Chapter 2

Mathematical Tools of Quantum Mechanics

Hilbert space

• Let’s recall for Cartesian 3D space:

• A vector is a set of 3 numbers, called components – it can be expanded in terms of three unit vectors (basis)

• The basis spans the vector space

• Inner (dot, scalar) product of 2 vectors is defined as:

• Length (norm) of a vector

zzyyxx AeAeAeA

zzyyxx BABABABA

AAA

Hilbert space

Hilbert space

• Hilbert space:

• Its elements are functions (vectors of Hilbert space)

• The space is linear: if φ and ψ belong to the space then φ + ψ, as well as aφ (a – constant) also belong to the space

David Hilbert(1862 – 1943)

Hilbert space

• Hilbert space:

• Inner (dot, scalar) product of 2 vectors is defined as:

• Length (norm) of a vector is related to the inner product as:

David Hilbert(1862 – 1943)

d)()(*)(),(

d)()(*

d2)(

Hilbert space

• Hilbert space:

• The space is complete, i.e. it contains all its limit points (we will see later)

• Example of a Hilbert space: L2, set of square-integrable functions defined on the whole interval

David Hilbert(1862 – 1943)

d)()(*

Wave function space

• Recall:

• Thus we should retain only such functions ψ that are well-defined everywhere, continuous, and infinitely differentiable

• Let us call such set of functions F

• F is a subspace of L2

• For two complex numbers λ1 and λ2 it can be shown that if

2.A

1),( trdP rdtrCtrdP 32),(),(

Fr )(1

Fr )(2

Frr )()( 2211

Scalar product

• In F the scalar product is defined as:

• Properties of the scalar product:

• φ and ψ are orthogonal if

• Norm is defined as

2.A.1

rdrr )()(*,

*,,

22112211 ,,,

,*,*, 22112211

0,

rdrrdrr 2)()()(*,

,

Scalar product

• Schwarz inequality

,,,

Karl Hermann Amandus Schwarz

(1843 – 1921)

2.A.1

Linear operators

• Linear operator A is defined as:

• Examples of linear operators:

• Parity operator:

• (Multiplication by) coordinate operator:

• Differentiation operator:

)(')( rrA

22112211 AAA

)()( rr

)()( rxrX

xrrDx

)()(

2.A.1

Linear operators

• Product of operators:

• In general:

• Commutator:

• Example:

)()( rBArAB

)(, rDX x

BAAB

BAABBA ,

)()( rXDXD xx

)(rxxx

x

)()( rxxx

rx

xxr

xrx

xrx

)()()(

)(r 1, xDX

2.A.1

Orthonormal bases

• A countable set of functions

• is called orthonormal if:

• It constitutes a basis if every function in F can be expanded in one and only one way:

• Recall for 3D vectors:

)(rui

ijji ruru )(),(

i

ii rucr )()(

2.A.2

,ju

iiij ucu ,

iiij ucu ,

iiji uuc ,

i

ijic jc rdrruuc jjj

)()(*,

ijji ee

iiiecC

ii eCc

Orthonormal bases

• For two functions

• a scalar product is:

• Recall for 3D vectors:

j

jji

ii rucrrubr )()(;)()(

,

2.A.2

i

ii cb *

i

iicbCB

jjj

iii ucub ,

jijiji uucb

,

,*

ji

ijji cb,

*

i

ii

ii ccc 2*,

i

ii cb *,

Orthonormal bases

• This means that

• Closure relation

i

ii rucr )()(

2.A.2

i

ii ruu )(,

i

ii rurdrru )(')'()'(*

')'()()'(* rdrruru

iii

')'()()'(*)( rdrrurur

iii

)'()()'(* rrrurui

ii

Orthonormal bases

• δ-function:

)()( afdxaxxf

Orthonormal bases

• A set of functions labelled by a continuous index α

• is called orthonormal if:

• It constitutes a basis if every function in F can be expanded in one and only one way:

)(rw

)'()(),( ' rwrw

drwcr )()()(

2.A.3

,w ')'(, ' dwcw ')'(, ' dwcw

',)'( ' dwwc '')'( dc )(c

rdrrwwc

)()(*,)(

Orthonormal bases

• For two functions

• a scalar product is: ')()'()(;)()()( ' drwcrdrwbr

,

2.A.3

dcb )()(*

')()'(,)()( ' drwcdrwb

')(),()'()(* ' ddrwrwcb

')'()'()(* ddcb

dcdcc 2)()()(*,

dcb )()(*,

Orthonormal bases

• This means that

• Closure relation

drwcr )()()(

2.A.3

drww )(,

drwrdrrw )(')'()'(*

')'()()'(* rdrdrwrw

')'()()'(*)( rdrdrwrwr

)'()(),'()()'(* rrrwrwdrwrw

)'()(),( ' rwrw

Orthonormal bases

• Useful relationship:

2.A.3

)(21 udkeiku

Examples of orthonormal bases

• Let us apply Fourier transform to function ψ(x):

• Using functions of plane waves

• we can write:

dpepxipx

)(

21)(

2.A.3

dxexp

ipx

)(

21)(

ipx

p exv21)(

dpxvpx p )()()(

dxxxvp p )()(*)(

drwcr )()()( rdrrwc )()(*)(

)(),( ' xvxv pp

dxe

ppxi '

21

)'( pp

Examples of orthonormal bases

• For two functions

• a scalar product is:

')()'()(;)()()( ' dpxvpxdpxvpx pp

,

2.A.3

dppp )()(*

')()'(,)()( ' dpxvpdpxvp pp

')(),()'()(* ' dpdpxvxvpp pp

')'()'()(* dpdppppp

dpp 2)(,

dppp )()(*,

Examples of orthonormal bases

• This means that

• Closure relation

')()'()( ' dpxvpx p

2.A.3

dpxvv pp )(,

dpxvdxxxv pp )(')'()'(*

')'()()'(* dxxdpxvxv pp

')'()()'(*)( dxxdpxvxvx pp

)'()(),'()()'(* xxxvxvdpxvxv pppp

)'()(),( ' ppxvxv pp

Examples of orthonormal bases

• Let us consider a set of functions:

• The set is orthonormal:

• Functions in F can be expanded:

)()( 00rrrr

rdrrrrrr rr

)'()()(),( 00'00

000 )()()( rdrrrr

2.A.3

,0r ')'()(, 00'00

rdrrrr

')()'(, 0'0 00rdrr rr

',)'( 0'0 00rdr rr

'')'( 0000 rdrrr )( 0r

rdrr rr

)(,)(000

)'( 00 rr

00 )()(0 rdrrr

Examples of orthonormal bases

• For two functions

• a scalar product is: ')'()()(;)()()( 00'00 00

rdrrrrdrrr rr

,

2.A.3

000 )()(* rdrr

')'()(,)()( 00'00 00rdrrrdrr rr

')(),()'()(* 00'00 00rdrdrrrr rr

')'()'()(* 000000 rdrdrrrr

)()()()(*, 02

000 rdrdrr

000 )()(*, rdrr

Examples of orthonormal bases

• This means that

• Closure relation

00 )()()(0

rdrrr r

2.A.3

0,)(00

rdr rr

0)(')'()'(00

rdrrdrr rr

')'()()'( 000rdrrdrr rr

')'()()'()( 000rdrrdrrr rr

)'()()'( 000rrrdrr rr

)'()(),( 00'00rrrr rr

State vectors and state space

• The same function ψ can be represented by a multiplicity of different sets of components, corresponding to the choice of a basis

• These sets characterize the state of the system as well as the wave function itself

• Moreover, the ψ function appears on the same footing as other sets of components

2.B.1

State vectors and state space

• Each state of the system is thus characterized by a state vector, belonging to state space of the system Er

• As F is a subspace of L2, Er is a subspace of the Hilbert space

2.B.1

Dirac notation

• Bracket = “bra” x “ket”

• < > = < | > = “< |” x “| >”

2.B.2

Paul Adrien Maurice Dirac(1902 – 1984)

Dirac notation

• We will be working in the Er space

• Any vector element of this space we will call a ket vector

• Notation:

• We associate kets with wave functions:

• F and Er are isomporphic

• r is an index labelling components

2.B.2

Paul Adrien Maurice Dirac(1902 – 1984)

rEFr )(

Dirac notation

• With each pair ok kets we associate their scalar product – a complex number

• We define a linear functional (not the same as a linear operator!) on kets as a linear operation associating a complex number with a ket:

• Such functionals form a vector space

• We will call it a dual space Er*

2.B.2

Paul Adrien Maurice Dirac(1902 – 1984)

rE

22112211

,

Dirac notation

• Any element of the dual space we will call a bra vector

• Ket | φ > enables us to define a linear functional that associates (linearly) with each ket | ψ > a complex number equal to the scalar product:

• For every ket in Er there is a bra in Er*

2.B.2

Paul Adrien Maurice Dirac(1902 – 1984)

,

Dirac notation

• Some properties:

2.B.2

Paul Adrien Maurice Dirac(1902 – 1984)

2211,

2211

,2211

2211 **

,*,* 2211

*

2211 ,,

, ,* *

*

Linear operators

• Linear operator A is defined as:

• Product of operators:

• In general:

• Commutator:

• Matrix element of operator A:

' A

22112211 AAA

2.B.3

BAAB

BAAB

BAABBA ,

A

Linear operators

• Example:

• What is ?

• It is an operator – it converts one ket into another

2.B.3

,

Linear operators

• Example:

• Let us assume that

• Projector operator

• It projects one ket onto another

2.B.3

P

1

PPP 2

P

Linear operators

• Example:

• Let us assume that

• These kets span space Eq, a subspace of E

• Subspace projector operator

• It projects a ket onto a subspace of kets

2.B.3

qP

q

iii

1

q

iii

1

q

iiiqP

1

ijji

qqq PPP 2

q

jjj

q

iii

11

q

jijjii

1,

q

iii

1

qP

qji ,...,2,1,

q

jijiji

1,

Linear operators

• Recall matrix element of a linear operator A:

• Since a scalar product depends linearly on the ket, the matrix element depends linearly on the ket

• Thus for a given bra and a given operator we can associate a number that will depend linearly on the ket

• So there is a new linear functional on the kets in space E, i.e., a bra in space of E*, which we will denote

• Therefore

2.B.4

A

A

AA A

Linear operators

• Operator A associates with a given bra a new bra

• Let’s show that this correspondence is linear

2.B.4

' A

2211 2211

AA AA 2211

AA 2211

AA 2211 AA 2211 ... DEQ

Charles Hermite(1822 – 1901)

Linear operators

• For each ket there is a bra associated with it

• Hermitian conjugate (adjoint) operator:

• This operator is linear (can be shown)

2.B.4

''

A'

†' A

*'' *† AA

Charles Hermite(1822 – 1901)

Linear operators

• Some properties:

2.B.4

†† * AA AA ††

†A

AB

††† BABA

A B

†B †AB

†A ††AB ††† ABAB

Charles Hermite(1822 – 1901)

Hermitian conjugation

• To obtain Hermitian conjugation of an expression:

• Replace constants with their complex conjugates

• Replace operators with their Hermitian conjugates

• Replace kets with bras

• Replace bras with kets

• Reverse order of factors

2.B.4

* †AA

†AA

Charles Hermite(1822 – 1901)

Hermitian operators

• For a Hermitian operator:

• Hermitian operators play a fundamental role in quantum mechanics (we’ll see later)

• E.g., projector operator is Hermitian:

• If:

2.B.4

0, BA

†AA

††† ABAB

* AA

P †† P

AA † BB †

BA AB ABAB †

Representations in state space

• In a certain basis, vectors and operators are represented by numbers (components and matrix elements)

• Thus vector calculus becomes matrix calculus

• A choice of a specific representation is dictated by the simplicity of calculations

• We will rewrite expressions obtained above for orthonormal bases using Dirac notation

2.C.1

Orthonormal bases

• A countable set of kets

• is called orthonormal if:

• It constitutes a basis if every vector in E can be expanded in one and only one way:

iu

ijji uu

i

ii uc

ju i

iji uuc i

ijic jc

jj uc

2.C.2

Orthonormal bases

• Closure relation

• 1 – identity operator

i

ii uc

2.C.2

i

ii uu

ii

i uu

iii uu

iii uu

1̂}{ i

iiu uuPi

1̂

Orthonormal bases

• For two kets

• a scalar product is:

i

iii

ii ucub ;

2.C.3

i

ii cb *

1̂

i

ii uu

i

ii

ii ccc 2*i

ii cb *

iiii ucub ;

iii uu

Orthonormal bases

• A set of kets labelled by a continuous index α

• is called orthonormal if:

• It constitutes a basis if every vector in E can be expanded in one and only one way:

w

)'(' ww

dwc )(

2.C.2

w ')'( ' dwcw ')'( ' dwwc

'')'( dc )(c

wc )(

Orthonormal bases

• Closure relation

• 1 – identity operator

dww

2.C.2

dwc )(

dww dww

dww

1̂}{ dwwPw

1̂

Orthonormal bases

• For two kets

• a scalar product is:

dwcdwb )(;)(

2.C.3

dcb )()(*

1̂

dww

dc 2)( dcb )()(*

wcwb )(;)(

dww

Representation of kets and bras

• In a certain basis, a ket is represented by its components

• These components could be arranged as a column-vector:

...

...2

1

iu

uu

2.C.3

...

...

w

Representation of kets and bras

• In a certain basis, a bra is also represented by its components

• These components could be arranged as a row-vector:

......21 iuuu

2.C.3

...... w

Representation of operators

• In a certain basis, an operator is represented by matrix components:

...............

......

...............

......

......

21

22221

11211

ijii

j

j

AAA

AAAAAA

2.C.4

'............)',(............

A

jiij uAuA ')',( wAwA

jiji uBAuuABu 1̂

jk

kki uBuuAu

kjkki uBuuAu

Representation of operators2.C.4

jjji uuAu

A'

'' ii uc Aui 1̂Aui

j

jji uuAu j

jijcA

...

...

...............

......

...............

......

......

...'...''

2

1

21

22221

11211

2

1

iijii

j

j

i c

cc

AAA

AAAAAA

c

cc

Representation of operators2.C.4

''' dwwAw

')(' wc Aw 1̂Aw

''' dwwAw

')'()',( dcA

A'

Representation of operators2.C.4

...

...

...............

......

...............

......

......

...*...**2

1

21

22221

11211

121

iijii

j

j

c

cc

AAA

AAAAAA

bbb

A

iiii ucub ;

Representation of operators2.C.4

...*...**

...

... 21

2

1

j

i

cccc

cc

ii uc

...............

...*...**

...............

...*...**

...*...**

21

22212

12111

jiii

j

j

cccccc

cccccccccccc

Representation of operators

• For Hermitian operators:

• Diagonal elements of Hermitian operators are always real

2.C.4

jiij uAuA ††

'†† )',( wAwA

*ij uAu *jiA

*' wAw ),'(* A

*jiij AA *),'()',( AA

*iiii AA *),(),( AA

1SSSS ††

Change of representations2.C.5

i

iik uut

iiik uutkt 1̂kt

i

iki uS †

k

kki ttu

kkki ttuiu 1̂iu

k

kik tS

i

kii tuuki

ii tuu

kt kt1̂

i

iki Su

Change of representations2.C.5

ji

ljjiik tuuAuut,

lj

jji

iik tuuAuut

lk tAt

ji

jiijki SAS,

†

lk

jllkki uttAttu,

jl

llk

kki uttAttu

ji uAu

lk

ijklik SAS,

†

Eigenvalue equations

• A ket is called an eigenvector of a linear operator if:

• This is called an eigenvalue equation for an operator

• This equation has solutions only when λ takes certain values - eigenvalues

• If:

• then:

2.D.1

A

A

*† A

Eigenvalue equations

• The eigenvalue is called nondegenerate (simple) if the corresponding eigenvector is unique to within a constant

• The eigenvalue is called degenerate if there are at least two linearly independent kets corresponding to this eigenvalue

• The number of linearly independent eigenvectors corresponding to a certain eigenvalue is called a degree of degeneracy

2.D.1

Eigenvalue equations

• If for a certain eigenvalue λ the degree of degeneracy is g:

• then every eigenvector of the form

• is an eigenvector of the operator A corresponding to the eigenvalue λ for any ci:

• The set of linearly independent eigenvectors corresponding to a certain eigenvalue comprises a g-dimensional vector space called an eigensubspace

2.D.1

i

iicAA

i

iiAc

i

iic

i

iic

i

iic

giA ii ,...2,1;

Eigenvalue equations

• Let us assume that the basis is finite-dimensional, with dimensionality N

• This is a system of N linear homogenous equations for N coefficients cj

• Condition for a non-trivial solution:

2.D.1

ii uAu A

ij

jji uuuAu i

jjji uuuAu i

jjij ccA

0j

jijij cA

0 1A

Eigenvalue equations

• This equation is called the characteristic equation

• This is an Nth order equation in and it has N roots – the eigenvalues of the operator

• Condition for a non-trivial solution:

2.D.1

0 1A

0

...............

...

...

21

22221

11211

NNNN

N

N

AAA

AAAAAA

Eigenvalue equations

• Let us select λ0 as one of the eigenvalues

• If λ0 is a simple root of the characteristic equation, then we have a system of N – 1 independent equations for coefficients cj

• From linear algebra: the solution of this system (for one of the coefficients fixed) is

2.D.1

00 1A 00 j

jijij cA

1; 011

0 cc jj

j

jj uc0 j

jj uc10

jjj uc 0

1

Eigenvalue equations

• Let us select λ0 as one of the eigenvalues

• If λ0 is a multiple (degenrate) root of the characteristic equation, then we have less than N – 1 independent equations for coefficients cj

• E.g., if we have N – 1 independent equations then (from linear algebra) the solution of this system is

2.D.1

00 1A 00 j

jijij cA

0;1; 01

02

02

012

01

0 ccc jjj

j

jjj

jj ucuc 02

010

Eigenproblems for Hermitian operators

• For:

• Therefore λ is a real number

• Also:

• If:

• Then:

• But:

2.D.2

A†AA

A †* AA A

0Im A 0Im

A

A A

A A

A A 0

Observables

• Consider a Hermitian operator A whose eigenvalues form a discrete spectrum

• The degree of degeneracy of a given eigenvalue an will be labelled as gn

• In the eigensubspace En we consider gn linearly independent kets:

• If

• Then

2.D.2

,...2,1; nan

ninn

in giaA ,...,2,1;

'nn aa

0' jn

in

Observables

• Inside each eigensubspace

• Therefore:

• If all these eigenkets form a basis in the state space, then operator A is called an observable

2.D.2

ijjn

in

1̂1

n

g

i

in

in

n

'' nnijjn

in

Observables

• For an eigensubspace projector

• These relations could be generalized for the case of continuous bases

• E.g., a projector is an observable

2.D.2

ng

i

in

innP

1

n

g

i

in

in

n

AAA1

1̂

n

g

i

in

inn

n

a1

n

nnPa

n

nnPaA

P 1

Observables

• If

• Then

• If a is non-degenerate then

• so this ket is also an eigenvector of B

• If a is degenerate then

• Thereby, if A and B commute, each eigensubspace of A is globally invariant (stable) under the action of B

2.D.3

0, BA aA

aBBA aBAB

BaBA

B

aEB

Observables

• If

• Then

• If two operators commute, there is an orthonormal basis with eigenvectors common to both operators

2.D.3

0, BA

111 aA

21121 BaAB

222 aA 21 aa

021 B

21221 BaBA

21212121 BaaBAAB

Observables

• A set of observables, commuting by pairs, is called a complete set of commuting observables (CSCO) if there exists a unique orthonormal basis of common eigenvectors

• If all the eigenvalues of a certain operator are non-degenerate, this operator constitutes CSCO by itself

• If one ore more eigenvalues of a certain operator are degenerate, there is no unique orthonormal basis of eigenvectors

• Then at least one more operator commuting with the first one is used to construct a unique orthonormal basis of common eigenvectors, an thus a CSCO

2.D.3

Examples of representations

• Let us consider a set of functions:

• The set is orthonormal:

• Kets can be expanded:

00 )()(0

rrrrr

rdrrrrrr

)'()(' 0000

2.E.1

)'( 00 rr

000 )( rdrr

0r '')'( 0000 rdrrr

'')'( 0000 rdrrr

'')'( 0000 rdrrr

)( 0r

00 )( rr

Examples of representations

• Closure relation

000 rdrr

2.E.1

000 )( rdrr

000 rdrr

000 rdrr

000 rdrr

1̂000}{ 0 rdrrP r

1̂

Examples of representations

• For two kets

• a scalar product is:

000000 )(;)( rdrrrdrr

2.E.1

000 )()(* rdrr

1̂

000 rdrr

02

0 )( rdr

000 )()(* rdrr

0000 )(;)( rrrr

000 rdrr

Examples of representations

• Let us consider a set of functions:

• The set is orthonormal:

• Kets can be expanded:

03

0

02

1)( pervrpi

p

rdepp

ppri 00 '

300 21'

2.E.1

)'( 00 pp

000 )( pdpp

0p '')'( 0000 pdppp

'')'( 0000 pdppp

'')'( 0000 pdppp

)( 0p

00 )( pp

Examples of representations

• Closure relation

000 pdpp

2.E.1

000 )( pdpp

000 pdpp

000 pdpp

000 pdpp

1̂000}{ 0 pdppP p

1̂

Examples of representations

• For two kets

• a scalar product is:

000000 )(;)( pdpppdpp

2.E.1

000 )()(* pdpp

1̂

000 pdpp

02

0 )( pdp 000 )()(* pdpp

0000 )(;)( pppp

000 pdpp

Change of representations

• Recall:

• Choosing

• we obtain:

2.E.1

00 )( rr

rpi

p ervp

0

0 302

1)(

00

3002

1 rpi

epr

rpi

epr

32

1

pdppr

pdpprr 1̂r

00 )( pp

rdrrp rdrrpp

1̂p

pdeprpi

)(2 2/3

R and P operators

• For

• we obtain:

• where

• Similarly

• “Vector” operator R:

2.E.2

),,()( zyxrr

X'

),,(')('' zyxrr

),,(),,(' zyxxzyx rxr

' rxXr

ryYr

rzZr

ZkYjXiR ˆˆˆ

XX 1̂

rdXrr

Xrdrr

rdrxr rdrxr )()(*

R and P operators

• “Vector” operator P:

• Then:

2.E.2

ppPp xx

ppPp yy

ppPp zz

zyx PkPjPiP ˆˆˆ

xx PrPr 1̂

pdPppr x

xPpdppr

pdpppr x

pdppe x

rpi

)(2

13

x

ri

)(

R and P operators

• Analogously:

• Then:

2.E.2

xr

iPr x

)(

yr

iPr y

)( zr

iPr z

)(

)(ri

Pr

ri

xx PP 1̂

rdPrr x

xPrdrr

rdxr

ir

)()(*

R and P operators

• Calculating a commutator:

• Similarly:

2.E.2

XPXPrPXr xxx ,

Xrxi

Prx x

XPrXPr xx

rxxi

rxi

x

xxr

ir

xixr

xix

ri

iPX x ,

0],[],[ jiji PPRR

ijji iPR ],[

R and P operators

• Calculating a matrix element:

• Similarly:

• Position and momentum operators are Hermitian

2.E.2

rdrxrX )()(*

* X *)()(* rdrxr

rdrxi

rPx

)()(*

dxr

xrrrdydz

ix

x)(*)()()(*

dxrxi

rdydz )()(*

dxr

xirdydz )(*)(

* xP

R and P operators

• Calculating a matrix element:

• Thus:

• Similarly:

• Since |r > and |p > constitute complete bases, therefore operators R and P are observables

• Sets of operators {X,Y,Z} as well as {Px,Py,Pz} comprise a CSCO each, however, separate operators don’t, since they are degenerate (in other directions)

2.E.2

00 rrxrXr )( 00 rrx

)( 0rrx

00 rrx

00 rxrX

rzrZ

ryrY

rxrX

pppP

pppP

pppP

zz

yy

xx

Tensor products of state spaces

• Spaces of square-integrable functions in 1D, 2D, and 3D are not the same (e.g., Er and Ex are different)

• How are those spaces related?

• In general, if there are two or more mutually isolated subsystems of a certain system, each of which has its own space, what is the space of the entire system?

• Such questions are resolved via introduction of tensor products of spaces

2.F.1

Tensor products of state spaces

• Let there be two spaces E1 and E2 with dimensions N1 and N2

• Tensor product of E1 and E2 is a vector field E with the following properties:

• Notation:

• If vectors belonging to E1 and E2 are

• Then vectors belonging to E are

• Tensor product is linear:

2.F.2

21 EEE

)2(;)1(

)2()1(

)2()1()2()1()2()1(

Tensor products of state spaces

• Let there be two spaces E1 and E2 with dimensions N1 and N2

• Tensor product of E1 and E2 is a vector field E with the following properties:

• Tensor product is distributive:

• Tensor product of bases is a basis

2.F.2

)2()1()2()1(

)2()2()1(

21

21

21 )2(;)1( EvEu ji 21)2()1( EEEvu ji

Tensor products of state spaces

• If:

• Then:

• Components of a tensor product of two vectors are products of the components

• Not all the vectors in E can be represented as tensor products of vectors from E1 and E2:

2.F.2

j

jji

ii vbua )2(;)1(

ji

jiji vuba,

)2()1(

ji

jiji vuc,

jiji bac

Tensor products of state spaces

• Scalar product:

• For orthonormal bases:

• Tensor product of operators:

• Projector:

2.F.2

)2()2()1()1()2()1()2()1( BABA

)2()2()1()1()2()1()2()1(

jliklkji vuvu )2()1()2()1(

)2()2()1()1(

)2()1()2()1(

Tensor products of state spaces

• If:

• Then:

• And:

• Eigenvectors of A(1) + B(2) are tensor products of eigenvectors of A(1) and eigenvectors of B(2)

• If there is one CSCO in E1 and another CSCO in E2 their tensor product is a CSCO in E

2.F.3

)2()1()2()1()2( nmnnm bB

)2()2()2(;)1()1()1( nnnmmm bBaA

)2()1()2()1()1( nmmnm aA

)2()1(

)2()1()2()1(

nmnm

nm

ba

BA

Tensor products of state spaces

• If the problem is strictly 1D (e.g. x-dependent), then the state space is Ex

• In the x-representation the basis kets:

• Similarly we can consider Ey and Ez:

• Introducing

• we get

2.F.4

xxxxxx x )();()( 00 0

zyxxyz EEEE

yyyyyy y )();()( 00 0

zzzzzz z )();()( 00 0

zyxzyx ,,

Tensor products of state spaces

• Exyz is the state space of a 3D particle

• A ket in 3D can be represented:

• In general, ψ cannot be factorized

• Operator X in Ex is a CSCO by itself, but in Exyz its eigenvalues would be infinitely degenerate, because Ey and Ez in are infinitely-dimensional

• On the other hand, the {X,Y,Z} set is a CSCO

2.F.4

)()()()( 00000 zzyyxxrrrr

zyxzyx

zyxzyxdxdydz

,,),,(

,,),,(

States of a two-particle system

• For two particles the state space is:

• and the basis is:

• A ket can be represented:

• If:

• Then:

• In this case there is no correlation between the particles

2.F.4

2121, rrrr

2121 rrrr EEE

2121

212121

,),(

,),(

rrrr

rrrrrdrd

21

2121 ,),( rrrr 2211 rr

)()( 2211 rr