Lecture 5 Vector Spaces and Linear Algebra: Vector...

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Lecture 5

Vector Spaces and Linear Algebra:

Vector Representation of Wave States in Hilbert Spaces

Reading:

Notes and Brennan Chapter 1.6-1.7


Basis vectors (a minimal set of orthogonal vectors that uniquely and completely describes all of vector space) can be added together to construct any point in the vector space.

Example: the unit vectors in Cartesian coordinates.

Similarly, we can think of orthogonal sets of wave functions as vectors and treat them similarly to simple vectors – i.e. add linear combinations of basis wave function sets to result in any arbitrary wave function within the wave space.

Lee Algebra is often used here: Lee was a pure mathematician who described his Linear Algebra as “finally a mathematical formulation that the physicists cannot corrupt with a useful purpose [paraphrase – not direct quote]”.

Fundamental Expansion Postulate


Y AxisX Axis

Consider the case of two orthogonal wave functions (two states):

If Ψn and Ψm are orthogonal, then

mnnm dv δ=ΨΨ∫∞

∞−

*

However, in general, if Ψn and Ψm are orthogonal, then nothing can be said about

mnnopm Kdv δξ?

* =ΨΨ∫∞

∞−

because ξop may rotate Ψn into a projection of Ψm . However, if Ψnis an Eigenfunction of ξop then

mnnnMnnopM dvdv δλλξ ∫∫∞

∞−

∞

∞−

=ΨΨ=ΨΨ **



A basis set is a minimum set of functions that completely spans the space (i.e. all regions in space can be derived from linear combinations of the basis set).

Example: Fourier Series or unit vectors x, y, z.

Since Eigenfunctions of a Hermitian operator having unequal Eigenvalues are mutually orthogonal if they are complete, they must form a basis set.

The Fundamental Expansion Postulate states that every physical observable can be represented by a Hermitian operator with a complete basis set of Eigenfunctions, Ψ1 ,Ψ2 , Ψ3, ... , Ψn and every physical state Ψ can be expanded as a linear combination of Eigenstates as

Where each coefficient is given as:


∑ Ψ=Ψi

iic

See discussion Brennan p. 42-46 for details

∫ ΨΨ= dvc ii*

This is sort of like a dot-product operation with a unit vector in normal vector Linear algebra: For example: if you want to know the yth coefficient of the vector r=2x+3y+7z, where x, y, and z are unit vectors (basis vectors) simply take r•y


Proof:

Let Aop be a Hermitian operator with Eigenfunctions, Ψ1 ,Ψ2 , Ψ3, ... , Ψn and Eigenvalues A1, A2, A3, ... An. The arbitrary physical state Ψ can be expanded as a linear combination of Eigenstates as,

And the expectation value, <A> is then given by,

Therefore...


∑ Ψ=Ψi

iic


∫ ΨΨ= dvAA op*


Proof (cont’d):Fundamental Expansion Postulate


( ) ( )

( )( )

( )

s' c theare weights the whereA s,Eigenvalue

theof average a weighted is A Thus,

cAA

,normalized isΨ since and

dvΨΨcAA

toleading ,cancel" termscross" all

δdvΨΨ

since and numbersjust are sc but the

dvΨAc...ΨAcΨAcΨc...ΨcΨcA

dvΨc...ΨcΨcAΨc...ΨcΨcA

dvΨcAΨcA

ΨdvAΨA

2ii

i

2ii

ii

*i

2ii

ijj*

i

i

nnn222111*

n*

n*

2*

2*

1*

1

nn2211op*

n*

n*

2*

2*

1*

1

iiiop

i

*i

*i

op*

∑

∑ ∫

∫

∫∫

∫ ∑∑

∫

=

=

=

++++=

++++=

=

=

Now using the Eigenvalue relationship


Proof (cont’d):

But how do we find the |cj|2 ‘s? Given:

Before a measurement, the particle can be in an unidentifiable state Ψ (but is made up of a linear combination of Eigenstates). However, once measured, the particle is in a known Eigenstate, Ψi ,with known measurable variable Ai. The term “collapsing into known state Ψi “ is often used. The |ci|2 ‘s are the probability of collapse (probability that a measurement corresponding to Hermitian operator Aop of the particle in arbitrary state Ψ will result in an observable Ai).



i*

j

iiji

*j

ii

*ji

*j

ii

*ji

*j

iii

cΨdvΨ

δcΨdvΨ

dvΨΨcΨdvΨ

ΨΨcΨΨ

ΨcΨ

=

=

=

=

=

∫

∑∫

∫ ∑∫

∑

∑Multiply by Ψj

*

Integrate over all space


Since we can treat orthogonal wave functions as vectors, we can define many linear algebra functions that apply to wave functions as well.

Hilbert Space: Linear Vector Spaces using “Vector Functions”

( ) ( )( ) ( ) ( )

i allfor 0Ψ then0 all If :States all of ceIndepencenLinear ΨΨΨ :State Nulla of Existance

ΨΨΨΨ and ΨµλλΨµ :ntiplicatioScalor MulΨΨΨΨΨΨ and ΨΨΨΨ :Addition veDistribiti

iiii

mom

nmnmmm

lmnlnmmnnm

≠≠

=++=+=

++=+++=+

∑λλ

λλλ


State projections onto each other:

While basis vectors are orthogonal, any two generalized states (each made up of linear combinations of basis states) may not be. Thus, we can define an “overlap operation” similar to a dot product in traditional linear algebra. Similarly to the dot product, this operation returns a scalar representing how much overlap is between the two states:

Linear Vector Spaces

( )( )

∑

∑∑

∫∑∑

∫ ∑∑

∫

∫

=

=

=

=

=

=

∞

∞−

∞

∞−

∞

∞−

∞

∞−

ii

*i

jj

*i

i

ji

j*i

jj

*i

i

jjj

i

*i

*i

n*m

n*m

bad

bad

,Ψ and Ψ functions, basis orthogonal used weif againbut

dvΨΨbad

dvΨbΨad

dvΨΨd

dvΨΨd

ijδ


What is an Operator in the Wave space formulism?

An operator merely maps one wave state to another with both initial and final states being in the Hilbert space and thus, both initial and final states can be represented by linear combinations of the basis vector states. Formally this is simply:

Thus an operator merely changes the state from one state to another. Consider the simple “5 times” operator: The new state is mathematically different for the old state by a factor of 5.

Linear Vector Spaces

oldopnew ΨΨ ξ=


The Fundamental Expansion Postulate can be rewritten in matrix form. Every physical observable can be represented by a Hermitian operator with a complete basis set of Eigenfunctions, Ψ1 , Ψ2 , Ψ3, ... , Ψn and every physical state Ψ can be expanded as a linear combination of Eigenstates as

Matrix Methods in Quantum Mechanics

[ ]

Ψ

ΨΨΨ

=Ψ

Ψ=Ψ ∑

n

n

iii

cccc

c

M

L 3

2

1

321

See discussion Brennan section 1.7 for details

∫ ΨΨ= dvc ii*

Integrals like these that we have been using all semester are often called matrix elements – now you can see why.


Thus, every generalized state can be represented by a vector, c, which are the coefficients of the basis wave states (coefficients of the basis vectors).

Since an operator merely changes from one state to another, the operator then can be represented as a matrix as well. Consider:


[ ] [ ]

=

=

Ψ

ΨΨΨ

=

Ψ

ΨΨΨ

=

== ∑∑

n

3

2

1

nnn1

31

2n2221

1n131211

n

3

2

1

copa

n

3

2

1

n321a

n

3

2

1

n321c

iiia

iiic

c

ccc

ξξ

ξξξξξξξξ

a

aaa

ΨξΨ sincebut

aaaaΨ and ccccΨ

ΨaΨ and ΨcΨ

MOM

OM

L

L

M

M

L

M

L



But how do we find the individual matrix elements ξij? Consider our two states Ψc and Ψa:


( )

( )

( )

( )

( )

∫∑

∑ ∫

∑ ∫

∑ ∫∑

∑ ∫∫∑

∑ ∫∫ ∑

∑ ∫∫

∑

∑∑

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

Ψ==

Ψ=

Ψ=

Ψ=

Ψ=Ψ

Ψ=Ψ

Ψ=Ψ

Ψ

=

===

-i

*ji

ij

ii -

i*jj

i -ii

*jj

i -ii

*j

ii

i -ii

*j

-i

*ji

i

i -ii

*j

- iii

*j

c

i -ii

*j

-a

*j

*j

iiia

iiia

iiiccopa

Ψ where,ca

or cΨa

as, rewritten be can side handright But the

Ψca

Ψca

ΨcΨa

ΨcΨa

side, handleft theon ngonly worki and ,Ψfor in gsubtitutinNow

ΨcΨ

space, allover gintegratin and by sides both gMultiplyin

ΨcΨ operator, theApplying

ΨaΨ and ΨcΨ whereΨξΨ

dv

dv

dv

dv

dvdv

dvdv

dvdv

opjiji

op

op

opij

op

op

op

op

ξξξ

ξ

ξ

ξδ

ξ

ξ

ξ

ξ



The two state vectors can thus be related by a transformation (operator) matrix made up of matrix elements ξij.



[ ] [ ]

∫∑

∑∑

∞

∞

Ψ==

=

=

Ψ

ΨΨΨ

=

Ψ

ΨΨΨ

=

==

-i

*ji

ij

n

3

2

1

nnn1

31

2n2221

1n131211

n

3

2

1

copa

n

3

2

1

n321a

n

3

2

1

n321c

iiia

iiic

Ψ where,ca

c

ccc

ξξ

ξξξξξξξξ

a

aaa

ΨξΨ sincebut

aaaaΨ and ccccΨ

ΨaΨ and ΨcΨ

dvopjiji ξξξ

MOM

OM

L

L

M

M

L

M

L

Thus, many Quantum Mechanics problems reduce down to calculating integrals for each matrix element and then solving NUMERICAL matrix equations.


Consider what happens to the operator matrix if the basis wave set are Eigenfunctions of the operator.



( )( )

( )

( )

=

==

=

=

=Ψ=Ψ=

Ψ=

=

∫∫

∫∞

∞

∞

∞

∞

∞

n

3

2

1

nn

333

222

111

n

3

2

1

-i

*j

-i

*j

-i

*j

n

3

2

1

nnn1

31

2n2221

1n131211

n

3

2

1

c

ccc

ξ0

0ξ00ξ0000ξ

a

aaa

ΨΨ

Ψ where

c

ccc

ξξ

ξξξξξξξξ

a

aaa

MMOM

M

L

L

M

MOM

OM

L

L

M

n

ijiiiji

opji

dvdv

dv

λ

λλ

λ

δλλλξ

ξξ

Thus, using a Eigenfunction basis set results in a diagonal matrix whose elements are the Eigenvalues. Working in reverse, if we can diagonalize the matrix, we can find all the Eigenvalues along the diagonal.

[ ] [ ]

Ψ

ΨΨΨ

=

Ψ

ΨΨΨ

=

n

3

2

1

n321a

n

3

2

1

n321c aaaaΨ and ccccΨM

L

M

L


Consider the case of the Schrödinger Equation (described in detail in the next lecture). This equation can be written as an Eigenvalue equation of the form:

HΨ=EΨ

where H is the Hamiltonian operator (total energy) and E is the energy Eigenvalue.

Considering Ψ to be a vector wave state, we can rewrite the Schrödinger Equation as,



( ) 0

0

becomes, thisleft, thefrom sideright thegsubtractin and Ψ defining

space, allover gintegratin and by gMultiplyin

-i

*j

-

*j

-

*j

-

*j

-

*j

*j

=−

=−

Ψ=

ΨΨ=ΨΨ

ΨΨ=ΨΨ

Ψ

Ψ=Ψ

∑

∑∑

∫

∫∑∫∑

∫ ∑∫ ∑

∑∑

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

iijiji

iji

iiji

i

ji

ii

iii

i

iii

iii

iii

iii

EHc

EcHc

dvHH

dvEcdvHc

dvcEdvcH

cEcH

δ

δ


Continued...Matrix Methods in Quantum Mechanics


( )

( )( )

( )

( )

0

c

ccc

HH

HHHHHHHHHH

form,matrix inor

0

0

becomes, thisleft, thefrom sideright thegsubtractin and Ψ defining

space, allover gintegratin and by gMultiplyin

n

3

2

1

nnn1

3n3331

2n2221

1n131211

-i

*j

-

*j

-

*j

-

*j

-

*j

*j

=

−

−−

−

=−

=−

Ψ=

ΨΨ=ΨΨ

ΨΨ=ΨΨ

Ψ

Ψ=Ψ

∑

∑∑

∫

∫∑∫∑

∫ ∑∫ ∑

∑∑

∞

∞

∞

∞

∞

∞

∞

∞

∞

∞

MMOM

M

L

L

E

EE

E

EHc

EcHc

dvHH

dvEcdvHc

dvcEdvcH

cEcH

iijiji

iji

iiji

i

ji

ii

iii

i

iii

iii

iii

iii

δ

δ


Example: Consider the system whose Hamiltonian matrix elements,

evaluate to:

What are the possible results when the energy is measured?

Writing the (H-E)Ψ=0 => (H-E)(c1Ψ1+c2Ψ2+c3Ψ3)=0 matrix and solving, we get:



Ψ -

i*j∫

∞

∞

Ψ= dvHH ji

−−−

−=

310121

013H

( )( )

( )

( )( )

( )0

310121

013DET

:for solution nontriviala has which

0310

121013

3

2

1

=

−−−−−

−−

=

−−−−−

−−

EE

E

ccc

EE

E


What are the possible results when the energy is measured?Matrix Methods in Quantum Mechanics


( )( )

( )

( )( )

( )

( )

4or 1, 3,E

0)4)(1)(3(0)45)(3(

01)2(

010

)3(101

)1()3(1

1)2(E-3

minors,by Expanding

0310

121013

DET

:for solution nontriviala has which

0310

121013

2

3

2

1

=

=−−−=+−−

=−−

−+

−−−

−−−−−−

=

−−−−−

−−

=

−−−−−

−−

EEEEEE

EEEE

EE

E

ccc

EE

E


What are the corresponding Eigenvectors ΨA, ΨB, and ΨC, for each of these energy Eigenvalues? For E=1,

Similarly for E=3 and E=4,



( )( )

( )

[ ]

ΨΨΨ

=Ψ

=+−=−+−=+−

=

−−−−−

−−

3

2

1

321

321

321

3

2

1

121

yields,unknowns 3 and equations 3 theseSolving

0200002

:of solutiona has which

01310

11210113

A

ccccccccc

ccc

[ ]

ΨΨΨ

−=Ψ

3

2

1

101B [ ]

ΨΨΨ

−=Ψ

3

2

1

111C

Clearly these Eigenvectors are not normalized.


If the ACTUAL state of the particle is described by the vector,

What is the probability of each Eigen-energy (the only possible measurable values) being measured?

After we normalize the states we have previously found and the state above, we must evaluate the overlap of these states. Mathematically this is either,

as described previously or:

For i=A, B and C or equivalently in matrix form,



[ ]

ΨΨΨ

−=Ψ

3

2

1

232actual

∫ ΨΨ= dvc Actualii*

ActualCCActualBBActualAA ΨΨc ΨΨc ΨΨc •=•=•=


Normalizing each of the relevant vectors we get,

Thus, the probability of measuring ...

...Energy=1 is 6/17

...Energy=3 is 8/17

...Energy=4 is 3/17



[ ]

ΨΨΨ

−=Ψ

3

2

1

232171

actual

[ ] [ ] [ ] [ ] [ ] [ ]

173c

178c

176c

232171111

31c232

171101

21c232

171121

61c

ΨΨcΨΨcΨΨc

CBA

CBA

ActualCCActualBBActualAA

===

−•−=−•−=−•=

•=•=•=

[ ]

ΨΨΨ

=Ψ

3

2

1

1216

1A [ ]

ΨΨΨ

−=Ψ

3

2

1

1012

1B [ ]

ΨΨΨ

−=Ψ

3

2

1

1113

1C

Lecture 5 Vector Spaces and Linear Algebra: Vector...

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