Moreton "

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(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 .

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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 > .

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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 .

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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.

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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.

'