time dep quantum mechanics.pptx

#$%&'(%')$%"*+(%&$),'-"
!"
TION
be observed itself, but through it we obtain the probability density
amiltonian operator which describes all interactions between particles and fields, and
which characterizes the probability that the particle described by
is between
a
of
the
kinetic
and
potential
energy.
For
one
particle
(1.3)
ution for time-independent Hamiltonians
time t.
n of the state of a quantum system is described,#+C*+)+-*+-6"5./-6.#"#+%0/-$%("
by the time-dependent
(1.2)
is between
and
atcovered
he probability that the particle described
Mostbyof what
you have previously
is time-independent quantum m
n (TDSE):
is assumed to be independent of time:
where we mean that the Hamiltonian
of the system is expressed
through
the
wavefunction
.
The
wavefunction
is
!
+5="*$"A%0&'*$-;+&"*$)+-*+-6+"*/1"6+#)'"
i! " covered
r , t = Ĥis
" time-independent
r,t
(1.1)
t you have previously
then assume
a solutionquantum
with a mechanics,
form in which the spatial and temporal variabl
!t
nd cannot be observed itself, but through it we obtain the probability density
is assumedwavefunction
to be independent
of time:
. We
he Hamiltonian
are separable:
an operator which describes
all interactions between particles and
fields, and
*
'((+&3/:$1+4"*+-($6D"*$")&':/:$1$6D"
(1.3)
!
r
,
t
!
r
,
t
d
r
=
!
r
,
t
!
r
,
t
P
=
"
tion with a form in
which the spatial and temporal variables in the
netic and potential energy. For one particle
1-2
arable:
(.))'-$/#'"%0+"1+"('1.2$'-$"($")'((/-'"(%&$3+&+"%'#+4"
racterizes the probability that the particle described by
is between
and
at
"
r " r
Ĥ
!
1 (1.2)
i!
=
t" of"5.$-*$"('(,6.+-*'4"
(1.4)Tist a =function of
the left-hand side! is"r a, t function
time (t), and the right-hand
side
" " r" Tonly
" r
T t !t
"
only ( , or rather position and momentum). Equation (1.5) can only be satisfied if both
st
of is
what
you%0+"7"('**$(8/9/"(+"+-6&/#:$"$"#+#:&$"('-'".;./1$"/*".-/"%'(6/-6+"<=""
have
previously
covered is time-independent
quantum mechanics,
. The wavefunction
is
stem
expressed
through
the wavefunction
are equal to the >/1"#+#:&'"/"*?"($"'@+-+"+5="*$"A%0&'*$-;+&"$-*$)+-*+-6+"*/1"6+#)'"
same constant, . Taking the right hand side we have (1.5)
mean that the Hamiltonian
is assumed to be independent of time:
. We
be observed itself, but through it we obtain the probability density
Ĥ r ! r
me a solution with a form= in
E which"the spatial
Ĥ r and
! r temporal
= E! r variables in the (1.6)
! r
(1.3)
on are separable:
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ()
()
( ) ( )
()
( ) ( )
( )
( )
is our beloved Time-Independent Schrödinger Equation (TISE). The TISE is an eigenvalue
is between
and
at
the probability that the particle described by
(1.4)
are the eigenstates and E is the eigenvalue. Here we note
ion, for which
B"
beloved Time-Independent Schrödinger Equation (TISE). The TISE is an eigenvalue
e of (1.5):
with the expectation value of the energy of the syste
mechanics we associate
or which
are the eigenstates and E is the eigenvalue. Here we note
the left hand side of (1.5):
(1.7)
, so
is the operator corresponding
to E and drawing on classical
,#+C*+)+-*+-6"5./-6.#"#+%0/-$%("
we associate
with the expectation value of the energy of the system. Now taking
So, in the case of a bound potential we will have a dis
d side of (1.5): F"($-$(6&/"$-3+%+4"
corresponding
energy eigenvalues
from thesetTISE
(1.8)
So, in the case
of a bound potential
we will have a discrete
of e
()
nergy eigenvalues
()
(
)
(
)
# ! iE &
toTthe
T t =eigenvalues
expTDSE.
!iEt
/ ! from
= T0 exp
" t and there a
+ corresponding
T t = 0 solutions
(1.7)
energy
the !i
TISE,
0
%
(
with
discrete
set
of
eigenfunctions
$ !t ! '
solutions to the TDSE.
So,
in
the
case
a bound
potential
we will have a discrete set of eigenfunctio
from the TISE, and thereofare
a set of
corresponding
1 !T
i!
=E "
of a bound potential
we
will
have a
!t
T t
()
where
is a (complex)
G+"('1.2$'-$"($")'(('-'"5.$-*$"(%&$3+&+"%'#+4"
fromand
the TISE,
and thereamplitude.
are a set Th
of
corresponding energy eigenvalues
(1.8)
"
2
can (complex)
= 1 . Sinceamplitude.
the only time-dependence
exp #ii$to
t the TDSE. set, so that n (1.9)
"! n r , t = cn" n r solutions
The n eigenfunc
n where ! n = En / ! and !is
with
case of a bound "potential we will have a discrete set of eigenfunctions
(1.3)an
is orthonormal
independent
of time
for the eigenfunctions
"
. Since the only
time-dependence
is a phase fac
set, so that form
and
is a (complex)
amplitude. The n eigenfunctions
(6/6'"(6/2$'-/&$'"
1-3
+"$-";+-+&/1+4"
ng energy eigenvalues
from the TISE, and there are a set of corresponding
not
with time
and areThe
called
stationary states.
whereis a phase
is change
a (complex)
n eigenfunctions
form
. Therefor
(1.3)factor,
is and
independent
of timedensity
foramplitude.
the eigenfunctions
.
Since
the
only
time-dependence
the
probability
the TDSE.
# i$ n t
more
generally, a system may exist as
! r , t = " cn! n r , t = " cn enot
% n r with time However,
(1.10)
change
and
are
called
stationary
.
Since
the
only
time-dependence
is states.
a phase factor, the pro
set,
so
that
dent of time for the eigenfunctions
do(1.9)
n
n. Therefore, the eigenstates
However,
more
generally, a system may exist
asE" a linear
. Therefore,
thecombi
eigen
(1.3)
is independent
of time
for
the
and are called
stationary
states.
etime
(complex)
amplitudes.
For such
a case,
the probability
density
willeigenfunctions
oscillate with
and
is a (complex) amplitude. The n eigenfunctions form an orthonormal
notaschange
time and are
called stationary states.
n, example,
consider
two eigenstates
more generally,
a system
may exist
a linearwith
combination
of eigenstates:
TDSE.
(
( )
)
()
( )
(
)
( )
1-3
where
(1.10)
(1.10)
are (complex) amplitudes. For such a case,
the probability density will oscillate with
,#+C*+)+-*+-6"5./-6.#"#+%0/-$%("
As an example,
consider
two aeigenstates
retime.
(complex)
amplitudes.
For such
case, the probability density will oscillate with
n example, consider
two eigenstates
)+&"*.+"/.6'(6/,4"
"
#i$1t
#i$ 2 t
!
r
,
t
=
!
+
!
=
c
"
e
+
c
"
e
2
1 1
2
For this state the probability1 density
oscillates
in 2time
as
(1.11)
( )
()
2
2 as
te the probability density oscillates
in time
P t = ! = !1 + ! 2
2
(1.11)
:
:
2
= ! 1 + ! 2 + 2 Re "#! 1*! 2 $%
2
2
= c1&1 + c2& 2 + c1*c2&1*& 2 e
2
2
(
' i ( 2 ' ( 1t
(
)
+ c2*c1& 2*&1e
)
= ! 1 + ! 2 + 2 ! 1 ! 2 cos ( 2 ' ( 1 t
(
)
+ i ( 2 ' (1 t
(1.12)
(1.12)
We refer to this as a coherence, a coherent superposition state. If we include momentum (a
1/")&':/:$1$6D"!"#$%%&"%'-".-/"8&+5.+-2/")/&$"/11/"*$I+&+-2/"*+11+"*.+"/.6'8&+5.+-2+="
wavevector) of ($")/&1/"*$"#!'(')*&4"('3&/))'($2$'-+"%'+&+-6+"*$"(6/,="
particle associated with this state, we often describe this as a wavepacket.
o this as a coherence,
a coherent superposition state. If we include momentum (a
"
'(%$11/2$'-$"J'((+&3/:$1+"()+&$#+-6/1+"%'--+(('"%'-"*$-/#$%0+"%'+&+-,"
) of particle associated
with this state, we often describe this as a wavepacket.
Time Evolution Operator
More generally, we want to understand how the wavefunction evolves with time. The TDSE is
olution Operator
H"
commutes with . Multiplying eq. (1.14) from the left by
, we can write
, we
getget
1-5 describes the time-evolution of the
, we
We are also interested in the equation of motion for
which
if we substitute the projection operator
(or identity relationship)
nvely,
Operator
acts to
conjugate wavefunctions. Following the same approach and recognizing that
,
the
left: operator
.
I have
the definition
the exponential
for The TDSE(1.17)
want
to used
understand
how the of
wavefunction
evolves
with time.
is
(1.17)
')+&/6'&+"+3'1.2$'-+"6+#)'&/1+"
(1.21)
,
ce the TDSE is deterministic, we will define an operator that describes the
vely, if we substitute the projection operator (or identity relationship)
and integrating
, we
get
em: we see
we get
1.19),
(1.18)
(1.18)
1L')+&/6'&+"+3'1.2$'-+"6+#)'&/1+"M"*+(%&$3+"*$-/#$%/"*+1"($(6+#/"
(1.21) "/1"6+#)'"6="
! t = Û t,tt0 ! t0
(1.13)
)&')/;/-*'"1/"8.-2$'-+"*L'-*/"*/11L$(6/-6+"$-$2$/1+"6
()
( ) ( )
N
(1.22)
time-propagator
is
that
evolves
system as a function of .time. For the timeethe
time-propagator
isquantum
1.19),
we see the($"*$#'(6&/"%0+"(+"O"-'-"*$)+-*+"*/"64"PQR"
tonian:
Evaluating U(t,t0): Time-independent Hamiltonian
# "iĤ t " t0 &
" !iĤ t ! t0 %
! r , t of=eqn.
exp
r ,bet0 expressed as:
(1.19)
( ! can
% suggests that
Û t, t0 = exp $
(1.19)
Direct
(2.40)
' , , integration
!
! So now
m is useful when
are characterized.
our$ time-developing
wave'
#
& we can write
.
(1.22)
(1.14)
(
( )
)
(
( )
)
(
)
So, we see that the time-propagator is
as
5.$-*$4"
, which
function
of will
an define
operator.
ll define an operator
"
is an
operator, we
this operator through the expansion:
Since is a
"i En (t "t0 ) / ! So now we can write
*+S-$2$'-+"*$"+()'-+-2$/1+"*$".-"')+&/6'&+4"
m is useful when "! arer ,characterized.
our
time-developing
t =e
!
.
(1.20)wave# i$ n ( tr#t,0 )t
n
n
0
.
(1.20)
2
! n r ,its
t expansion
= " n % ein a Taylor
" n series:
! n r , t0
2
,
erator is defined through
"
%
H
t
!
t
( 0 )& + …
( !i +
" iH
%
i
!iH
i
iH
"
n
t ! t0 ) + * - #
exp $ ! ( t ! t0 ) ' = 1 +
(
as
2
!
) !,
')).&+4"
# !
&
# i$ t #t
= e n( 0) c
.
(1.23)
(( ))
(
%
)
(
)
( )
n
(2.42)
(2.43)
(1.17)
(1.18)
(2.44)
(1.19)
(2.45)
n
and therefore
# i$ n ( t #t0 )
! n r , t == %
" n c%
e
! n r , t0
t
" n Note " n commutes
n
at all
n
(1.16)
(
()
n
)
. You can confirm the expansion satisfies the equation of motion
.
(1.20)
K"
= %e
cn
.
(1.23)
PQR"*$#'(6&/2$'-+"/11/"1/3/;-/"
To evaluate U for the time-independent
s written in eq. (1.13), we seen that the time-propagator
acts to theHamiltonian,
right (on we expand in a set of eigenkets:
= c t "
(
# i$ n t #t0
()
)
for .
2-9
.
(2.38)
typically not
This is a reflection of the importance of linearity in quantum systems. While
equal to
(1.23
acting to the left:
the Hermetian conjugate of
(2.41) (1.18)
,
(1.2
')+&/6'&+"+3'1.2$'-+"6+#)'&/1+"
Note, since U acts to the right, order matters:
of U(t,t0) is As written in eq. (1.13), we see that the time-propagator
actssee
tothat
the right (o
e Properties
that the time-propagator
as Hermetian,
you can
From its definition as an expansion and recognizing
(2.38)
(
)
(2.42)
kets)
evolve
the
system
The evolution
of the
conjugate
wavefunctions (bras) is und
1) Unitary. Note that for
eq.to
(2.38)
to hold
to be conserved,
U must
" !and
t !probability
tin %time. density
iĤ for
"
%
( )
0
Û t, t0 = exp $
',
be unitary
the Hermetian conjugate
of
!
&
#
Properties of U(t,t0)
(
)
iiH
Ĥ t ! t0
(1.19)
Ûacting
t,tt0 to=the
expleft:
$
'
!
$#
'&
†
( )
(1.2
Equation (2.41) is already very suggestive of an exponential form. Furthermore, since time(2.39)
()
( ) ( )
†
ore
!
(t)
=
Û(t,
t
)
!
(t
)
!
t
=
!
t
Û
t,tt
0 also suggests
0
0 the time0
is continuous
and
the
operator
is
linear
it
what
we
will
see
that
1) Unitary. Note that for eq. (2.38) to hold and for probability density to be conserved, U must
(1.24
which holds only if
. In fact, this is the reason that equates unitary operators with
propagator is only
dependent
on a time interval
be
unitary
. and recognizing
(1.20)
probability conservation.
as Hermetian,
you can see that
From its definition as an expansion
')+&/6'&+".-$6/&$'4"
2) Time continuity:
() ()
P= ! t ! t
( )
( )
= ! t0 U U ! t0
†
(2.43)
(2.39)
"
2-9
.
In
fact,
this
is
the
reason
that
equates
unitary
operators with
which holds only if
(2.40)
%'-,-.$6D"-+1"6+#)'4" U ( t,t ) = 1 .
"probability conservation.
(2.43
Note,
since
U
acts
to
the
right,
order
matters:
3) Composition
property. If weUtake
then it stands to (2.41)
reason
("t2 ,t0the
) "= Usystem
(t"2 ,t1 )toU"(bet1,tdeterministic,
"
%'#)'($2$'-+4"
"T=U="'&*$-+"-'-"7"$-*$I+&+-6+4
0 )"
Time
continuity:
Time-reversal.
The "get
inverse
the
time-propagator
is the time
reversaltooperator.
From
eq.one step
" of
"" wavefunction
that we2)should
the
same
whether
we evolve
a
target
time
in
!
t
=
U
t
,t
U
( 2 ) ( 2 1 ) (t1,t0 ) ! (t0 )
Note, since U
acts to the right, order matters:
"
(2.41):
or
:
.
(2.40)
= U ( t2 ,t1 ) ! ( t1 )
" multiple steps
,#+C&+3+&(/14" U ( t,t0 )U ( t0 ,t ) = 1
(2.32)
(2.42)
2-9
(1.25
and
(2.41)
"
(2.42)
3) Composition property. If we take the system to be deterministic, then it stands to reason
Equation (2.41) is already very suggestive of an exponential form. Furthermore, since time
V"
.
(2.33)time in one step
that we should get the same wavefunction
whether we evolve to a target
is continuous and the operator is linear it also suggests what we will see that the time
Equation (2.41) is already
very suggestive
of an exponential
form. Furthermore, since time
or multiple
steps
:
has units of frequency. Since (1) quantum mechanics says
and
(2.44)
nics the Hamiltonian generates time-evolution, we write
n operator, we will define this operator through the expansion:
')+&/6'&+"+3'1.2$'-+"6+#)'&/1+"
(2.39)
(2.45)
ction of time. +5./2$'-+"*+1"#'6'"*$"M4"
Then
( )
( )
!
U t,t0 = ĤU t,t0
(2.40)
+5.$3/1+-6+"/"+5="*$"A%0="*$)="*/1"6+#)'X"*D"1+"(6+((+"$-8'Y"
mutes at all . You !t
can confirm
the expansion satisfies
the equation of motion
t by
i!
gives
the TDSE
!R"(+"O"-'-"*$)+-*+"*/1"6+#)'"PQR4"
(
aluate U for the time-independent Hamiltonian, we expand in a set of eigenkets:
H n = En n
! nandn = 1
(2.46) 2-12
n
( )
(
)
U t,t0 = & exp "# !iH t ! t0 / ! $% n n
n
(
)
= & n exp "# !iEn t ! t0 / ! $% n
n
()
( ) ( )
! t = U t,t0 ! t0
(2.47)
# "i
&
= ) n n ! t0 exp % En t " t0 (
!#"#
$
n
$%
'
( )
(
)
(
( )
cn t0
()
= ) n cn t
n
Expectation values of operators are given by
(2.48)
W"
(
integration is
2-13
')+&/6'&+"+3'1.2$'-+"6+#)'&/1+"
he time-evolution operator: Time-dependent Hamiltonian
ression for U describes all possible paths between initial and final state. Each of these
or U describes all possible paths between initial and final state. Each of these
t may seem
straightforward
dealacquired
with. If phaseis of
a function
of time, under
then the
nterfere
in ways
dictated bytothe
our eigenstates
the timen ways dictated by the acquired phase of our eigenstates under the timeon
nt of
Hamiltonian.BR"(+"O"*$)+-*+"*/1"6+#)'"1L$-6+;&/2$'-+"8'&#/1+"*D4"
The gives
solution for U obtained from this iterative substitution is known as
onian. The solution for U obtained from this iterative substitution is known as
itive) time-ordered exponential
1L+()&+(($'-+"*$"
$ !i t
'
e-ordered exponential
M"*+(%&$3+"6.@"
(2.51)
U t,t0 = exp & # H t " dt " )
t
$")'(($:$1$"
% "i t % ! 0 (
(
U
t,t
!
exp
d
#
H
#
*
%/##$-$"*/11'"
0
+ '
$
ext Step:
& ! t0
)
%0+"($"&$('13+"%'-"#+6'*'"$6+&/,3'4"PQR"
(6/6'"$-2$/1+"1"
e this exponential as an expansion in a series, and substitute into the equation
of
t
%
(
"i
/11'"(6/6'"S-/1+"
"d!i
# %H t#d(* H (
!UTˆ t,t
exp '= 1 +
(2.58)
m it:
$
[X"/9&/3+&('"
t
0$
0 & !
)
(2.58)
# ! '& t0 )
(6/,"$-6+&#+*$"
n
1
#"+*"-"
#
t
+ "i . t2
The expression
for U describes all possible paths between initial and final state. Eac
= 1 + 2 -+ " 0!i $% d#t nd$( d(#dn(…H$ (d#H
H
#
H
#
…
H
#
(2.52)
n"1
(2.57) by the acquired phase of our eigenstates under
* t0 1 ( * n
t0)
t
paths1 interfere in ways dictated
n=1 , !$ /! ' 0 )t0
t0
# &
( )
( )
()
( )
() ()
( )
( ) (() ) ( )
( )
dependent Hamiltonian. The solution for U obtained from this iterative substitution is
# i" !i % t &(
(*
the (positive) time-ordered
exponential
the Tyson
time-ordering
operator.)
In
this
expression
time-ordering
is:
wown
the as
eigenstates
H,+ %we
eq.
(2.46)
to
express
U (as* the
= of
exp
!+ $could
d' ! use
H
(
!
)
d
(
d
(
d
(
H
(
H
H
(
U
(
,t
*
**
**
**
(
0
)'t0 expression
)t0
e Tyson time-ordering $operator.)
the time-ordering is:
!# !t & )t0In this
t'
"
3
() ( ) ( ) (
)
(2.59) For
(2.53)
"1 !
" 2 !be
" 3 aware
.... " n !that
t there is a time-ordering to the
t0 !you
om this expansion,
should
interactions.
(2.59)
!
…
"
!
"
!
"
t
##
#
before , which acts before :
.
e third term, acts
0
expression tells you about how a quantum system evolves over a given time interval, and
sn dangerous;
we how
are not
treating system
as evolves
an operator.
aretime
assuming
that
the
tells Imagine
you about
a quantum
over aWe
given
interval,
and
you
are
starting
in
state
and
you
are
working
toward of
a target
s for any possible trajectory from an initial state to a final state through any number
differenttrajectory
times commute!
. Itstate
is only
the any
casenumber
for special
possible
from
an
initial
state
to
a
final
through
of between
diate
states.
Each
term
in
the
expansion
accounts
for
more
possible
is known as the
Tyson time-ordering operator.)
( transitions
.
The
possible
paths
and
associated
time
variables
are:
ate
th aEach
highterm
degree
of symmetry,
which the
havetransitions
the same between
symmetry
s.
in the
expansion in
accounts
for eigenstates
more possible
t intermediate quantum states during this trajectory.
Z"
In this expression the time-ordering
2.3
SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
The mathematical formulation of the dynamics of a quantum system
is not unique. So far we
&/))&+(+-6/2$'-$"*$""
have described the dynamics by propagating the wavefunction,
which encodes probability
A%0&'*$-;+&"+"O+$(+-:+&;"
densities. This is known as the Schrödinger representation of quantum mechanics. Ultimately,
since we can’t measure a wavefunction, we are interested in observables (probability amplitudes
associated with Hermetian operators).
Looking at a time-evolving expectation value suggests an
A%0&'*$-;+&")$%6.&+4"*$)+-*+-2/"6+#)'&/1+"$-%1.(/"$-"
8.-2$'-+"*L'-*/""
alternate interpretation of the quantum
observable:
()
 t
( ) ( ) = ! (0) U
= ( ! ( 0 ) U ) ˆ (U ! ( 0 ) )
= ! ( 0 ) (U ÂU ) ! ( 0 )
= ! t  ! t
†
()
ÂU ! 0
†
(2.65)
†
The last two expressions here suggest
alternate transformation that can describe the dynamics.
O+$(+-:+&;")$%6.&+4"*$)+-*+-2/"6+#)'&/1+"$-%1.(/"
$-"')+&/6'&+"
These have different physical interpretations:
1) Transform the eigenvectors:
2) Transform the operators:
(1)
. Leave operators unchanged.
. Leave eigenvectors unchanged.
\"
Schrödinger Picture: Everything we have done so far. Operators are stationary.
to this part of the Hamiltonian, that we may be able to account for easily.
s described by
2-21
Setting V to zero, we can see that the timeWe
evolution
of the
exact partofofmotion
the Hamiltonian
are after
an equation
that describes the time-evolution of the in
(2.84)
is described by
THE INTERACTION PICTURE
picture wave-functions. We begin by substituting
eq. (2.87) into the TDSE:
e, most generally,
&/))&+(+-6/2$'-+"*$"$-6+&/2$'-+"
The interaction picture is a hybrid representation that is useful in solving problems with time(2.84)
(2.85)
dependent Hamiltonians in which we can partition the Hamiltonian
as
()
()
H t = H0 + V t
(2.83)
.,1+"5./-*'"O"($/")/&,2$'-/:$1+"%'#+""""""""""""""""""""""""""""""X"*'3+"]P6R"7".-/")+&6.&:/2$'-+"(.I="
(2.85)
where, most
generally,
(2.86)
or a time-independent
)$%%'1/"*/";/&/-,&+"%0+";1$"/.6'(6/,"*$"O "($/-'"/-%'&/".-/":.'-/":/(+="
N
is a Hamiltonian for the degrees of freedom we are interested in, which we treat exactly, and
"
! iH ( t !t ) !
efine
wavefunction
in the interaction picture
^'(6'"]P6RJN"($"6&'3/4"
U 0 ( t,t0 ) = ethrough:
(2.86)
buta for
a time-independent
is a time-dependent
potential
can be (although for us generally won’t be) a function of time.
"
which canin
complicated.
In!the
picture we will treat(2.87)
each part of the Hamiltonian in
!bethe
t " U 0 ( t,t0 )picture
t interaction
We define A$"*+S-$(%+4"
a wavefunction
through:
S ( )interaction
I( )
0
0
a different representation. We will use the eigenstates of
as a basis set to describe the
! I = U 0† ! S
(2.88) (2.87)
wherethat
dynamics induced by
, assuming
is small enough that eigenstates
of
are a
basis to describe
If
is not ainfunction
time,
time-dependence
ctively
representation
definesH.
wavefunctions
suchthe
a of
way
thatthen
the there
phase
"is#simple
or the interactionuseful
(2.88)
satisfies
Schrödinger
equation
with a new Hamiltonian: the inte
I
= VI # I
! i!
('(,6.+-*'"$-"+5="*$"A%0="7")'(($:$1+"*$#'(6&/&+"%0+"3/1+"PQR4"
to this
part of the Hamiltonian,
we may
able to account
for easily.
"t
is removed.
For small V,that
these
are be
typically
high frequency
mulated under
, which
is thethat the
unitary
. Note: Ma
Effectively" the interaction representation defines Hamiltonian,
wavefunctions in such
a way
phasetransformation of
lations relative "to the slower Setting
amplitude
changes
in coherences
V.
V to
zero, where
we
can see thatinduced
the timebyevolution
of the exact part of the Hamiltonian
!i
"
t
is removed. For
small
accumulated
under
k andfrequency
l"/.6'(6/,"*$"O
are eigenstates of
H.
VI = V,
k Vthese
l = eare lk typically
Vkl wherehigh
T=U="+1+#+-6'"*$"#/6&$%+"*$"]
I
_"7""""""""""""""""""""""""""""""""""""""""""""%'-"k"+*"l
N=" 0
is described by
oscillations""relative to the slower amplitude changes in coherences induced by V.
satisfies the Schrödinger equation with a new Hamiltonian: the interaction
We can now define a time-evolution operator in the interaction picture:
(2.84)
, which is the
unitary transformation of
. Note: Matrix ele
Hamiltonian,
where, most generally,
where k and l are eigenstates of H0.
where
!N"
(2.85)
.
(2.92)
he Schrödinger equation with a new Hamiltonian:
, which is the
the interaction picture
. Note: Matrix
elements in
&/))&+(+-6/2$'-+"*$"$-6+&/2$'-+"
unitary transformation of
where k and l are eigenstates of H0.
*+S-$2$'-+"*$"M"-+11/"&/))&+(+-6/2$'-+"*$"$-6+&/2$'-+4"
now define a time-evolution
operator in the interaction picture:
at
()
( )
( )
! I t = U I t,t0 ! I t0
(2.93)
$ !i t
'
U I t,t0 = exp+ & # d" VI " ) .
% ! t0
(
( )
()
( )
(2.94)
2-23
( ) ( )
! U t,t0 = U 0 t,t0 U I t,t0
2-23
(2.96)
(2.96)
Using the time ordered exponential in eq. (2.94), U can be written as
dered exponential in eq. (2.94), U can be written as
( )
(2.95)
( )
U t,t0 = U 0 t,t0 +
" !i %
) $# ! '&
(
n=1
(
n
+
t
t0
*n
*2
t0
t0
( ) ( )(2.97)
(
)
d* n + d* n!1 … + d* 1 U 0 t,* n V * n U 0 * n ,* n!1 …
) ( ) (
U 0 * 2 ,* 1 V * 1 U 0 * 1 ,t0
)
where we have used the composition property of
(2.97)
. The same positive time-ordering
. The same positive time-ordering
applies. Note that the interactions V(!i) are not in the interaction representation here. Rather we
at the interactions V(!i) are not in the interaction representation here. Rather we
used the composition property of
used the definition in eq. (2.92) and collected terms.
!!"
after
a few
Thisin isamplitude
perturbation
theory,
where
the with small coupling matrix
we
partition
aterms.
time-dependent
Hamiltonian,
(2.107)
well
for
small
changes
of
the
quantum
states
wavefunctions
(2.104)
ctly, but the influence of
on
is truncated. This
s relative to the energy splittings involved (
) As we’ll see,
Alternatively we can express the expansion coefficients in terms of the interaction (2.108)
picture
ndent
Hamiltonian
andcouplingis2-26
a time-dependent
plitudeexact
of thezero-order
quantum states
with small
matrix
2-26
2-26
lts we
obtain from perturbation theory are widely used &/))&+(+-6/2$'-+"*$"$-6+&/2$'-+"
for spectroscopy,
condensed
wavefunctions
+"6+'&$/"*+11+")+&6.&:/2$'-$"
eigenkets
and
eigenvalues
:
) As Notice
we’ll see,
gs
involved
(relaxation.
(This
notation
followsofCohen-Tannoudji.)
ynamics,
and
PERTURBATION
THEORY
TION
THEORY
RY
(2.105)
()
(2.108)
( )
, we
, :we
()
()
(2.109)
( )
: == ek Ufrom
on of the
wavefunction
thatcwavefunction
results
culate
the
evolution
of=the
:t ) n from that
unction
that
results
Ub (!
c ( t results
t ) ( t :) (2.106)
)
! ( t from
)
(
"
ke the specific case where we have a system prepared in , and we want to know the
=e
k U ! (t )
(2.109)
This
is
the
same
identity
we
used
earlier
to
derive
the
coupled
differential
equations
that
describe
!
t
=
b
t
n
(2.111)
nts
are given
(2.111)
( by
)system
" in( ) at time , due to(2.111).
ity of observing
the
= e theb ( t )
want to know
have athe
system
prepared
in , and we amplitude
change
in the time-evolving
of the eigenstates:
(2.107)
. timeFor
a (complex
timeed
differential
equations
amplitudes
of
.used
Forearlier
a for
complex
quations
for
the
amplitudes
For a complex
time-that describe
using
the
coupled
amplitudes
of) ! . differential
(2.114)
= identity
b for
bthe
=
k Uthe
t,t
Psame
(tto)differential
(t. ) oftheweequations
(t )derive
This
is
the
to
coupled
equations
at
time
, due
ion theory are widely used for spectroscopy, condensed
ck t = k U 0U I ! t0
Hwhere
t follows
=eigenkets
Hwe
+know
VCohen-Tannoudji.)
t the
awhere
Hamiltonian
where
the
for
eigenkets
n know
= for
EnNotice
we know
the
for
: Hwe
we
(This
notation
"ni# k,t:eigenkets
0
0
=
e
k
U
!
t0
stateProbability
of the system as a superposition of these eigenstates: I
ion
" i# k t
0
k
n
n
kI
0
" i# k t
I
I
n
0
" i# k t
n
k
2
k
k
k
I
0
express
expansion
inamplitude
terms
interaction
the
change
in the
time-evolving
ofthe
the
eigenstates:
m with
many
states
tocoefficients
be
considered,
solving
these
equations
isn’t equations isn’t
states
considered,
solving
these
equations
isn’t
ence
ortothe
abesystem
with
many
states
to
be of
considered,
solvingpicture
these
2-27
(2.114)
b
t
=
k
U
t,t
!
I , calculate
0 with
, calculate
,towe
can directly
choose ktowe
work
work
,as:
calculate
al.
Alternatively,
candirectly
choose
to workofdirectly
with
expansion
coefficient
the as:
interaction
picture
wavefunctions.as:Remember
So, is thewith
()
( )
(2.110)
(2.110)
(2.108)
$ i t
'
(2.112) (2.115)
bk t = and
k exp+ & ! # d" V. I If" necessary
" (2.112)
we can calculate
and (2.112)
then add in the
)
t
% ! 0 of the (interaction picture wavefunctions. Remember
So, is the expansion coefficient
Cohen-Tannoudji.)
Notice
extra oscillatory
term at the end.
(2.113)
(2.113)
and
. If necessary
we can calculate
and then
add in the
(2.113)
()
extra oscillatory term at the end.
()
(2.109)
!B"
(2.115)
&/))&+(+-6/2$'-+"*$"$-6+&/2$'-+"
+"6+'&$/"*+11+")+&6.&:/2$'-$"
i t
d" k VI " !
#
t
0
"
()
()
bk t = k ! !
$ !i '
+& )
% "(
#
t
d" 2
t0
()
#
"2
t0
()
bk t = ! k! "
( ) ( )
()
m
!i" !k t
( )
2
+
t
t0
d) 2
+
)2
t0
d) 1 e
!i* mk ) 2
+…
()
(2.117)
Vk! t
i t
"i$ !k #1
d
#
e
Vk! # 1
1
%
t
" 0
" !i %
+( $ '
# !&
(2.116)
d" 1 k VI " 2 VI " 1 !
k VI t ! = k U 0† V t U 0 ! = e
using
So,
2
“first order”
( ) e!i*
Vkm ) 2
"m) 1
( )
Vm" ) 1
(2.118)
+…
(2.119)
“second order”
The first-order term allows only direct transitions between
and
!E"
, as allowed by the matrix
element in V, whereas the second-order term accounts for transitions occuring through all
&/))&+(+-6/2$'-+"*$"$-6+&/2$'-+"
+"6+'&$/"*+11+")+&6.&:/2$'-$"
2-42
2-42
This indicates that the solution doesn’t allow for the feedback between
2.5 FERMI’S GOLDEN RULE
2-42
($;-$S%/6'"S($%'4"
2.5 FERMI’S GOLDEN RULE
for of
changing
populations.
Thisin
is the
reason we after
say that
validity adictates
"
FERMI’S GOLDEN
RULE
The transition rate and probability
observing
the system
a state
applying
2.5 FERMI’S
GOLDEN RULE
$1")'6+-2$/1+"*$"$-6+&/2$'-+"]")&'#.'3+"6&/-($2$'-$"6&/"/.6'(6/,"*$"O
The transition rate and probability of observing
the system in a state
N="($"(.))'-;/"
the in
constant
perturbation
allow
forto the
feedback
perturbation
to !thefrom
ansition rate and%0+"1'"(6/6'"$-2$/1+"($/"""""""""+"5.+11'"S-/1+"($/""""""""""=""
probability
of observing
system
a state first-order
applying
a doesn’t
Ifk after
is not
an eigenstate,
we only
need
express
it as a superpos
! from
the constant
perturbation doesn’t allow
perturbation
k after
The transition rate
and probability of observing the
system in to
a state
applyingfirst-order
a
$1")&$#'"'&*$-+"%'-($*+&/"6&/-($2$'-$"*$&+9+"*/""""""/"""""""X""#+-6&+"$1"(+%'-*'"'&*$-+"
between
quantum
states,
so
it
turns
out
to
be
most
useful
in
cases
where
we
are
interested just the
bation to
from
the constant first-order perturbation doesn’t allow for the feedback
%'-($*+&/"1+"6&/-($2$'-$"/9&/3+&('"6.@"$")'(($:$1$"(6/,"$-6+&#+*$""""""="
! 0 = " bn ( 0 ) n and
.
between
quantum
states,
it turns
out
to bewemost
useful the
in cases
where we are
of leavingfirst-order
a state. perturbation
This
question
shows
up so
commonly
when
calculate
transition
perturbation to
from rate
the constant
doesn’t
allow
for
nthe feedback
en quantum states, so it turns out to be most useful in cases where we are interested just the
rate ofeigenstate,
leaving a state.
This question
shows up commonly
probability
not
to most
an individual
but
a distribution
of the
eigenstates.
Often thewhen
set ofwe calcu
between
quantum
states,
so
it
turns
out
to
be
useful
in
cases
where
we
are
interested
just
Now
there
may
be
interference
effects
between
the
pathways
initiating from
leaving a state. This question shows up commonly when we calculate the transition
probability
not
to
an
individual
eigenstate,
but
a
distribution
of
eigenstates.
eigenstates
form
a
continuum
of
accepting
states,
for
instance,
vibrational
relaxation
or
rate of
a state. eigenstate,
This question
up commonly
when we Often
calculate
bility
notleaving
to an individual
but shows
a distribution
of eigenstates.
the the
set transition
of
eigenstates form a continuum of accepting states, for instance, vibratio
ionization.
probability
not
to
an
individual
eigenstate,
but
a
distribution
eigenstates.
Often the
tates form a continuum of accepting states, for instance, of
vibrational
relaxation
or set of
ionization.
eigenstates
form
a
continuum
of
accepting
states,
for instance,
vibrational
relaxation
Transfer
to
a
set
of
continuum
(or bath)
states forms
the basisorfor a describing irreversible
ion.
TransferAlso
to a note
set ofthat
continuum
(or bath)
states
forms
theinto
basis
ionization.
relaxation. You can think of the material
Hamiltonian
forif our
partitioned
,
the problem
system
isbeing
initially
prepared
in two
a for
statea desc
Transfer to a set of continuum (or bath) states forms the basis for a describing irreversible
relaxation. You can think of the material Hamiltonian for our problem being p
, the
where
you are
interested
in the
of amplitude
in the
portions,
Transfer
to
a
set
of
continuum
(or
bath)
states
basis
aisdescribing
perturbation
turned
onirreversible
and
thenloss
turned
off over the
time interval
ion. You can think of the material Hamiltonian for ourforms
problem
beingfor
partitioned
into
two
, where you are interested in the loss of amp
portions,
relaxation. You can think states
of the as
material
Hamiltonian
for
our
problem
being
partitioned
into two
is just
the Fourier
transform of
complex
amplitude
inthe
the target
state
it leaks
into
. Qualitatively,
expectindeterministic,
oscillatory
feedback
ns,
, where you
are interested
in the loss
of you
amplitude
!H"between
states asinitthe
leaks
. Qualitatively,
you expect deterministic, oscillatory
portions,
, where
youstates.
are interested
lossinto
of amplitude
in the state
discrete
quantum
However,
the amplitude
of one discrete
coupled to a continuum
energy
gap
.
as it leaks into
. Qualitatively, you expect deterministic, oscillatory feedback between
turned
onanatoscillating
time
. This describes how a light
action of a system
with
perturbation
n of a system with an oscillating perturbation
transitions in a system
d on at time field .(monochromatic)
This describes induces
how a light
. This describes how a light
n at time
through
dipole
interactions.
Again, we are looking to
(monochromatic)
induces
transitions
in a system
nochromatic) induces transitions in a system
calculate the
transition
gh dipole interactions.
Again,
we areprobability
looking to between states
ipole interactions. Again, we are looking to
and k:probability between states
late the transition
the transition probability between states
:
(2.151)
(2.151)
(2.151)
()
V t = V cos ! t = " µ E0 cos ! t
(2.152)
()
Vk! t = Vk! cos ! t
V
To first order, we have:
= k! #$ ei! t + e" i! t %&
2
rst order, we have:
rder, we have:
"i t
i% #
bk = k ! I t = $ d# Vk" # e k"
! t0
, using
ng
+(+#)$'4""
!`"'&*$-+a")+&6.&:/2$'-+"/&#'-$%/"
()
(2.152)
(2.152)
()
setting
=
Now, using
"iVk"
2!
"V
= k"
2!
i % +% #
i % "% #
d# & e ( k" ) " e ( k" ) (
0
'
)
$
t
(2.153)
setting t 0 ! 0 (2.153)
setting
(2.153)
& ei (% k" + % )t "1 ei (% k" " % )t "1 (
+
+
*
% k" " % +)
*' % k" + %
as before:
as before:
as before:
# i(" k! !" )t / 2 sin # " ! " t / 2 % ei(" k! +" )t / 2 sin # " + " t / 2 % %
!iVk! ' e
$ k!
&+
$ k!
&(
(2.154)
bk =
(
" '
" k! ! "
" k! + "
&
$
(2.154)
(2.154)
Notice that these terms are only significant when
. As we learned before, resonance is
(
)
(
ce that these terms
are to
only
. As we learned before, resonance is
required
gainsignificant
significantwhen
transfer of amplitude.
. As we learned before, resonance is
at these terms are only significant when
red to gain significant transfer of amplitude.
o gain significant transfer of amplitude.
)
!K"
+(+#)$'4""
!`"'&*$-+a")+&6.&:/2$'-+"/&#'-$%/"
2-39
First Term
Second Term
max at : ! = +! k!
! = "! k!
E k > E!
E k < E!
Ek = E! + "!
Ek = E! ! ""
Absorption
Stimulated Emission
(resonant term)
(anti-resonant term)
For the case where only absorption contributes,
, we have:
!V"
(2.155)
Absorption
Stimulated Emission
By expanding
, we see that on resonance
(resonant term)
(anti-resonantwhich
term)is the Lorentzian lineshape centered at
Limitations of this formula:
.
calculate the adiabatic limit,
setting
+(+#)$'4""
Pk! = bk =
s of this formula:
2
(
Vk!
We
w
!`"'&*$-+a")+&6.&:/2$'-+"/&#'-$%/"
transitions
, we see
that on resonance . But let’s restrict ourselves to long
, we have:
perturbation has cycled a few times (this allows us to neglect cro
This clearly will not describe long-time behavior. This is a result of 1st o
By expanding
where only absorption contributes,
2
with
2
" ! k! " !
)
2
! . However,
not
treating
the
depletion
of
, so we
1
%" (# k! $it#will
' (2.157)
#
%
=
V
+ "hold
# k! for
+ # small
w
sin $ 2 ! k! " ! t &
)
(
)
k!
k!
2
&
(
2"
(
2
2
)
2-40
(2.155)
2
2
E
µ
This
clearly
will
notresonance
describe
long-time behavior. This is a result of 1st order perturbation theory
ng
, we see
on
0 that
k!
2 1
or
sin #$ 2 ! k! " ! t %&
2
" ! k! the
" !depletion of . However, it will hold for small , so we require
not treating
At the same time, we can’t observe the system on too short a time scal
pare this with the exact expression:
make several oscillations for it to(2.157)
be a harmonic perturbation.
(2.158)
(
)
(
)
theory
y will not describe long-time behavior. This is a result of 1st order perturbation
(2.159)
(2.156)
At the same time, we can’t observe the system on too short a time scale. We need the field to
the depletion of . However, it will hold for small , so we require
make several oscillations for it to be a harmonic perturbation.
Limitations of this formula: These relationships imply that
s out that this is valid for couplings
that are small relative to the
2!
1
1
(2.158)
By expandingt <<
, we seet that
(2.159)
"
> on resonance
1$#$6/,'-(4""""""""""""""""""""""""/-*""""""""""""""""""""""""""""""""""60+&+8'&+4"
Vk! << "! k! (2.160)
Vk"
!
!
k!
. The maximum probability for transfer is on resonance
e time, we can’t observe
system onimply
too short
These the
relationships
that a time scale. We need the field to(2.157)
al oscillations for it to beThis
a harmonic
clearly will notperturbation.
describe long-time behavior.
This is a result of 1st order perturbation theory
(2.160)
!W"
Transfer to a set of continuum (or bath) states form
w those variables to be factored out of integral
ministic, oscillatory feedback between
So, using
the same
relaxation. You can think of the material
Hamiltonian
forid
discrete state coupled to a continuum Absorption
cillating frequencies for each member
(resonant term)
SheGOLDEN
RULE since
limits
is broad relative to
, where you are intereste
portions,
(2.164)
distribution of final states:
8+&#$";'1*+-"&.1+"
2-42
Stimulated Emission
states as it leaks into
. Qualitatively,
2-43 you expect determ
2-43
(anti-resonant
discrete quantum states.
However, term)
the amplitude of one
. decay
Usingduethe
will
to destructive interferences between the os
( )
of the continuum.
!(2.162)
Ek :
to asystem in a state k after
he
transition
probability
from ! the
6&/-($,'-"8&'#""""""""""6'"/"#/-$8'1*"'8"(6/6+(""""""""""*+(%&$:+*":b"*+-($6b"'8"(6/6+("""
e and
probability
of observing
applying a
For the case where only absorption contributes,
For a constant perturbation:
, we have:
So, using the same ideas as before, let’s calculate t
from the constant first-order perturbation doesn’t allow for the2 feedback
Vk! (2.165)
2
distribution
of final2 states:
.
1
P k = " dEk ! Ek Pk
(2.162)
#
% in a state , the t
sin
!
"
!start
t(2.163)
Pk! = bk =
(
)
If
we
k!
2
2
$
&
states, so it turns out to be most useful in cases where we are"interested
just
the
2
!
"
!
( k! )
itude in discrete eigenstate of
(2
Probability of observing amp
Now, let’s make two
assumptions
to evaluate thisthe
expression:
2
state.
This
question
shows
up
commonly
when
we
calculate
transition
tion:
E02 µ k!
2-44
1)
varies slowly with frequency
and there is a sin 2 # 1 (!
or
" ! ) t %&
k!
an individual
eigenstate,
but
a
distribution
of
eigenstates.
Often
the
of
2
2 set $
, describes distribution
of
final
Density of states—units in
continuum
of final states. (By slow what
is k ) :
.-*+&"60+"/))&'?$#/,'-"60/6"""""""""
" (we
! k!are"saying
!!)( E
2
2!
the observation point t is relatively long).
=
" Vk"states,
t thatfor
(2.166)
$("(1'c1b"3/&b$-;"/-*"]
a continuum ofPk accepting
instance, vibrational
states—all eigenstates of
[1"$(" relaxation or
(2.163)
!
$-3/&$/-6"/%&'(("60+"S-/1"(6/6+(""
2) The
matrix element
is invariant across the final
We are just interested in the
We
this
with the exact
expression:
!P can compare
( )
wk! =
k!
states.
If we start in a state
, the total transition probability is a
ability
is linearly proportional
time. For relaxation processes, we will
um of probabilities
!t toThese
distribution:
(2.167)
assumptions
allow those
to be factored
out of integral
a set of continuum (or bath) states forms
the basis
forvariables
a 60+"6&/-($,'-"&/6+"*+)+-*("'-"60+"(6&+-;60"'8"
describing
irreversible
sumptions
to evaluate
this
2
2" expression:
. (2
:
ansition
rate,
60+"%'.)1$-;":+6c++-"60+"$-$,/1"/-*"S-/1"(6/6+"
def4"
(2.164)
wkk!! =Hamiltonian
# Vkk!! for our problem being
an think of the material
partitioned into two
][1"/-*"'-"60+"-.#:+&"'8"c/b("60+"6&/-($,'-"
" (2.161)
%/-"0/))+-"P$=+=X"60+"*+-($6b"'8"60+"S-/1"(6/6+(R="
owly with frequency and there is a
and occup
We
in the
of leaving
, where you are interested
inchosen
the loss
of amplitude
injust
theisinterested
Here, we have
the limits
sinceare
broad relative
to rate
. Using
the
!Z"
points isout that this is valid for couplings
that are small relative t
states. (By slow what wewhich
are saying
identity
distribution:
ying
any
state
or
for
a
continuous
point
t
is
relatively
long).
nto
. Qualitatively, you expect deterministic, oscillatory feedback between
detuning
. The maximum probability for transfer is on resonance