vectors, matrix transormations, dirac notation

5/20/2018 Vectors, Matrix Transormations, Dirac Notation

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Matrix Representation

. .

Convenient method of working with vectors.

Superposition Complete set of vectors can be used to

express any other vector.

Complete set of Nvectors can form other complete sets ofNvectors.

Eigenvectors and eigenvalues

.A u u

Matrix method Find superposition of basis states that are

eigenstates of particular operator. Get eigenvalues.

Copyright Michael D. Fayer, 2009


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Orthonormal basis set inNdimensional vector space

e basis vectors

AnyNdimensional vector can be written as

1

N

jj

j

x x e

jjx e xwith

To get this, project out

from

piece of that is ,j j j jj

e e x

x e e e x x e

then sum over all .e



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Operator equation

1 1

N Nj j

j j

j j

e A x e

Substituting the series in terms of bases vectors.

1

Nj

j

j

x A e

ie

N

Left mult. by

1

i j

j

y e A e x

i je A e

eN sca ar pro uctsNvalues ofj ;Nfor eachyi

and the basis set .j

A e

are comp e e y e erm ne y



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je

Writingi j

ija e A e Matrix elements ofA in the basis

N

a x

gives for the linear transformation

1j

Know the aijbecause we knowA and j

e

1

2

x

x

1

2

y

7 5 4

7

Q x y z

vector

Nx

Ny

5

4

,

must know basis

The set ofNlinear al ebraic e uations can be written as

(Set of numbers, gives you vectorwhen basis is known.)

y A x

double underline means matrixCopyright Michael D. Fayer, 2009


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A

11 12 1Na a a

array of coefficients - matrix

21 22 2N

ij

a a aA a

The aijare the elements of the matrix .A

1 2N N NN

A x

1 11 12 1 1 Ny a a a x

vector representatives in particular basis

2 21 22 2

1 2

N N N NN N

y a a x

a a a x

AThe product of matrix and vector representativex

is a new vector re resentative with com onents

1

N

i ij j jy a x



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Matrix Properties, Definitions, and Rules

A BTwo matrices, and are equalA B

if aij = bij.

1 0 0

The unit matrixThe zero matrix

0 1 01

ij

ones down

principal diagonal

0 0 0

0 0 00

Gives identity transformation0 0 0

1

i ij j i

j

y x x

Corresponds to

x

1x x



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Matrix multiplication

ConsiderA x B y operator equations

B A x

Using the same basis for both transformations

1

N

k ki i

i

b y

z B y matrix ( )kiB b

By B A x Cx

C B A

N

has elementsExample

1

kj ki ij

i

Law of matrix multiplication3 4 5 6 41 39



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Multiplication Associative

Multiplication NOT Commutative

BA B A

Matrix addition and multiplication by complex number

A B C

ij ij ij

A

1 1 1

Inverse of a matrix

nverse oCT

1 AA

transpose of cofactor matrix (matrix of signed minors)

0 If 0 is singularA A A



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Reciprocal of Product

1 1 1A B B A

( )ijA aFor matrix defined as

jiA a interchange rows and columnsTranspose

* *

Complex Conjugate

ija comp ex con uga e o eac e emen

*

jiA a

Hermitian Conjugate

complex conjugate transpose



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Rules

| | | |A A determinant of transpose is determinant

* **( )A B A B complex conjugate of product is product of complex conjugates

* *| | | |A A determinant of complex conjugate is

complex conjugate of determinant

( )A B B A

Hermitian conjugate of product is product of

Hermitian conjugates in reverse order

*| | | |A A determinant of Hermitian conjugate is complex conjugate

of determinant



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Definitions

A A

Symmetric

A A

Hermitian

*

A A

*

Real

1A A

Unitary

ij ij ij a a Diagonal

0 1 21A A A A A A

Powers of a matrix

1

2!

A Ae A



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1x

Column vector representative one column matrix

2

N

xx

x

vector representatives in particular basis

y A x

thenbecomes

1 11 12 1 1

2 21 22 2

Na a a x

y a a x

1 2 N N N NN Ny a a a x

row vector transpose of column vector

1 2, Nx x x x

y A x x A trans ose

y A x y x A

Hermitian conjugate



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Change of Basis

orthonormal basis

ie

then

, 1,2,i j ije e i j N

ie

i

Superposition of can formNnew vectors

linearly independent

1, 2,N

i k

ike u e i N

1k

complex numbers



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j i

New Basis is Orthonormal

if the matrix

coefficients in superposition

1 2N

i ke u e i N

1U U

1k

meets the condition

1U U

is unitaryU

ieImportant result. The new basis will be orthonormal

U

1U U U U

if , the transformation matrix, is unitary (see book

and Errata and Addenda ).



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e

eUnitary transformation

substitutes orthonormal basis for orthonormal basis .

x

i

ix x e

Vector

i

i

i

i

x x e

abstract space)written in terms of two basis sets

x Same vector different basis.

UThe unitary transformation can be used to change a vector representative

of in one orthonormal basis set to its vector representative in another

x vector rep. in unprimed basis

x' vector rep. in primed basis

x U x

x U x

change from unprimed to primed basis

change from primed to unprimed basisCopyright Michael D. Fayer, 2009


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, ,y z

Example

Consider basis

y

x

sVector - line in real space.

7 7 1s x y z

In terms of basis

, ,

7

7s

1



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Change basis by rotating axis system 45 around .z

sCan find the new re resentative of , s'

s U s

y z

U is rotation matrix

sin cos 0

0 0 1

U

For 45 rotation aroundz

2 / 2 2 / 2 0

2 / 2 2 / 2 0U

0 0 1



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Then

2 / 2 2 / 2 0 7 0

0 0 1 1 1

s

7 2

vector re resentative of in basiss

e

1

.

Properties unchanged.

1/ 2

s sExam le len th of vector

1/ 2 1/ 2 1/ 2 1/ 2*( ) (49 49 1) (99)s s s s

1/ 2 1/ 2 1/ 2 1/ 2*( ) (2 49 0 1) (99)s s s s



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Can o back and forth between re resentatives of a vector bx

change from unprimed change from primed

to un rimed basis

x U x

x U x

xcomponents of in different basis



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y A x

Consider the linear transformation

operator equation

y A x

eIn the basis can write

i ij j a xor

e U

Change to new orthonormal basis using

y U y U A x U AU x

or

A U AU

Awith the matrix given by

U

1

A U AU

Because is unitaryCopyright Michael D. Fayer, 2009


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Extremely Important

1A U AU

Can change the matrix representing an operator in one orthonormal basis

into the equivalent matrix in a different orthonormal basis.

Called

Similarity Transformation



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y A x A B C A B C

In basis

e

Go into basis

e

y A x A B C A B C

Relations unchanged by change of basis.

A B CExample

U A BU U C U

U U A BCan insert between because1

1U U U U

U AU U BU U C U

Therefore

A B C



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Isomorphism between operators in abstract vector space

an ma r x represen a ves.

abstract vectors and operators

.

can be used in place of the real things.



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Hermitian Operators and Matrices

x A y y A x

Hermitian operator Hermitian Matrix

+ - complex conjugate transpose - Hermitian conjugate



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Theorem (Proof: Powell and Craseman, P. 303 307, or linear algebra book)

1 2 N

,

there exists an orthonormal basis,

,

relative to whichA is represented by a diagonal matrix

1

20 0

0

A

.0 N

The vectors, , and the corresponding real numbers, i, are thei

U

A U U

solutions of the Eigenvalue Equation

and there are no others.



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Application of Theorem

in some basis . The basis is any convenient basis.

In eneral the matrix will not be dia onal.

ie

There exists some new basis eigenvectors

i

in which represents operator and is diagonal eigenvalues.A

To get from to

unitar transformation.

ie

iU

.i iU U e

1A U AU

Similarity transformation takes matrix in arbitrary basis

into diagonal matrix with eigenvalues on the diagonal.Copyright Michael D. Fayer, 2009


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Matrices and Q.M.

Previously represented state of system by vector in abstract vector space.

Dynamical variables represented by linear operators.

pera ors pro uce near rans orma ons.

Real dynamical variables (observables) are represented by Hermitian operators.

y x

Observables are eigenvalues of Hermitian operators. A S S



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Matrix Representation

In proper basis, is the diagonalized Hermitian matrix and

A A

A

the diagonal matrix elements are the eigenvalues (observables).

A suitable transformation takes (arbitrary basis) into1

U AU

A (diagonal eigenvector basis)

1

.A U AU

Diagonalization of matrix gives eigenvalues and eigenvectors.

U takes arbitrary basis into eigenvectors.

Matrix formulation is another way of dealing with operators

and solving eigenvalue problems.



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All rules about kets, operators, etc. still apply.

Exam le

Two Hermitian matrices

can be simultaneously diagonalized by the same unitary

andA B

transformation if and only if they commute.

All ideas about matrices also true for infinite dimensional matrices.



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Example Harmonic Oscillator

(already diagonal).

2 21H P x

1a a a a

2

2

1a n n n 1 1a n n n

0 1 2 3

00 1 0 0 0

0 0 0a

ma r x e emen s o a

1

2

3

0 0 2 0 0

0 0 0 3 0

0 2 0a

a

1 0 01 1 0

aa

1 3 0a



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0 0 0 0

1 0 0 0

0 2 0 0

0 0 3 0a

1

2H a a a a

0 0 04

0 1 0 0

0 0 0 0

1 0 0 0

0 0 0 3

40 0 0 0a a

0 2 0 0

0 0 3 0

0 2 0 0

0 0 3 0

0 0 0 4

0 0 0 4



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1

2H a a a a

0 0 0 0

1 0 0 0

0 1 0 0

0 0 2 0

0 0 0 0

0 1 0 0

0 2 0 0

0 0 3 0a a

0 0 0 3

40 0 0 0

0 0 2 00 0 0 3

0 0 0 4



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a a

a a HAdding the matrices and and multiplying by gives

1 0 0 0 1 2 0 0 0

0 3 0 0 0 3 2 0 0

0 0 5 0 0 0 5 2 02 0 0 0 7 0 0 0 7 2

H

.

the matrix would be multiplied by .

s examp e s ows ea, ut not ow to agona ze matr x w en youdont already know the eigenvectors.



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Diagonalization

Ei envalue e uation ei envalue

uAu 0uAu

matrix representing

operator

representative of

eigenvector

0 1,2

N

ij ij j a u i N

This represents a system of equations

We know the aij.

11 1 12 2 13 3

21 1 22 2 23 3 0

a u a u a u

a u a u a u

We don't know - the eigenvalues

31 1 32 2 33 3 0a u a u a u i- ,

one for each eigenvalue.



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Besides the trivial solution

1 2 0Nu u u

a a a

A solution only exists if the determinant of the coefficients of the uivanishes.

21 22 23

31 32 33 0

a a aa a a

know aij, don't know 's

Expanding the determinant givesNth degree equation for the

the unknown 's (eigenvalues).

Then substituting one eigenvalue at a time into system of equations,the ui(eigenvector representatives) are found.

Nequations for u's gives onlyN- 1 conditions.

* * *

1 1 2 2 1N Nu u u u u u

Use normalization.



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Example - Degenerate Two State Problem

Basis - time independent kets orthonormal.

0H E

and not ei enkets.

0H E Coupling .

These e uations defineH.

The matrix elements are And the Hamiltonian matrix is

H

H

0E

0H E

0E



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0E

The corresponding system of equations is

0( ) 0E

the determinant of the coefficients vanish.

0

0 0E

0

0

E

E

Take the

matrix

a e n o e erm nan .

Subtract from the diagonalelements.

2 Dimer Splitting

2

Expanding

E0 Excited State

0

2 2 2

0 02 0E E

0E

Energy Eigenvalues

Ground State

E = 00E


T bt i Ei t


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To obtain Eigenvectors

Use system of equations for each eigenvalue.

a b

a b

Eigenvectors associated with + and -.

,a b

,a b

and are the vector representatives of and

in the , basis set.

We want to find these.


First for the


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0E First, for the

eigenvalue

write s stem of e uations.

11 12( ) 0H a H b

21 22a

H11 12 21 22; ; ;H H H H H H H H Matrix elements of

0H E

The matrix elements are0 0 0( )E E a b

H

H

0 0a

0a b

The result is0

0a b



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An equivalent way to get the equations is to use a matrix form.

0

0 0E b

u s u e 0

0 0 0E E a

0 0 0E E b

0b

Multi l in the matrix b the column vector re resentative ives e uations.

0a b



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0a b

The two equations are identical.

a b

Always getN 1 conditions for theNunknown components.

Normalization condition gives necessary additional equation.

1a b

Then

2a b

and

1 12 2

Eigenvector in terms of thebasis set.


For the eigenvalue


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For the eigenvalue

0E

0 0

0

E a

b

Substituting 0E

0

a

b

0a b

0a b

a

1 1

2 2a b

ese equat ons g ve

Using normalization

1 1

2 2

Therefore


Can diagonalize by transformation


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1H U H U

Can diagonalize by transformation

diagonal not diagonal

Transformation matrix consists of representatives of eigenvectors

in original basis.

1/ 2 1/ 2U

b b

1 1/ 2 1/ 2

1/ 2 1/ 2U

complex conjugate transpose


Then


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1/ 2 1/ 2 1/ 2 1/ 2E

Then

01/ 2 1/ 2 1/ 2 1/ 2E

1/ 2Factoring out

0

0

1 1 1 112 1 1 1 1

EHE

after matrix multiplication

0 0

0 02 1 1H

E E

0 0EH

diagonal with eigenvalues on diagonal0


vectors, matrix transormations, dirac notation

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