Download quantum system .

Lecture
10
Linear
-
time
last
From
10-1
out
c
.
want
we
:
to
linear
study
in
rcspouie
Hamiltonian
with
system
quantum
a
response
H=HotV
Towards
this
picture
end
use
to
relevant
The
.
[
)
id+A±lt
these
an
preliminaries
for
expression
system
The
.
notation
Ho
,
in
)
we
:
]
1
subscript
the
quantum
a
done
be
will
drop
we
in
response
derive
to
ready
are
,
calculation
following
the
in
the
from
I
.
the
classical
H=
we
,=lH
A
are
.
}
,=lH
linear
the
picture
interaction
As
[
=
is
,
eats
response
With
an
✓
appropriate
Re
is
evolution
time
iot9±=
Linear
picture
interaction
the
,
are
case
,
Hot
interested
Perturbation
8<131+77
V
in
Ho
=
at
=
Tr
{
Bitch
)
t
time
%)
-
}
t
=
f
to
Tr
(
Bltlglt
'
)
)
described
)
B
observable
in
on
system
a
fltlalt
-
response
switched
V
study
we
'
dt
=
to
.
with
the
y
10
t
8431+17
=
if
-
Trf
[
BHI
'
vlt
)
,
)}
't
glt
-
z
'
dt
to
NOW
Trl
use
d)
A[ B.
get
ABC
(
Tr
=
To
(
Tr
=
)
ACB
-
=
Tr (
ABC
-
BAC
)
the
equilibrium
C)
[ Ai B)
t
Tr{
}dt
=
if
-
[
Bit
)
's
vet
,
91T
]
'
't
to
ifotdt
=
API
to
first
in
g-
°
'd
[
'
BH
+
i
790
< [
del
)
Alt
,
't
FHYSIEY
]
,Alt 't ] 7
Bltl
'
oflt
}
)
to
which
(
write
we
to
taking
-7
b
-
)
t
8<131+17
)
=
-
where
cfpalt
Because
of
DA
and
commentator
A
=
<
-
dtl
)
A 7
by
given
Ali
)
,
we
] )
for
A
sB=
and
°
B
<
-
then
¢
,
alt
)
=
it
[
oalo
DBIH
,
)
] )
.
t
subtract
also
can
,
and
B
from
averages
[ BIH
it
=
the
'
fit
1
is
response
)
'
t
p
linear
the
4,3 alt
-
1370
>
o
10-3
form
Alternative
had
We
d. a
iTr{BH[a.
=
] }
÷zTr{
↳
#
't
%
=
-
)[
Bit
]
)
Hfettlo
PHOA
)etHo
ipltia
PHO
13*0
commututor
the
Rewrite
EP
,
A
]
follows
as
pH
.
e-
=
Ae
-
.
:
(
=
-
A)
e-
EPHAEPHO
tltoaetko
§dt
=
=
=
of
§dt
§de
e-
if
=
di
.
,
it
P
PH
e-
e-
(
HA
-
AH
+
o
"
e-
[ A.
ni (
e-
)
ii
e-
0
0
where
in
the
picture
⇒
Alt
%a
't
)
step
last
)
eiltota
=
Tr{
=
=
=
Bet
§
4
i
{
Bit
e-
[ BH
in
the
interaction
ittot
)§dt
)A'
that
used
we
ni
lit
)o
(
it
)
,A6
)
] )
.
)%
de
}
Kubo
formula
§
10
Recap
-
y
D
8<131+17
/
=
X
,3a(
t
'
t
-
)
'
flt
dtl
1
Ict
b
-
Where
It )
XBA
the
ensures
Note
¢
,
,n
)
th < [
=
fs
Then
:
p
Alo
,
=)
] >
)
<
BH
) A.
Lite )
)
A
limit
of
{
0
→
the
0
into
goes
quantum
second
result
classical
the
observable
an
)dt
Alttitt
¥§
=
tame "tde
"
e-
~
¢Balt
Since
Note
"
{
so
Bltlitlol
)
p
=
)
{
equilibrium
Blttslnts
XBAH )
.
correlation
) >
< Blt )
=
=
-
de
°
.
manifestly
=
BH
formula
Kubo
A.
13<131+7
=
It )
transform
Kubo
)
classical
formula
The
the
by
given
is
.
1072g
!
the
In
:
+70
Causal
is
lopalt
:
quantum
a
+
it
1
response
classical
it
°
response
linear
The
admittance
the
is
)
function
,={
at
OH
)
step
Heaviside
The
.lt
4,3
=
Pok
Alo )
)
)a¥
( Bh
⇒
4
functions
Bit
)
)
Flo ) )
stationary
are
) >
Bk
nilo
.
.
-
<
)
Alo ) >
.
10-5
i
Electric
)
B
J
=
=
SE
-
where
V
.
Ehl
I
d3r
-
=
volume
=
it
tr §
)
qiri
:
F.
E✓
polarization
frill
electric
uniform
,
,
I
field
electric
applied
to
Elw
olw )
density
current
=
response
V
jlw7=
conductivity
D=
note
qq.fi
I
d-
e
T
Examples
=
=
,
=
-
=)
A
VP
=
get
then
we
Oijltl
Xjpct
=
The
d-
)
C
jiltlisjlo
jikljnjlo
<
)
>
alt
)
) )
that
means
oijlw
<
pv
=
PVOIH
=
This
)
=
prop
jiltlj
<
Conductivity
.
is
;
(
o
eiwtdt
>
)
obtained
in
the
w→o
a
cry
.
=
Oijlw
→
o
)
=
PV )
.
<
Iiltljjlo
)
>
dt
.
limit
10.6
ii
Dynamic magnetic susceptibility
)
B=
✓
=
at
vk.is
-
,
a
,
VK
=
p
)
Xlw
Xmmlw )
=
pv )
=
)
Mit
<
kilo
)
)
eiwtdt
0
In
classical
static
the
case
:
p
Xlw
-
01
=
=
PVS
(
pvf
<
-
wit
0
=
since
<Mcas
(
pu
=
)Ml°
) )
uncorrelated
This
result
may
be
-
(
o
Mlo
)
mlo )
472
(
pu
(
in
)
Mit
obtain
we
)
)
)
)
dt
)ml°
)
min
-
dt
) >
mt )
<
become
=
hula
as
familiar
)
t
> < mlo
→
) >
since
s
.
to
you
.
Mlt
)
and
Mo )
10-7
,
perturbation
has
,
the
response
,
of
propertiei
analytical
Namely
.
the
after
comes
the
for
consequences
susceptibility
dynamical
the
that
fact
the
Causality
properties of
analytical
and
Causality
X
if
t
S<
)
Bit
7
ftp.alt
=
'
-
t
fplt )
1
'
dt
'
is
-
that
such
XBACW )
=
Blw ) )
<
8
Then
•
Xlw
)
)
iwt
4h
S
=
falw
)
dt
e
0
We
function
this
continue
can
a
Xlz
)
izt
f
=
freqnuccic
complex
to
,
dt
date
0
Inz
.
-
For
In
?
't
0
,
function
y
analytic
in
upper
the
-
and
o
→
e
as
half
plane
that
means
§dw
with
behaves
integrand
therefore
is
This
the
t
-70
'
×#-iy
'
w
-
w
this
see
To
,
.
integration
÷
@
into
=o
+
the
upper
-
need
just
we
half
of
extend
to
the
complex
the
plane
{ dtI¥+I=o
,
since
the
only
pole
is
at
2-
=
w
.
in
this
10-8
Now
the
use
can
we
,
Note
:
P
the
To
see
real
§
The
"
××I÷y
defines
,¥yfH
d
.
part
principal
the
'
#
is
it
°
{
limit
the
Y
-70
ex
o
it
P
X
=o
xtiyldx
-
1°×¥f←=
.§×¥,,←iId×=÷fsT¥I
=
-
Ziti
Iz
I
=
thrfore
and
in
t¥'dxt§±I'
'¥d×=
¥
!
=P
¥IIi¥i
=
part
imaging
×÷f
value
)
write
this
part
-
tiitocx
principal
×÷j=
The
(k )
=P
×±÷j
with
expression
hypo
¥j
=
Tax
)
and
him
y
-70
×t÷j
=
Ptx
Iitck
)
,
III
Introducing
the
XCW )
Note
get
we
double
"
(
w
)
.
the
dw
here
are
"lw
Xiw
)
)
=
Kramer
Generally
response
through
's
-
and
the
"
gives
)
' '
w
-
'
(
w
1
-
IT
X
the
,
equation
w
'
-
w
p
×
Kramer
'
w
this
'
W
-
part
real
of
-
X
relations
kronig
the
'
'(
X
'
Tt
dw
Pf
=
dw
)
p
-
-
(
part
imaginary
is
X
"ii
-
=
and
-
dispersion
or
'
is
reactive
the
dissipative
krouig
'
9
derivatives
not
:o)
itslwwiflxtwytix
'
,
real
10
as
)
w
Prime
X(
.§[pC±
:
o=
Taking
{
it
+
of
part
imaginary
'
X
=
and
pine
:
and
real
relations
part
.
.
relations
part
These
of
are
the
related
:
)
]
l
Time
reversal
-
Most
symmetry
quantitie
time
reversal
→
5
last
→
reversal
Ea
±
=
T
operator
.
(
T
x
[
-
=
that
A
transform
)
"
T
Ea
=
Al
-
t
)
.
anti
An
F)
,
Tale
transformation
The
a
→
)
-
operators
consider
we
Alt
where
Fxl
-7
FXF
=
time
under
an
transtrm
should
therefore
and
is
5
since
understood
be
and
generally
:
( spin )
5
,
L
Note
-
momentum
one
,
under
I
-
can
one
angular
So
sign
I
→
I
like
change
or
:
I
The
invariant
either
are
desciped
is
T
operator
unity
147+131+7 )
x*Tl9
=
anti
is
)
+
p×
antiuuikz
an
by
linear
.
It
>
satisfies
and
<
Warning
It's
:
Ttlte
The
safe
adjoints
)
notation
Dirac
to
that
use
+147
<
=
the
involve
is
*
designed for
equations
above
T
.
but
linear
be
operators
way
of
.
o
-10
Q
What
Uk
T=
where
basis
K
,
usually
taken
(
conjugation
complex
is
be
IF ?
4=1
and
to
fermions
spintess
For
gives
write
always
Can
that
operator
the
is
.
)
K
spintnl
with
o
I
-
acts
that
U
For
.
spin
Pauli
?t
(
-
=
a
th
,
,
)k
,
need
freedom
of
=(%)
IB
-
we
operators
ox
where
KC
=
degrees
spin
the
K
some
( 1<2--1 )
r
=
fermions
spin
the
#
K
to
respect
with
.
>
5
K
For
Lo -11
?
T
-
tz
only
we
)
represent
can
,og,sz )
Cox
E=
matrices
on
a=⇐ ;)
are
need
We
'
uE*i
T2=
while
tr
spin
T2=
There
therefore
are
systems
.
Those
K
.
5
2
=
does
isy
U=
the
job
.
fomious
spin less
for
that
Note
-
that
yourself
Convince
=
1
tz
igkiogk
two
wwe
=
classes
T2=+|
#
iogti
of
time
and
=
.
-
oj2=
reversal
those
-
1
.
symmetric
with
T2=
-
1
.