Notas-1-Mecanica-Cuantica-Postgrado.pdf

8/17/2019 Notas-1-Mecanica-Cuantica-Postgrado.pdf

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1. FundamentalConcepts

1.1 Introduction

Theoreticalframeworkof classicalphysics:

∗ state(atfixedt)isdefinedbyapoint{xi, pi} inphasespace

∂F ∂G −∂F ∂G∗ Poisson bracket {xi, p j} = δ ij {F G} = i ∂xi ∂pi ∂pi ∂xi (flat space, symplecticmfld)

2∗ ObservablesarefunctionsO(xi, p j)onphasespace [ex: x +y2+z 2, p2/2m , . . .] ∗

HamiltonianH (x, p)definesdynamics

q ˙ =

{q, H

}forq =xi, p j, . . .

q q M

z

q q M

I

B I

phasespace

(x, p)

Frameworkdescribesallof mechanics. E&Mincludingfluids,materials,etc. ...

•

Simple,

intuitive

conceptual

framework

• Deterministic• Time-reversibledynamics

Example: Classicalsimpleharmonicoscillator(SHO)

1 2H =k

x2 + p2 2m

p

!

x

X �$ X "

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1ẋ = {x,H }= p

m ṗ = { p,H }=−kx

(=mẍ)

Theoreticalframeworkof quantumphysics

∗ statedefinedby(complex)vector |vinvectorspaceH. (Hilbertspace)

∗ ObservablesareHermitianoperatorsOdagger =O∗ Dynamics: |v(t)=e−iHt/�|v(0)(H hermitian,� constant)

∗ “Collapsepostulate”

[(subtleties:

degeneracy,

(simpleversion) cts.spectrum)]If |φ= αi|λi,

withA|λi=λi|λi,λi, measure A = λi,

jλ=Thenwithprob.|αi|2

|φ=| λiaftermeasurement

This

framework

[(with

suitable

generalizations,

i.e.

field

theory)]

describes

all

quantum

systems,andallexperimentsnotinvolvinggravitationalforces.

–

Atomic

spectra

(quantization

of

energy)

– Semiconductorstransistors,(quantumtunneling)

• Counterintuitiveconceptualframework• Nondeterministic

• Irreversibledynamics

Bothclassical&quantumconceptualframeworksusefulincertainregimes.

Classical

picture

not

fundamental

—

replaced

by

QM.

IsQMfundamental?

Perhapsnot,butdescribesallphysicsof currentrelevancetotechnology&society.

Simpleexampleof QM:2-statesystem(spin 1 particle)2

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S � �

Stern&Gerlach1922

- - -

- - - - -

-

N

�

� � I B - - -

- -

- -

Agatoms � � � � � � � � � � � � � �

� � � � � � � � � � � � � �

�Oven

Silver: 47 electrons, angular momentum =⇒electron.

Energy U =

F z =

Givesforcealong ẑ dependingonµz.

Classically,expect

screen

µ (mag. dipole moment) from spin of 47th

−µ

·B

∂U

∂Bz−

=

µz∂Z ∂Z

&

Actuallysee

# %

#

%

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�

� � � � � � � � � �

� � � � � � � � � �

e� eµz

≈ ±

≈ S z

2mc mc

S z = ± [� = 1.0546× 10−27 ergs]2

So

measuring

S z =⇒ discrete

values

(2

states)

CanbuildsequentialS − Gexperiments, usingcomponents

S z = + splitter Z

S z =−

S z = +

Z

� � � � � � � � � �

� � � � � � � � � � Z

filters

S z =−

(Canalsoformsplitter,filtersonx-axis,etc.)

Single

particle

experiments

i) S z = +

Z Z

S z

= + 100%

0%

ii)

S z = +

Z X

S x = +

50%

50%

S x

=−

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� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

iii) S z

= + S x = +

Z X Z 100%

S z

= +

0% S x =− S z =−

BUT

iv) S z = + S x = +

Z X Z

S z = + 50%

50%

� � � � � � � � � � S z =−

v) S z

S z

= += +

Z

X

� � � � � � � � � �

50%

50% Z

S x =− S z =−

Cannot

simultaneously

measure S z

, S x

“incompatibleobservables” S zS x S x= S zAnalogousto2-slitexperiment forphotons.

In

iii),

not

measuringS x.

in

iv),

v)

measuring

S x.

iii)needs“interference”of probabilitywave—noclassical interpretation.

iii),iv)=⇒Irreversible,nondeterministicdynamics(assuming locality)

1.2 MathematicalPreliminaries

1.2.1

Hilbert

spaces

Firstpostulateof QM:

∗ Thestateof aQMsystemattimetisgivenbyavector(ray)|αinacomplexHilbertspaceH. [[willstatemorepreciselysoon.]]

Vector

spaces

Avector space V isacollectionof objects(“vectors”)|αhavingthefollowingproperties:

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A1: |α+|β givesauniquevector|γ inV . A2: (commutativity) |α+|β =|β +|α A3:

(associativity)

(|α+|β ) + |δ = |α + (|β + |δ ) A4:

∃vector

|φ

suchthat

|φ

+

|α

=

|α

∀|α

A5:

For

all

|α

in

V

,

−|α

is

also

in

V

so

that

|α

+ (−|α) =

|φ.

[(A1–A5): V isacommutativegroupunder+]

ForsomeFieldF (i.e.,R,C, with +, ∗ defined)scalar multiplication of anyc∈F withany|α ∈ V givesavectorc|α ∈ V . Scalarmultiplicationhasthefollowingproperties

M1: c(d|α) = (cd)|αM2: 1|α=|αM3: c(

|α+

|β ) = c

|α

+ c|β

M4: (c+d)|α=c|α+d|α. V iscalleda“vectorspaceoverF .” F = R: “realv.s.” F = C: “complexv.s.”

Examples

of

vector

spaces

a1

.a) EuclideanD-dimensionalspace isarealv.s. .

.aD

b) Statespaceof spin 1 particle is 2D cpx

v.s.

2states:

c+|++c−|−,c± ∈ C [[note: certainstatearephysicallyequivalent]]c) spaceof functionsf � : [0, L] → C

Henceforth

we

always

take F =C

Subspaces

V ⊂W isasubspace of av.s.W if V satifiesallpropsof av.s.and isasubsetof W .

[suffices

for

V

to

be

closed

under

+,

scalar

mult.]

Ray

A ray in V isa1Dsubspace{c|α}.

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Linear

independence

&

bases

|α1, |α2, . . . , |αn are linearly independent iff c1|α1 +c2|α2 +· · · + cn|αn = 0

has

only

c1 =

c2 =

· · ·

=

cn =

0

as

a

solution.

If |α1, . . . , |αn in V are linearly independent, but all sets of n+1 vectors are linearlydependent,then

• V isn-dimensional (nmaybefinite,countably∞,oruncountably∞.)• |α1, . . . , |αn formabasis forthespaceV .

If |α1, . . . , |αn formabasisforV ,thenanyvector |β canbeexpanded inthebasisas|β = ci|αi . (Thm.)

i

Unitary

spaces

AcomplexvectorspaceV isaunitary space (a.k.a.innerproductspace)

if

given

|α, |β ∈ V there is an inner product α|β ∈ C with the followingproperties:

∗I1: α|β =β |α I2:

α

|(

|β

+

|β �

) =

α

|β

+

α

|β �

I3:

α|(c|β ) =

cα|β

I4: α|α ≥ 0 I5: α|α= 0 iff |α = |0

α|β isasesquilinear form(linearinβ ),conj.lin.in|αExamples:

z1

.a) V =CN ,N -tuples |z = .. ,z i ∈C.

zN ∗ ∗

z

|w

=z I

∗w1 +z 2w2 +

· · ·+ z N wN

b) V ={f : [0, L] → C} Lf |g = f ∗(x)g(x) dx

0

Terminology:

⎧ α|β = norm of |α(sometimes,|α

if α|β = 0, |α, |β areorthogonal

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Dual

Spaces

∗Fora(complex)vectorspaceV , the dual space V isthesetof linearfunctionsβ | :V → C.Giventheinnerproductβ |α,canconstruct isomorphism.

|α ←→ α|∗V ←→ V ∗note: c|α ←→ c α| .

Physicsnotation(Dirac”bra-ket”notation)

|α ∈ V ketβ | ∈ V ∗ bra

Hilbert

space

A space

V

is

complete

if

every

Cauchy

sequence

{|αn}

converges

in

V ∀�∃N :|αn − |αm < � ∀m, n>N (i.e.,∃|α :limn→∞|α − |αn = 0.)

Acompleteunitaryspaceisa(complex)Hilbert space .

Note: anexampleof anincompleteunitaryspaceisthespaceof vectorswithafinitenumberof nonzeroentries. Thesequence { (1, 0,

12

,0, . . . ),

(1, 0, 0, . . . ), (1, 1

2

, 13

, 0, . . . ) . . .

}isCauchy,butdoesn’tconvergeinV . This is not aHilbertspace.

A

Hilbert

space

can

be:

a) Finitedimensional(basis1Ex.spin

|α1, . . . , |αn) particleinStern-Gerlachexpt.

2

b) Countably infinitedimensional(basis|α1, |α2, . . .) Ex.QuantumSHO

c) Uncountably infinitedimensional(basis|αx,x∈ R)[Technicalaside:

¯A space

V

is

separable

if

∃

countable

set

D

⊂ V

so

that

D

=

V .

(Ddense inV : ∀�,|x ∈ V ∃|y ∈ D : |x − |y < � ) (a),(b)areseparable,(c)isnot.

non-separableHilbertspacesareverydiceymathematically.

Generally, separability implicit in discussion — e.g., label basis |αi, i takes discretevalues ]

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�

Orthonormal

basis

Anorthonormalbasisisabasis|ϕiwithϕi|ϕ j=δ ijAnybasis |α1, |α2, . . .can be made orthonormal by Schmidt orthonormalization

|φ1 = ⎧ |α1α1|α1|α� = |α2 − |φ1φ1|α22

2|φ2 = ⎧ |α�α�|α�2 2|α� = |α3 − |φ1φ1|α3 − |φ2φ2|α33

...

Gives|φi

with

φi

|φ j

=δ ij.

If {|φi}areanorthonormalbasisthenforall|α,

|α= ci|φi, ci =φi|α.i

Canwriteas|α= |φiφi|α. (Completenessrelation)i

Schwartz

inequality

α|αβ |β ≥

|α|β |2

∀|α,

|β

.

Proof: write|γ =|α+λ|β

γ |γ = α|α+λα|β +λxβ |α+|λ|2β |β −β |α

setλ = β |β |α|→ γ |γ = α|α − |β ≥0β |β

[Second

postulate

of

QM:

∗ Observablesare(Hermitian)operatorsonH[self-adjoint]

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1.2.2

Operators

Linear

operators

AlinearoperatorfromaVSV toaVSW isatransformationsuchthat

A|α +

|β =

A|α +

A|β

∀|α,

|β .

We write A=B iff A|α =B|α ∀|α. ∗A acts on

V through(β | ∀A)|α = β |(A|α)(actsonrightonbras.)

Outer

product

Asimpleclassof operatorsareouterproducts|β α|(|β α|)|γ =|β α|γ

Adjoint

∗V ↔ V Recallcorrespondence |α ↔ α| Given an operator A,defineA† (adjointof A)(Hermitianconjugate)byA†|α ↔ α|A

∗Example(|β α|)† =α|β |. Followsthatα|A†|β = (β |A|α) .

Hermitian

operators

A is

Hermitian

if

A

=

A† (∼ self-adjoint)⎛

Technicalaside: mathematically,Hermitiancalled”symmetric”. Self-adjointiff symmetric

+A&A† havesamedomain of definition ,relevanttoi.e.,Diracop.inmonopolebackground(symmetricop.withseveralself-adjointextensions)more: Reed&Simon,Jackiw

∗Exampleof domainof def: ConsiderH =L2(R) = {funsf :R→ C: 0 f f < 0 e−x2 ∈ H,−∞2xO = mult

bye ⎞

Oe−x2 /∈ H, so e−x2 notindomainD(O).LinearoperatorsAformavectorspaceunderaddition(+commutative,associative)

(A+B)|α =A|α +B|αMult.definedby

(AB)|α =A(B|α)GenerallyAB BA=But(AB)C =A(BC )Note: (XY )+ = Y +X +

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Identityoperator: 11 11|α=α|.Functionsof oneoperatorf (A) = C nA

ncanbeexpandedaspowerseries(mustbecareful

outsideROC—candoingeneralif diagonalizable)

Diagonalizableoperators: canalwayscomputef (A) if f definedfordiagonalelements(evalues)

Functions f (A,B) must have definable ordering prescription. (e.g. eABe−A =B+AB−BA +· · · ) Inverse A−1 satisfiesAA−1 =A−1A= 11 Doesnotalways exist. (Ex. if Ahasanev. =0.) Note: BA = 11 does not imply AB = 11 (Ex.later)

Isometries

U isanisometry if U +U = 11,sincepreserves innerproduct(β |U +)(U |α) = β |α

Unitary

operators

U isunitaryif U † =U −1.

Example: non-unitaryisometries. (HilbertHotel) Consider the shift operator S |n = |n+ 1 acting on H with countable on basis {|n, = 0, 1, . . . } S = |n + 1n|satisfiesS +S = 11butnot SS + = 11. (SS + = 11 − |00|). S hasno(2-sided)inverse.

Projection

operators

Aisaprojection if A2 =A.

Ex.A=|αα|forα|α= 1.

Eigenstates

&

Eigenvalues

If A|α=a|αthen|αisaneigenstate (eigenket)of Aandaistheassociatedeigenvalue .

Spectrum

Thespectrumof anoperatorAisitssetof eigenvalues{a} [technical aside: this is the “point spectrum”, mathematically, spectrum of A = set of λ : A − λ11 isnot invertible]

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=�

Important

theorem

If A=A†,thenalleigenvaluesai

of Aarereal,andalleigenstatesassociatedwithdistinctai areorthogonal.

Proof:

∗A|a=a|a ⇒ a|A† = a|a ⇒ b|(A − A+)|a = (a − b ∗ )b|a = 0

∗if a = b , a = a isreal.if

Consequence

of

theorem:

a b , b|a = 0

For

any

Hermitian

A,

can

find

an

O.N.

set

of

eigenvectors

|ai

A|ai=ai|ai , (ai notnecc.distinct— can be degenerate )

[Proof: useSchmidtorthog.foreachsubspaceof fixedevaluea—OKaslongascountable#of

(indep.)

states

for

anya (e.g.inseparableH)[caution: thissetspansspaceof eigenvectors,

butmay not becompletebasis]

Completeness

relation

If φi areacompleteonbasisforH.

|α= |φiφi|α ∀|α ,i

so |φiφi|= 11 (completeness )i(sumof projectionsonto1Dsubspaces)

Matrix

and

vector

representations

If

H

is

separable,

∃

a

countable

on

basis,

|φi

φ1|αcanwrite|α= |φiφi|α ⇒ φ2|α i ...

β |= β |φiφi| ⇒ (β |φ1β |φ2 · · ·) i

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�

Diagonalization

of

Hermitian

operators

Theorem.AHermitian matrix (finite dim)H ij =φi|H |φ jcan always be diagonalized by

a unitary transformation.

Proof.

if

|φi

a

general

ON

basis,

ON

eigenvectors

|hi

related

to

|φi

through

|hi = U |φi, unitary.hi|H |h j = δ ijhi =φi|U †HU |φ j

soU +H k�U �j

isdiagonal.ik(generalizestoanyobservable)

Algorithmforexplicitdiagonalizationof amatrixH (finitedimensional):

1) Solvedet(H −λ11)=0forN ×N matrices,N solutionsareeigenvaluesof H .2) SolveH ijc j =λci forci’sforeachλ. N lineareqns. inN unknowns.

Giveseigenvalues&eigenvectors.

Invariants

Somefunctionsof anoperatorAareinvariantunderU :

TrA = φi|A|φi, |φi ONbasisU †TrU †AU = ij

A jk U kj =δ jk A jk = Tr A i,j,k

[Technical

note:

careful

for

∞

matrices

—

need

all

sums

converging.]

Anotherinvariant: detA: det(U †AU ) = det U detAdetU † = det UU † detA= det A

[Note: fullspectrumof ev’sis invariant!]

Simultaneous

diagonalization

Theorem.Twodiagonalizable operators A,B are simultaneously diagonalizable iff

[A,B] = 0

⇒ say A

|αi

= ai

|αi

, B

|αi

=bi

|αi

AB|αi

=

BA|αi

=

aibi|αi

.

⇐ Say AB=BA , A|αi=ai|αi.

AB|αi=aiB|αi,

so B keeps state in subspace of e.v. ai. Thus, B is block-diagonal, can be diagonalized ineachai subspace

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1.3 Therulesof quantummechanics

[[Developedovermanyyearsinearlypartof C20. Cannotbederived— justifiedbylogicalconsistency&agreementwithexperiment.]]

4basicpostulates:

1) Aquantumsystemcanbeput intocorrespondencewithaHilbertspaceHsothatadefinitequantumstate(atafixedtimet)correspondstoadefiniterayinH.

so |α ≈ c|α representsamephysicalstateconvenienttochooseα|α =1,leavingphasefreedomeiφ|α

• Note: stillaclassicalpictureof statespace(“realistapproach”). Pathintegralapproachavoidsthispicture.

• “state”reallyshouldapplytoanensembleof identicallypreparedexperiments(“pure

ensemble”=purestate.)Ex.statescomingoutof SGfilter

S z = + Z

|α =|+ . � � � � � � � � � �

2) ObservablequantitiescorrespondtoHermitianoperatorswhoseeigenstates forma

complete

set.

Observablequantity=somethingyoucanmeasure inanexperiment.

[[Note: book constructs H from eigenstates of A: logic less clear as HA = HB for someA,B.]]

3) AnobservableH =H † definesthetimeevolutionof thestateinHthrough

i� d|ψ(t) =i� lim |ψ(t+ ∆t) − |ψ(t) = H |ψ(t) . dt ∆t

→0

∆t

(Schrödinger

equation)

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4) (Measurement&collapsepostulate)

If anobservableAismeasuredwhenthesystemisinanormalizedstate|α, where A hasanONbasisof eigenvectors |aiwitheigenvaluesai.

a)

The

probability

of

observing

A

=

a

is

|a j|α|2 =α|P a|α j:aj

=a

whereP a = j:aj

=a|a ja j|istheprojectorontotheA=aeigenspace.b) If A=aisobserved,afterthemeasurementthestatebecomes|αa=P a|α= ⎧

˜|a ja j|α(normalizedstateis|αa=|αa/ αa|αa). j:aj

=a

Discussionof rule(4):

Simplest

case:

nondegenerate

eigenvalues

|α= ci|ai, ai

= a j

.

Thenprobabilityof gettingA=ai is|ci|2.Normof α|α= 1 ⇔ |ci|2 = 1.

˜AftermeasuringA=ai,statebecomes|αi=|ai.

This postulate involves an irreversible ,nondeterministic , and discontinuous change in the

state

of

the

system.

– sourceof considerableconfusion

– lesstroublesomepicture: pathintegrals.

– alternatives: non-localhiddenvariables(’tHooft?),stringtheory—newprinciples(?)

Forpurposesof thiscourse,take(4)asfundamental,thoughcounterintuitive,postulate.

Todiscussprobabilities,needensembles .

Consequenceof (4):

Expectation

value

of

an

observable

A

in

state

|α

is

A= |ci|2ai

=α|A|αsinceA= |aiaiai|.i i

So

for:

4basicpostulatesof QuantumMechanics:

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�

1) State=rayinH [incl.def of H space]2) Observable=Hermitianoperatorwithcompletesetof eigenvectors

3) i� d

|ψ(t)

=H

|ψ(t)

dt

4) Measurement&collapse

ProbabilityA=a: α|P a|α ⎧

˜Aftermeasurement,system−→ |αa = P a|α/ α|P a|α

P a = |a ja j

| j:aj

=a

⇒Expectationvalueof A: A=α|A|α= |ci|2ai

if |α=

ci|ai

4

⇒

Thesearetherulesof thegame.

Restof thecourse:

Examplesof physicalsystems,toolstosolveproblems.

An

example

revisited

in

detail

Backtospin- 12

system.

Statespace

H

=

{|α

=

C +|+

+

C −|−

, C ± ∈

C}P 1 Unitnormcondition

2

α|α=|C +|2 +|C −| = 1

eiθ|α, |αarephysicallyequivalent

Operators:

�

2�

2

1 0 0 −1S = σz measuresspinalongz -axis= z

0 11 0

�2

�2S =

σx measures

spin

along

x-axis

=

x

0 −ii

0

�

2�

2S y

= σy

measuresspinalongy-axis =

For

general

axisn̂:

S n =S ·n̂ (HW#2)haseigenvalues±�2 .eigenstates |S n;±: S n|S n;±=±�2 |S n;±formcompletebasis.

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�

|S x;± = √ 1 |+ ± √ 12 |−2|−

2

|S y;± = √ 1 |+ ± √ i inS z basis.2 |S z

;± = |+

Some

further

properties

of

S i:

[S i, S j ] = i�ijk� S k

{S i, S j

} = S iS j

+ S j

S i

= 1� 2δ ij2

S 2 =S ·S =S 2 +S 2 +S 2 = 3� 211 = 3� 2

1 0

x y z 0 14 4And

[S 2, S i] = 0 .

Measurement

If |α=c+|++c−|−, α|α= 1, 2+ |

| ||2

prob.thatS is = z 2

c+c−−�prob.thatS is =z 2

Considersingleparticleexperiments fromlecture1.

�

2�

2S S = + = +i) z z 100% S z

S z

� � � � � � � � � � S z =− 0%�2

|α=|+�

2Repeatedmeasurementof S z gives+ 100%of thetime.

ii) S z = + S x = + 50%

S z S x

50% �

� � � ��

� � �

S x =−

|α

=

|+

|S x;±= √ 12 (|+ ± |−) so |α=|+= √ 1 [|S x; + + |S x; −]2

is 12 (50%)so prob. S x = +�

2

prob. S x

=−�2 is 12 (50%)

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20/28

iii) S z

= + S x

= + S z

= +

100%

S S S

z x z �

� � � � � � � �� 0% B S =− S =−x z

|−)

1

|α=|α++|α−=|+ |α+= 2 (|++|−)|α=

|+

1

2

(|α

−=

|+

−

Combinedstate|α=|+enterslastmeasurementapparatus,sinceS x

notmeasured.

= +z�

2GivesS 100%of time.

iv) S

z = + S x = + S z = +

50%

x

� ��

S S S z z �

� � � � � � � �� 50% S z

=−

|α+

=

√ 12

(|+

+

|−)|α

=

|+

state|α+= √ 12 (|++|−)enters lastapparatus.Prob. S = +x

Prob. S =−x

�

2

�

2

:

(50%)

: (50%)

Compatible

vs.

incompatible

observables

ObservablesA, B are:

Compatible if [A, B] = AB−BA = 0 incompatible if [A, B] = 0.

Examples: S 2, S z arecompatibleS x, S y arenotcompatible.

Theorem.Compatible observables A, Bcan be simultaneously diagonalized,and have eigen

vectors

|ai, bi

with

A|ai, bi = ai|ai, bi B|ai, bi = bi|ai, bi .

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�

(Proof inlastlecture: AB|α =aB|αif A|α =a|αsoB=Ha

→ Ha,diagonalizeineachblock.)

Acomplete set of commuting observables (CSCO)isasetof observables{A, B, C , . . .} suchthatallobservables inthesetcommute:

[A,

B] = [A,

C ] = [B,

C ] =

· · · = 0

andsuchthatforanya, b , . . . atmostonesolutionexiststotheeigenvalueequations

A|αB|α

=

=.

a|αb|α

..

Tensor

product

spaces

usefulformany-particlesystems [+quantumcomputing, . . . ]bigr)

Given two Hilbert spaces H(1), H(2), with complete ON bases |φ(1)i, |φ(2) j

, the tensorproduct

H =H(1)⊗ H(2)istheHilbertspacewithONbasis

φ(1) ⊗ |φ(2)|φij =| i j

and

inner

product φ

(1)1,φ(2)|φ(2)φi,j|φk,� =φ(1)| 2.i k

j �

If H(1),H(2) havedimensionsN, M , then H = H(1)⊗ H(2) hasdimensionNM . If H(1),H(2) separable,H isseparable.[inparticular,if eitherorbothof H(1),H(2) havecountablebasis&bothcountableorfiniteHhascountablebasis.]

Tensor

product

of

kets

and

operators

Kets :

If

|α =

ci|φ(1)i ∈ H(1)

|β = di|φ(2) j

∈ H(2)

22


22/28

,H(2)areketsinH(1) .then

|α ⊗ |β = cid j|φi,j ∈ H i,j

is

in

H

=

H(1)

⊗ H(2)

.

[Note:

not

all

vectors

in

H

are

of

tensor

product

form.

Ex.

|φ1,1 +|φ1,2 +|φ2,1]

Operators:

If A,BareoperatorsonH(1),H(2),thenwecanconstructA⊗ B asanoperatoronH =H(1)⊗ H(2) through

(A⊗ B)|φi,j

= (A|φ(1))⊗ (B|φ(2)).i j[defines

A⊗

B onallof H

by

linearity]

If A,Bareobservables,thenA⊗ B isanobservable.

Summary

of

Tensor

product

spaces

H

dimHBasis: |φi,j

φi,j

C i|φ(1)Kets: |α |β i

|α ⊗ |β

Brassame

= H(1)⊗ H(2)= (dim H(1))(dimH(2))= |φ(1) ⊗ |φ(2) j=

φ(1)

| ⊗ φ

(2)

| j

= d j|φ(2) j

= C id j

|φi,j

φ(2))Operators

(A⊗ B) C i,j|φi,j = C i,j(A|φ(1))⊗ (B|i j

Simple

class

of

operators

on

H:

A

⊗ 11 ,

11 ⊗ B .

If A,B actonH(1),H(2),willoftenrefertotheseas justA,Bwhencontext isclear.

Useful

relation:

(A⊗ B)· (C ⊗ D) = (AC )⊗ (BD).

23


23/28

Note: [(A ⊗11), (11⊗B)]=0. Notation: inmanybooks,tensorproductsymbol isomitted

|α ⊗ |β ⇒ |α|β A

⊗B

⇒ AB .

CSCO’s

in

tensor

product

spaces

If {A1, A2, . . . , Ak} areaCSCOforH(1), & {B1, . . . , B�} areaCSCOforH(2), then{A1, . . . , Ak

, B1, . . . B�} areaCSCOforH(1)⊗ H(2) [[Ex.of notationA1 =A1⊗11.]]

Example

of

tensor

products:

Two

spin- 1

particles2

, H(2)Considertwospin- 1 particleswithHilbertspacesH(1)2 2 2 .Thetwo-particleHilbertspace is

H

=

H(1)⊗ H(2)2 .2AbasisforH is:

|++ = |+1⊗ |+2

|+

− =

|+

1

⊗|−2

|−+

=

|−1⊗ |+2 |−− = |−1⊗|−2

Operators:

Acompletesetof commutingobservables is

S (1) = S (1)⊗11z zS (1) = 11 ⊗ S (2).z z

24


24/28

Consideroperators

S z =(2)S + z

(1)S z .

� 0 0 0

0 0 0 0 =

0 0

0

0

0 0 0 −� = (S (1))z

1⊗S (2))⊗11)(1(2)S z(1)S z(2)⊗S z

(1)S z

=

�

2�

2�

2−

�

2= �2

�

2

−

−�2

�

2−

+1

� 2 −1 =

4

−1

.

+1

Incompatible

observables

If [A,B] .A,B=0,thencannotsimultaneouslydiagonize

Experiments

� � �

� � � �

� � � ��

� � � �

A

=

a B

=

b1B=b2

B=bk

(AssumeA,B,C nondegenerate)

i) Allowallbi tocombinewithoutmeasuringB

Probability

(C

=

c) =

|c|a|2

[B notmeasured]

C

=

c

A B . C ..

25


25/28


26/28

�

� �

�

�

Example: Instate|+

�2

4∆S 2 = 1 0 �2

1z 0

4

212=

−

1 0 − � 02

� 2 � 2

= − = 0 . 4 4

�2

4∆S 2 = 1 0 �2

1z 0

4

2� 0 � 12= − 1 0 � 0 0

2

� 2

=

.

4

Uncertainty

relation

If A,B areobservables,2∆A2∆B2 ≥ 1 |[A,B]|4

Proof.

Schwartz: (α|∆A)(∆A|α)(α|∆B)(∆B|α)(on

∆A|α,∆B|α)

≥

(α|∆A)(∆B|α)(α|∆B)(∆A|α)

1 = α|�2

[∆A,∆B] + 1 {∆A,∆B}|α22

= 1 2[A,B] + {∆A,∆B}4 ↑ ↑imaginary real

([A,B]skew-Hermitian) ({∆A,∆B} Hermitian)[prob.1–1.]

1 1 2=4|[A,B]|2+

4|{∆A,∆B}|

1

≥

4|[A,

B]|2

.

Example: Instate|α=|+.

∆S 2 = 0 z� 2∆S 2 =x

4

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27/28

14 |[S z, S x]|2 = 14 |S y|2 = 0 .

1.4 Position,momentumandtranslation

Until

now,

all

explicit

examples

involved

finite-dimensional

matrices.

Generalize

tocontinuous degrees of freedom .

Wanttodescribeparticlein3Dbywavefunctionψ(x,y,z )

Simplyto1D:ψ(x)

Want |ψ(x)|2 dx=probabilityparticleisinregiondx.

x

|ψ(x)|2

� � � � � � � � �

� � � � � � � � �

� � � � ⎨⎩⎫⎬

dx

NaturalHilbertspace: L2(R): ∞

Squareintegrablefunctions −∞|ψ(x)|2


28/28

in no way satisfies the requirements of mathematical rigor — not even if these arereducedinanaturalandproperfashiontotheextentcommonelsewhereintheoreticalphysics. Forexample,themethodadherestothefictionthateachself-adjointoperatorcan be put in diagonal form. In the case of those operators for which this is notactually the case, this requires the introduction of “improper” functions with self-

contradictory

properties.

The

insertion

of

such

a

mathematical

“fiction

is

frequently

necessary in Dirac’s approach, even though the problem at hand is merely one of calculatingnumerically theresultof aclearlydefinedexperiment. Therewouldbenoobjection here if these concepts, which cannot be incorporated into the present dayframework of analysis, were intrinsically necessary for the physical theory. Thus, asNewtonianmechanicsfirstbroughtaboutthedevelopmentof theinfinitesimalcalculus,which,initsoriginalform,wasundoubtedlynotself-consistent,soquantummechanicsmight suggest a new structure for our “analysis of infinitely many variables” — i.e.,the mathematical technique would have to be changed, and not the physical theory.But this is by nomeans the case. It should rather bepointed out that the quantummechanical “Transformation theory” can beestablished in amannerwhich is justasclear and unified, but which is also without mathematical objections. It should beemphasized

that

the

correct

structure

need

not

consist

in

a

mathematical

refinement

andexplanationof theDiracmethod,butratherthatitrequiresaproceduredifferingfromtheverybeginning,namely,therelianceontheHilberttheoryof operators.

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