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Moreton "
�1�
(1) Schrodinger t Heisenberg
pictures(2) Time - dependent Hamiltonians and time - ordered evolution operator
(1) Schrodinger +
Heisenbergpictures .
So far we've studied time evolution using the
SchrodngerpictureaeStates evolve in time by 14147 = UIH 14110) >
,
or equivalently ikdlylt ) > = HIHHD.
• Observables are time independent .
There is a completed way at looking at
time evolution that is often useful :
HeismbegpoctureoeStates are time - independent : I 410 ) >
,
• Observables depend on time by : Att ) = UYHAUHI.
* These pictures are equivalent ( he 'll show itl,
but youcannot mix them together within a single calculationof a time - dependent property .
Equivalence of Schrodinger + Heisenberg pictures "
• Expectation values match :
( HIHIAIYIH > = ( 41°11 UHHAUHI 1410) >/
'
= ( 41011 AHI 141017Schrodinger K Heisenberg
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• What about measurement ?
Let A have eigenvalues a, ,
... , an,
with projectors P, ,
... ,PK
projecting on to corresponding eigenspaus .
Then A=§a ; Pi .
Schrodinger Measure A in state 14 Hl ) .
Prob ( at= ( HH ) IPIIHH ) ) = ( No ) IUHHP; UK ) 1410 ) ) .
Collapsed wavefwetan is P;IHIH ) = P; UIH 14107 )
= UHUHHP ; UH ) 1410 ) )
= UIH Pitt ) 141° ) ) .
We've written out these results to compare with the Heisenberg picture . . .
Hew : Measure AH ) in state 146 ) ).
AH = Utttltfaipi ]UH = ?aiPiHT
Projector onto the
t.me#duteigenspaceforeign value Ai
.
Pnblaili ( 41011 Pitt ) 141017
= ( 41011 UTIHPIUIH 1410 ) > - > same ✓
Collapsed wave function : Pitt 114101 > ✓Since I Ysur ; dinger > = UIHIY Heisenberg >
,
this is the same as the collapsed wave functionin the Schrodinger picture .
�3�
Heinen :
dAE= TEUTHAUH = duty AUH + UHHA that
= i¥ UHHAUH . UTHIAUHIHZ
= in [ H,
AH ]
dFz=izlH,AHۥ Looks more like classical equations et notion .
This will be useful
later when we make contact with classical mechanics .
• You will get some practice working on the Heisenberg picture on HW3
in a specific example
mm
TINHame40
Sometimes we want to study problems where H → HIH,
i. e. there is time - dependence in He Hamiltonian . We will lookat a specific example shortly .
How de we deal with such problems ?
. Schrodinger equation becomes its # HIH > = HHHYHD
• We can just g ahead and solve this equation . But what
happens to the term at the tone erohrtan operator .We don't
have time translation symmetry anymore ,so we expect
UH,
t' ) 7 UH - El = expfizhtlt - EI) .
• Still, we just need to find a solution to a 1st - order linear
equation ,so it shouldn't be to . bad
. . .
• 1st guess : Utt,
o ) = expfizfottlt ') at ' )
"
\for simplicity ,
Cheon initial time to be zero.
This guess encodes the idea that HIHfrom t'
:O up to t'
#
should play a role in time evolution between these times.
Deeth ?
DUH , o )First at all
,should have
a=
- iHf± Ult ,o )
,
because then the Schrodinger equation holds :
iktahtltl > = ihdudltf' 146 ) ) = HHIYHI > .
�5�
Ok. . .
da UH, o ) = ffexpfizftnhtldt ' )
= # 1 - izftuitiat '. tzfotltttilati fotmttil dti + . . . }
= - izttltl - to HHI # that .
'- he #ltildt'THft
( This is a problem !
[ Hltil , Hlth ] to in general ,we cannot bring HIH
out through to the left ,which is what needs to happen
for the answer to be - in HIHUH , o ) .
So our 1st guess
deesnetword .
�6�
Indeedconsider breaking tin evolution into infinitesimal steps . -
UH, o ) = Utt ,t - al - - a Uha
,e) Ula
,0 )
,take limit a → ° (many time steps )
Over time interval t'
→ the,
Hk ' ) approximately time independent
⇒ U ( the,
t' ) a eypfiz EH HA.
⇒ Ultio ) a expfiez HH ) expfiez Hit . a) . . . exp f 'Inno ) )
• In this expression HHI with later times always appear to the leftat HIT ) with earlier times
,it we Taylor expand .
• Ok,
but this expression is awkward .However
,it motivates a nicer
and correct guess to the solution :
UH ,o ) = 'T expfizftnieiat ' )T
time ordering symbol ...
let's see what it means .
1T HH , ) Hit , )=. {HHIHIH it t
,> t
.
Hltx) HIT, ) it t
,< t
,
"later on the left
"
( t ,= tn is at measure zero and we don't care about it ) .
H It, )H It ,
) HIT , ) if t,
> t. > t
,THHIIHKIH Its ) =
| Hit , ) Hit , ) Hlhl
,
if t ,> t
,> to
etc.
�7�
Then :
Texpfiz # that' ) = 1 - fifth # lat
'.
t.IT/dtiftdtiHttilHkiI+..Tf.tdtiftdtiuhyHtti)=2f.dtifoYtiHttilHHil
Generally ...
Texpfizfouhat ' ) : 1 - fifth that'
+ fizp f.tw#tiHttilHHil
+ tietfotatiftiatiftoidt's HKIIHHIIHHIH '"
( You should Convince yourself this is true . )
Then it U Hit .) - Texpfizftt
.Htttdt'
) ,simple to check that UH ,
t.KHHUH.to)
t.int#....m.TuegMeanuait.
'
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