Post on 11-Feb-2022
Lecture 6: Time Propagation
Outline:
• Ordinary functions of operators
– Powers
– Functions of diagonal operators
• Solving Schrödinger's equation
– Time-independent Hamiltonian
– The Unitary time-evolution operator
– Unitary operators and probability in QM
– Iterative solution
– Eigenvector expansion
Ordinary Functions of Operators
• Let us define an `ordinary function’, f(x), as a
function that can be expressed as a power
series in x, with scalar coefficients:
• When given an operator, A, as an argument,
we define the result to be:
• Examples:
• THM: A functions of an operator is defined by
its power series
n
n
n xfxf !=)(
n
n
n AfAf !=:)(
!"
=
=
++++=
+#+#=
0
32
753
!
....!3!2
...!7!5!3
)sin(
n
n
A
n
A
AAAIe
AAAAA
Powers of Operators
• An operator raised to the zeroth power:
• Positive integer powers:
• Operator inversion:
– The operator A-1 is defined via:
• Negative powers:
• Fractional powers:
...
:
:
:
3
2
1
etc
AAAA
AAA
AA
=
=
=
IA =:0
( ) AA
IAA
=
=
!!
!
:
:
11
1
nnAA )(: 1!!
=
...
:2/12/1
etc
AAA =
Functions of Diagonal Operators
• Diagonal operators have the form:
• They can be expressed in Dirac notation as:
• Every operator is diagonal in the basis of
its own eigenvectors
• They have the property:
– let C and D be diagonal matrices
• From which it follows that:
!!!!!!
"
#
$$$$$$
%
&
=
Md
d
d
d
D
L
MOMMM
L
L
L
000
000
000
000
3
2
1
nndD
M
n
n!=
=1
nnd
mmnndcDC
M
n
n
M
n
m
M
m
n
!
!!
=
= =
=
=
1
2
1 1
( )
( )( )
( )
( )!!!!!!
"
#
$$$$$$
%
&
=
Mdf
df
df
df
Df
L
MOMMM
L
L
L
000
000
000
000
3
2
1
Solving Schrödinger’s Equation
• When the Hamiltonian is not explicitly time-
dependent, Schrödinger's Equation is readily
integrated:
– Proof:
)()( tHh
it
dt
d!! "=
)0()( / !! hiHtet"
=
)0(!
)0(0
/ !!m
tHi
dt
de
dt
dmm
m
iHt "#
=
$%&
'()
*$=h
h
)0(!
1
1
!m
mtHi
mm
m
"#
=
$ %&
'()
*"=h
)0()!1(
11
1
!"
#$
%&'
(""=
"")
=
*m
tHi
Hi
mm
m hh
)0(!0
!n
tHi
Hi
nn
n
"#
=
$%
&'(
)**=hh
)0(/ !h
h
iHtHei ""=
)(tHi
!h
"=
The Unitary Time-Evolution Operator
• In general, the time-evolution operator is
defined as:
– The operator U(t,t0) must be Unitary ( U†=U-1 ) topreserve the norm of |!(t)"
• For the case where H is not explicitly time-
dependent, we see from the exact solution
that:
• In the more general case where H=H(t), the
above is not necessarily valid
– In this case we must find an equation for U(t,t0).
– We start from Schrödinger's Equation:
– Which we now write as:
– Since this must be true for any initial state,|!(t0)", it follows that:
)(),()( 00 tttUt !! =
h/)(
00),(ttiH
ettU!!
=
)()( tHh
it
dt
d!! "=
)(),()(),( 0000 tttUHh
itttU
dt
d!! "=
),(),( 00 ttUHh
ittU
dt
d!=
Unitary Operators and probability in QM
Solving the Time-Evolution Operator
Equation
• Since |!(t0)"=U(t0,t0)|!(t0)", it is clear that:
• The equation of motion:
• Can be formally integrated:
• Or re-expressed via the definition of thederivative as:
• With t0=t, this gives infinitesimal time evolutionoperator:
• So that (for numerical purposes):
– Where and
1),( 00 =ttU
),()(),( 00 ttUtHh
ittU
dt
d!=
),()(1),( 00
0
ttUtHtdh
ittU
t
t
!!!"= #
),()(1),(
),()(),(),(
00
000
ttUdttHh
itdttU
ttUtHh
i
dt
ttUtdttU
!"
#$%
&'=+
'='+
dttHh
itdttU )(1),( !=+
!"
#$%
&'!"
#$%
&'!"
#$%
&'=
+++=
()
()
dttHi
dttHi
dttHi
tdttUtdttUtdttUttU
NN
NNN
)(1)(1)(1lim
),(),(),(lim),(
01
00110
hhK
h
K
dtmttm
+=0
Nttdt /)( 0!=
Can this be simplified further?
• We have found the most general result is
• This can be re-written as:
• Note that:
– Only in the case [A,B]=0
• Thus can we write:
– ONLY if the Hamiltonian satisfies:
!"
#$%
&'!"
#$%
&'!"
#$%
&'=
+++=
()
()
dttHi
dttHi
dttHi
tdttUtdttUtdttUttU
NN
NNN
)(1)(1)(1lim
),(),(),(lim),(
01
00110
hhK
h
K
dttHi
dttHi
dttHi
N
eeettUN )()()(
0
12
...lim),( hhh!!!
"#=
BABAeee
+=
)(
00),(
tHtdit
t
ettU
!!" #=
h
[ ] tttHtH !"=! ,0)(),(
Iterative solution:
• We have:
• Start with:
• The iterative form of the equation is:
• Which gives
– Note: the “I” is an integration constant fitted to the
initial conditions
• The final solution is:
),()(),( 00 ttUtHh
ittU
dt
d!=
IttU =),( 00
),()(),( 001 ttUtHh
ittU
dt
dnn
!=+
!"=t
t
tHdti
IttU0
)(),( 1101h
( )
( )
( )
...
)()()(
)()(
)(),(
123123
3
1212
2
110
2
0
3
00
2
00
0
+
+
+
+=
!!!
!!
!
"
"
"
tHtHtHdtdtdt
tHtHdtdt
tHdtIttU
t
t
t
t
t
t
i
t
t
t
t
i
t
t
i
h
h
h
),(),( 00 ttUttU !=
Eigenvector expansion
• For the case where H is not explicitly
time-dependent, it is most common to
use the eigenvector basis to express
the evolution operator.
– The eigenvectors of H are defined by the
eigenvalue equation:
– Note the following:
nnnH !!! h=
n
ti
n
iHt nee !!!""
=h/
mnnm
n
nn
!""
""
=
=# 1
Eigenvector Expansion cont.
• Start from:
• Apply the bra #$n|!:
• Integration then gives:
• We can express the state vector as:
!! Hdt
di =h
)(
)()(
t
tHtdt
di
nn
nn
!""
!"!"
h
h
=
=
)0()( !"!" "
n
ti
n
net#
=
)0(
)()(
!""
!""!
"n
ti
n
n
n
n
n
ne
tt
#$
$
=
=
Summary
• Two approaches to solvingSchrödinger's Equation:– Time-Evolution Operator:
• Case I: H(t)=H(0)=H:
• Case II: H(t)%H(0), but [H(t),H(t_)]=0:
• Case II: H(t)%H(0), but [H(t),H(t_)]=0:
– Eigenvalue expansion:
)0()( / !! hiHtet"
=
)0(),(),(),(lim)( 0011 !! tdttUtdttUtdttUtNN
N
+++="#
K
)0()( !""! "n
ti
n
n
net#$=
)()( 0
)(
0 tet
tHtdit
t !!""# $
=h