particles. The simplest case is when d = 2.
notation and describe the single-particle basis with {|+(cid:105), |−(cid:105)}. The resulting basis for Sym2 C2 is
{| + +(cid:105), |+−(cid:105)+|−+(cid:105)... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
:105) = |ψ(cid:105) ∀i (cid:54)= j}
AntiN V ≡ {|ψ(cid:105) ∈ V ⊗N : F ij|ψ(cid:105) = −|ψ(cid:105) ∀i (cid:54)= j}
The corresponding wavefunctions are those satisfying
ψ((cid:126)r1, . . . , (cid:126)ri, . . . , (cid:126)rj, . . . , (cid:126)rN ) = ±ψ((cid:126)r1, . . . , (cid:126)rj, . . . , (cid:126)ri, . . . , (cid:... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
In other words if π, ν ∈ SN then
(denoted e) and is closed under multiplication and inverse.
applying ν then π is another permutation (denoted πν) and there exists a permutation π−1 satisfying
ππ−1 = π−1π = e. Additionally F π is a representation meaning that F πν = F πF ν. Verifying these
facts is a useful exercise. O... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
1
N !
(cid:88)
π∈SN
where the second equality used the fact that F π|ψ(cid:105) = |ψ(cid:105) for all |ψ(cid:105) ∈ SymN V .
The argument for the antisymmetric projector is similar, but we first need to define sgn(π),
which is called the sign of a permutation. It is defined to be 1 if π can be written as a product
of an e... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
obeys sgn(νπ) = sgn(ν) sgn(π) for any permutations ν, π. This can be used to prove that Panti is
the projector onto the antisymmetric subspace, using an argument similar to the one used for Psym
and the symmetric subspace.
As a result, we can write a basis for SymN V consisting of the states
|ψsym
α1,...,α =
(cid:105) ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
:105) = det
N
ψα1((cid:126)r1)
.
.
.
· · · ψα1((cid:126)rN )
.
.
.
ψαN ((cid:126)r1)
· · · ψα
N ((cid:126)rN )
.
(16)
This is called a Slater determinant. For example when N = 2, the wavefunction is of the form
ψα1((cid:126)r1)ψα2((cid:126)r2) − ψα2((cid:126)r1)ψα1((cid:126)r2)
√
2
.
(17)
1.3 Non-in... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
1(cid:105) ⊗ · · · ⊗ |αN (cid:105),
it follows that
H|α1(cid:105) ⊗ · · · ⊗ |αN (cid:105) = (Eα1 + . . . + EαN )|α1(cid:105) ⊗ · · · ⊗ |αN (cid:105).
(20)
N
N (cid:105)} is an orthonormal basis of eigenstates of H. The ground state is |0(cid:105)⊗ ,
Thus {|α1(cid:105) ⊗ · · · ⊗ |α
which has energy N E0. The first excite... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
and one 1, but these all refer to
exactly the same state. Similarly all the same energies Eα1 + . . . + EαN still exist in the spectrum of
symψα1,...,αN (cid:105)
H restricted to SymN V , but the degeneracy of up to N ! is now gone. Specifically state |
has energy Eα1 + . . . + EαN . Since these are a basis for SymN V w... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
e.g. |ψ0,1,...,N −3,N,N +1(cid:105)). Holes are studied in solid-state
physics, and were the way that Dirac originally explained positrons (although this explanation has
now been superseded by modern field theory).
0,1,...,N 2,N +1(cid:105)), moving the hole to lower energies (e.g. |ψ
0,1,...,N 2,N (cid:105) which has e... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
n(cid:48)
Either way this factorizes as
i ranges over 0,1 (for fermions)
or over all nonnegativ
e integers (for bosons).
Pr[n0, n1, . . .] =
(cid:89)
i≥0
e−βni(Ei−µ)
(cid:80)
n(cid:48)
i
e−βn(cid:48) (Ei−µ)
i
.
In other words, each occupation number is an independent random variable.
For fermions this results in the Fe... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
could be written either as |n, l, m, s(cid:105) or (dividing
into spatial and spin parts) as |n, l, m(cid:105) ⊗ |s(cid:105).
More generally, suppose the state space of a single particle is V ⊗ W . Then the state of N
distinguishable particles is
(V ⊗ W )⊗N ∼= V ⊗N ⊗ W ⊗N .
This isomorphism is proved by simply rearrang... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
12:34|α1, α2, α3, α4(cid:105) = |α3, α4, α1, α2(cid:105).
(31)
(32)
What if we would like to understand Anti2(V ⊗ W ) in terms of the symmetric and antisymmetric
subspaces of V ⊗2 and W ⊗2? Then it will be convenient to rearrange (31) and write (with some
small abuse of notation)
Anti2(V ⊗ W ) = {|ψ(cid:105) ∈ V ⊗ V ⊗ ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
pace
of F 3:4. It also contains states in the −1 eigenspace of F 1:2 and the +1 eigenspace of F 3:4, as well
as superpositions states in these two spaces. Putting this together we have
Anti2(V ⊗ W ) ∼= (Sym2 V ⊗ Anti2 W ) ⊕ (Anti2 V ⊗ Sym2 W ).
Similarly the symmetric subspace of two copies of V ⊗ W is
Sym2(V ⊗ W ) ∼ (... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
:105), . . . , |N (cid:105) corresponding to wavefunctions ψ1((cid:126)r), . . . , ψN ((cid:126)r). If we had one electron
in position |1(cid:105) with spin in state |s1(cid:105), another electron in position |2(cid:105) with spin in state |s2(cid:105), and
so on, then the overall state would be
1
√
N !
(cid:88)
π∈SN
s... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
2((cid:126)r) is nonzero and ψi((cid:126)r) = 0 for i = 2. (Here we use the
assumption that the electrons are well separated.) Suppose this field is Bzzˆ in this region and zero
elsewhere. This field would correspond to a single-particle Hamiltonian of the form |2(cid:105)(cid:104)2| ⊗ ωzSz
9
(cid:54)
for ωz = −µeBz, wh... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
model of N distinguishable spins. Spatial position can be used to
distinguish particles, but it does not have to in every case.
2 Degenerate Fermi gas
2.1 Electrons in a box
Consider N electrons in a box of size L × L × L with periodic boundary conditions. (Griffiths
discusses hard-wall boundary conditions and it is a go... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
the k with the lowest values of k2. These k vectors are contained in
(cid:126)
(cid:126)
(cid:126)
10
a sphere of some radius which we call kF (aka the Fermi wave vector). Since each wavevector can
(cid:126)
be thought of as taking up “volume” (2π/L)3 in k-space, we obtain the following equation for kF :
4
3
πk3
F ·
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
)
(cid:126) k2
2
2m
=
2 (cid:90) kF
(cid:126)
2m 0
4
ck dk =
(cid:126)2
k5
F
c
2m 5
=
(cid:126)2k2
3
F
2m 5
3
N = N EF .
5
(48)
In hindsight we should have guessed that Egs would be some constant times N EF . There are N
electrons, each with energy somewhere between 0 and EF depending on their position within the
spher... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
EF = O(1)eV .
This is (cid:29) kBT for room-temperature T , justifying the assumption that the state is close to the
ground state as room temperature, and corresponds to a vF = (cid:112)2EF /m that is (cid:28) c, justifying a
non-relativistic approximation.
Can we justify neglecting the Coulomb interaction? See the pse... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
be ≈ 1.1 × 10−9 watt - ohm
and modeling them
K2
as a Fermi gas predicts ≈ 2.44 × 10−8 watt - ohm
. In real metals, this number is much closer to the
2
K
Fermi gas prediction, which provides some support for the Drude-Sommerfeld theory.
σT
There are some other aspects of the theory which are clearly too simplistic. The ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
istic electrons in their ground
state. There are a lot of assumptions here. The ground-state assumption is justified because
photons will carry away most of the energy of the star. The non-interacting assumption will be
discussed on the pset. We consider only electrons and not nuclei because degeneracy pressure scales
w... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
3
Edegen(R) = Egs = NeEF (R)
5
(cid:126)2 (cid:18) 3π2Ne
2me
V
3
= Ne5
(cid:32)
= (cid:126) f
2 5/3
2/3
(cid:19)
(cid:18) 3 (cid:19) (cid:18)
10
9π
4
2/3
(cid:19) (cid:33) 5/3
N
m R2
e
, =
λN 5/3
meR2
,
(cid:124)
(cid:123)(cid:122)
≡λ
(cid:125)
where again λ is a universal constan
t. For f =
0.6 (as is the
case for our... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
3
(cid:126)N
m c λ
e
meκ
−1/3
N
G
∼ (cid:126)c
N N 2/3
m2
p
=
2
m
p
m2
Pl
N 2/3.
In the last step we have introduced the Planck mass mPl which is the unique mass scale associated
with the fundamental constants GN , (cid:126), c. Thus the critical value of N at which the non-relativistic
assumption breaks down is
Ncrit ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
=
3V k
F
23π
(cid:19)
3
4πk2dk (cid:126)ck
(cid:125)
=
= κ(cid:48)
V (cid:126)c
k4
4π2 F
N 4/3
V 1/3
where κ(cid:48) ≡ 3
4
(cid:0) 9π
4
(cid:1)1/3 f 4/3(cid:126)c. Now the total energy is
Etot = −
κN 2
R
+
κ(cid:48)N 4/3
R
Λ
≡ .
R
(56)
The situation is now much simpler than in the non-relativistic case. If Λ < 0 then t... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
by mp. It turns out that neutron stars are stable up to masses
of roughly 3.0MSun (although this number is fairly uncertain in part because we don’t know the
15
structure of matter in a neutron star; it may be that the quarks and gluons combine into more
exotic nuclear matter). Above 3.0MSun there is no further way to... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
(x + a). Since [Ta, V ] = 0 we have [T , H] = 0 and thus Ta and H can be simultaneously
diagonalized. Bloch’s theorem essentially states that we can take eigenstates of H to be also
eigenstates of Ta.
ˆ ˆ
ˆ
ˆ
ˆ
ˆ
To see what this means let’s look more carefully at Ta. We have seen this in 8.05 already:
(cid:19)
(cid:18... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
π/a, it has some momentum-like properties. To see this, first look at the
eigenvalue equation satisfied by un,k.
(En,k − V (x))eikxun,k(x) =
p2
2m
eikxun,k(x) = eikx
2
(p + (cid:126)k)
2m
un,k(x).
Rearranging we have
(cid:18) (p + (cid:126)k)2
2m
(cid:19)
+ V
|
(cid:124)
(cid:123)(cid:122)
H
k
(cid:125)
By the Hellmann-F... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
(cid:126)a1, (cid:126)a2, (cid:126)a3. The
theory of crystallographic groups includes a classification of possible values of (cid:126)a1, (cid:126)a2, (cid:126)a3. The “re-
ciprocal lattice vectors” b1, b2, b3 are defined by the relations
(cid:126) (cid:126) (cid:126)
a a
This is equivalent to the matrix equation
(cid:12... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
1(cid:105)(cid:104)n| − ∆|n
− (cid:105)(cid:104)n|.
1
−∞
This can be rewritten in terms of the translation operator T = (cid:80)
n=
n |n + 1(cid:105)n as
By Bloch’s theorem the eigenstates can be labeled by k, with
H = E0I − ∆(T + T †).
T |ψk(cid:105) = eika|ψk(cid:105)
H|ψk(cid:105) = Ek|ψk(cid:105)
From (67) we can c... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
one of these points corresponds to a delocalized state.
Band structure. Now suppose each site can support multiple bound states, say with energies
E0 < E1 < E2. Then tunneling opens up a band around each energy. If ∆ is small enough then
there will be a band gap between these.
Kronig-Penney model Another model that can... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
a
n = 0
- π
a
k
Figure 3: Band diagram for free electrons.
eika for k and integer multiple of 2π . If we let φ := 2πa/L then there is a basis in which we have
L
1
. . .
1
eiφ
T =
. . .
eiφ
e2iφ
. . .
and
V =
∗ ∗ ∗
∗ ∗ ∗
∗ ∗ ∗
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
k which are vertical lines in Fig. 3. Treat V as a
perturbation. For a typical value of k the kinetic energy of these points is well-separated and so
V does not significantly mix the free states. However, near ±π/a, the kinetic energy term has a
degeneracy, so there the addition of V will lead to a splitting, which will... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
duction band. A small electric field is enough to move this into the conduction
band. The resulting material is called an “n-type semiconductor.” On the other hand, aluminum
has one few electron than silicon, and so is an “acceptor.” Adding aluminum will increase the
number of holes in the valence band and reduce the nu... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
ian. Either way the magnetic field turns out to enter the Hamiltonian
2m ((cid:126)p − q (cid:126)A((cid:126)x))2. We thus obtain the Pauli Hamiltonian
replacing the kinetic energy term with 1
(neglecting an additional spin term):
c
H =
(cid:16)
(cid:126)p −
1
2m
q
c
(cid:17)
2
(cid:126)A((cid:126)x) + qφ((cid:126)x).
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
is gauge invariant if (cid:104)ψ(cid:48)|O(cid:48)|ψ(cid:48)(cid:105) = (cid:104)ψ|O|ψ(cid:105).)
Remark: This gauge freedom will appear many times in the coming lectures and often leads
to seemingly strange results. It is worth remembering that we are already used to a simpler form of
f (t(cid:48))dt(cid:48)/(cid:126)... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
:16)
1
m
pi −
(cid:17)
Ai
q
c
=⇒
(cid:126)p = m ˙(cid:126)x +
q (cid:126)A,
c
(79)
obtaining a generalized momentum p(cid:126). Next we should remember that A((cid:126)x), φ((cid:126)x) depend on position
so that
(cid:126)
p˙i = −
∂H
∂xi
(cid:88)
= −
j
(cid:16)
q
mc
pj −
Aj
q
c
(cid:17) ∂Aj
∂xi
− q
= −
∂φ
∂xi
(cid:88)
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
k
(cid:15)ijkBk
(cid:88)
x˙ j
j
(cid:18) ∂Aj
∂xi
−
∂Ai
∂xj
(cid:19)
(cid:88)
=
j,k
(cid:15)ijk ˙xjBk = ( ˙(cid:126)x × (cid:126)B)i.
22
(80)
(81)
(82)
We can now rewrite (82) in vector notation as
¨m(cid:126)x = q
−∇(cid:126) φ −
(cid:123)(cid:122)
(cid:124)
(cid:126)E
1
c
˙(cid:126)A
(cid:125)
+
q ˙
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
make things simple, namely
(cid:126)A = (−By, 0, 0), φ = 0.
(84)
(cid:126)
Another variant of the Landau gauge is A = (0, Bx, 0). On the pset you will also explore the
(cid:126)
“symmetric gauge,” defined to be A = (− 1
2 By, 1 Bx, 0). It is nontrivial to show that these result
in the same physics, but this too will be ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
fined l0 to be characteristic length scale of the harmonic oscillator,
(cid:126)
mωL
(cid:113) (cid:126)c .
=
qB
We can now diagonalize H from (85). The eigenstates are labeled by kx and ny, and have
energies and wavefunctions given by
Ekx,ny = (cid:126)ωL(ny + 1/2)
ψ
kx,ny = e
ikxx
1 4
/
−
l0
φny
(cid:18) −
y
l0
(cid:1... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
finite region
in the plane, say with dimensions L × W . For simplicity we impose periodic boundary conditions.
This means that
kx =
nx, nx ∈ Z.
2π
W
We also have the constraint that y0 should stay within the sample. (Assume that l0 (cid:28) L, W so we
can ignore boundary effects.) Then 0 < y0 < L, or equivalently
This im... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
ωL(n + 1/2) =
(cid:126)eB
2mec
(2n + 1) ≡ µBB(2n + 1)
µB =
e(cid:126)
2mec
.
Thus the induced dipole moment is
µI = −
∂E
B
∂
=
−(2n
+
1)µB.
The minus sign means diamagnetism, corresponding to the fact that the circular orbits caused by
a magnetic field will oppose that field. (We neglect here the contributions from spin.... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
, according to (90).
The number of fully filled LLs is j ≡ (cid:98)ν(cid:99), meaning the largest integer ≤ ν. Thus j ≤ ν < j + 1.
The energy of the ground state is then
E =
j−1
(cid:88)
n=0
(cid:124)
(cid:123)(cid:122)
filled levels
j
−1
(cid:88)
(cid:32)
n=0
= N
(cid:126)
ωL
2
D(cid:126)ωL(n + 1/2) +
(N − jD)(cid:126)
... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
uities at integer values of ν. Also observe that for ν < 1 all electrons are in a
single LL, so we observe the simple classical prediction of diamagnetism, which we refer to here as
“Landau diamagnetism” even though it is the only part of this diagram where the Landau levels
do not really play an important role.
26
Fi... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
cid:28) B.
To see that this assumption is reasonable, note that in units where (cid:126) = 1, c = 1 an electric field of
1V /cm is equal to 2.4 · 10−4eV2 while a magnetic field of 1 gauss is equal to 6.9 · 10−2eV2.
If the velocity is the one induced by the electric field then the magnetic field causes a drift in
the positi... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
This has solution
d
dt
(cid:126)v = A(cid:126)v + b = A((cid:126)v + A−1(cid:126)b)
(cid:126)
d
dt
((cid:126)v + A−1(cid:126)b) = A((cid:126)v + A−1b).
(cid:126)
(cid:126)v(t) + A−1b = eAt((cid:126)v(0) + A−1b),
(cid:126)
(cid:126)
28
Figure 6: Motion of a charged particle subject to crossed electric and magnetic field... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
is moving
(cid:126) (cid:126)
at velocity (cid:126)v = vH xˆ. (Suppose for now that we do not know vH .) To leading order in v/c the E, B
fields transform as
(cid:126) (cid:48)
E = E +
(cid:126)
× (cid:126)
B = (E
−
(cid:126)B(cid:48)
= B − × E = (B
−
(cid:126)
(cid:126)
v
c
v
c
v
H
c
v
H
c
B)yˆ
E)zˆ
(97a)
(97b)
(cid:12... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
.) We
can also show that jy = −σH Ex. We then conclude that
=
jx
jy
0
σH
−σH
0
.
Ex
Ey
(This matrix perspective is also useful when computing
resistivity which is ρ = σ−1.)
(cid:124)
(cid:123)(cid:122)
σ
(cid:125)
While the formula we have found is entirely classical, we can interpret in terms o... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
L = (cid:112)(cid:126)c/qB, we then
obtain
H|kx =
p2
y
2m
+
1
2
mω2
L(y − y0)2 − qEy
y0 ≡ −(cid:96)2
0kx
+
1
2
p2
y=
2m
p2
1
y
+
2m 2
=
(cid:18)
y2 − 2yy0 + y2
0 −
mω2
L
(cid:19)
2qE
mω2
L
y
mω2
L (y − y0 − y1)2 −
1
2
mω2
L(2y0y1 + y2
1)
y1 ≡
qE
mω2
L
=
qE
mωL
qB
mc
=
E
B c
ωL
=
(100)
(101)
vH
ωL
(102)
This looks again... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
. We
can think of the Landau levels as being “tilted” as follows.
(cid:126)E = 0
(cid:126)E (cid:54)= 0
Note that earlier we broke the symmetry between x and y somewhat arbitrarily, and using a
different gauge would’ve resulted in wavefunctions that were plane waves in the y direction, or even
localized states. This fre... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
− vH ,
v
ωL
which is precisely the classical result we would expect from a particle undergoing oscillations with
frequency ωL on top of a drift with velocity vH .
We can now calculate the group velocity vg = ∂ω of the eigenstates of H. From (103) we
∂k
immediately obtain vg = vH , suggesting that each eigenstates moves... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
y1 and
32
so we can write
(cid:90) L
0
Sxdy =
ωL
W
ω
≈ L
W
ωL=
W
ωL=
W
(cid:90)
L
|φn(y − y0 − y1)|2(y − y0)
0
(cid:90)
∞
0
(cid:90) ∞
0
y1
|φn(y − y0 − y1)|2(y − y0)
|φn(y − y0 − y1)|2
(y − y0 − y1
(cid:125)
(cid:123)(cid:122)
(cid:124)
(cid:124)
(cid:125)
(cid:123)(cid:122)
even
odd
+
y1)
We conclude that (cid:126)j... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
LL then we rapidly fill a LL and
the conductivity jumps up.
This explains the plateaus in the conductivity but not why the conductivity should be an integer
multiple of σ0. We will return to this point later after discussing the Aharonov-Bohm effect.
3.5 Aharonov-Bohm Effect
To write down the Hamiltonian for a charged par... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
1 = (cid:126)x2. In this case the path becomes a loop and
(cid:73)
P =
(cid:126) · (cid:126) Stokes’ thm (cid:90)
=
A dl
∇(cid:126)
(
(cid:126)
× A) · d(cid:126)a =
(cid:90)
loop
surface
surface
(cid:126)B
· d(cid:126)a = Φ.
(113)
In the last step we have defined Φ to be the magnetic flux through the surface. The last tw... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
particular it only moves through a region where B = 0.
(cid:126)
However, the vector potential cannot be zero outside the solenoid. Indeed consider a curve C
enclosing the solenoid such as the dashed line in Fig. 8(b). The loop integral of A around C is
(cid:126)
(cid:73)
C
(cid:126) · (cid:126)A d(cid:96) = πR2B = Φ
(... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
a nonzero vector potential but zero magnetic field.
We will need a prescription for solving the Schr¨odinger equation in a region where B = 0 but
(cid:126)A may be nonzero. To do so, let us fix a point (cid:126)x0, which in our scenario will be the location of the
source S. Then given a curve C from (cid:126)x0 to (cid:1... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
)x0→(cid:126)x
along C2
(cid:90)
(cid:126)x→(cid:126)x0
along C2
(cid:126)A · d(cid:126)(cid:96)
(cid:126)A · d(cid:126)(cid:96)
(cid:126)
A
· (cid:126)d(cid:96)
(cid:126)x0→(cid:126)x→(cid:126)x0
along C1 then C2
(cid:73)
=
q
(cid:126)c
q
= Φ
(cid:126)c
(121a)
(121b)
(121c)
(121d)
In this last equation, Φ is the flux e... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
)
(cid:126)
eig((cid:126)x)∇g
= (cid:126)
= (cid:126)eig((cid:126)x) q
(cid:126)c
q (cid:126)Aeig((cid:126)x)
c
=
(cid:126)A((cid:126)x)
This means that when p(cid:126) − q (cid:126)A “commutes past” eig it turns into simply p(cid:126). In other words
c
This implies that
(cid:17)
(cid:16)
q
p(cid:126) − (cid:126)A
c
ei... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
and region B contains point B.
This will allow us to use (122) separately in each region.
Let gA and gB denote the functions g restricted to regions A and B respectively, and let
ψ(0)(S → A → y) and ψ(0)(S → B → y) be the solutions of the free Schr¨odinger equation in those
regions. This means that
q
gA(y) = (cid:126)c... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
2 ∝= cos2
(cid:18) 1
2
(cid:18) kay
L
− 2π
(cid:19)(cid:19)
.
Φ
Φ0
(131)
Figure 11: Observed interference pattern when performing the two-slit experiment with a magnetic
flux Φ enclosed between the two paths.
The resulting interference pattern is depicted in Fig. 11. Observe that the experiment is only
sensitive to the ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
the Thouless charge pump. Suppose we have a
1-d system containing some number of electrons. As 0 ≤ t ≤ T suppose we adiabatically change
the Hamiltonian from H(0) to H(T ), and suppose further that H(T ) = H(0). In other words we
return to the Hamiltonian that we start with. Then the net flux of electrons from one end o... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
Now suppose we adiabatically increase Φ from 0 to Φ0. By the results of the pset, the final
Hamiltonian is the same as the initial Hamiltonian (up to a physically unobservable gauge trans-
form), and by the Thouless charge pump argument, this means that an integer number of electrons
must have flowed from the inner loop ... | https://ocw.mit.edu/courses/8-06-quantum-physics-iii-spring-2016/6e02558ecf4b7af37d2cdff550fc0f71_MIT8_06S16_chap4.pdf |
6.776
High Speed Communication Circuits and Systems
Lecture 8
Broadband Amplifiers, Continued
Massachusetts Institute of Technology
March 1, 2005
Copyright © 2005 by Hae-Seung Lee and Michael H.
Perrott
The Issue of Velocity Saturation
(cid:131) We often assume that MOS current is a quadratic
function of Vgs:
(cid:1... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/6e414dbc7052cda18a7b4f76fd566ced_lec8.pdf |
. Perrott
0
0.4
0.6
0.8
1
1.2
(Volts)
V
gs
1.4
1.6
1.8
2
MIT OCW
Example: Gm Versus Voltage for 0.18µDevice
Vgs
Id
M1
W
L
=
1.8µ
0.18µ
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
)
s
t
l
o
V
/
s
p
m
A
i
l
l
i
m
(
m
g
H.-S. Lee & M.H. Perrott
0
0.4
0.6
0.8
1
g
versus V
m
gs
1.2
(Volts)
V
gs
1.4
1.6
1.8
2
MIT OCW
Exam... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/6e414dbc7052cda18a7b4f76fd566ced_lec8.pdf |
directly with W
(cid:131) This is independent of bias regime
- We can therefore relate gmof devices with different
widths given that they have the same current density
H.-S. Lee & M.H. Perrott
MIT OCW
A Numerical Design Procedure for Resistor Amp – Step 1
Vdd
αIbias
Vin+
M5
R
Vo-
Ibias
M1
M2
2Ibias
M6
(cid:131) Two k... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/6e414dbc7052cda18a7b4f76fd566ced_lec8.pdf |
current density =
115 µA/µm
(cid:131) Note that current
density reduced
as gain increases!
- ft effectively
decreased
MIT OCW
600
700
Example (Continued)
(cid:131) Knowledge of the current density allows us to design
the amplifier
- Recall
- Free parameters are W, Ibias, and R (L assumed to be fixed)
(cid:131)... | https://ocw.mit.edu/courses/6-776-high-speed-communication-circuits-spring-2005/6e414dbc7052cda18a7b4f76fd566ced_lec8.pdf |
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