Matrices & Projection Operators: Elements, Eigenvalues - Prof. M. W. Bromley, Study notes of Quantum Mechanics

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Lecture 3 Outline - Matrices

projection operators [Sections 1.6]

start of eigenproblems [Sections 1.8]

matrix elements and algebra [Sections 1.6]

Matrix elements

So

j

| i ′ 〉 = 〈 j | Ω | i

ji

and

β

| α 〉 = ∑ i b

i ′ | i ′ 〉

or

b i = 〈 i | β 〉 = 〈 i | Ω | α 〉

b i = 〈 i | Ω ( ∑ j a j

j 〉 ) = ∑ j a j

i | Ω | j 〉 = ∑ j a j

ij

so a linear transform has a matrix representation

 

b 1

b 2

b n

 

〈 1 | Ω | n 〉

〈 2 | Ω | n 〉

〈 n | Ω | 1 〉 〈 n | Ω | 2 〉

〈 n | Ω | n 〉

a

1

a

2

a

n

 

ie. the first column is the components of

′ 〉

consecutive Projection operators

consider two projection operators (with basis vectors)

P i P j = | i

i | j

〉〈

j

|

=

δ

ij

(^) | j

〉〈

j

|

=

δ

ij

(^) P

j

insert example of electric field polarisers

Projection example

Stating the Eigenproblem

General operator has

Ω | γ 〉 = | β 〉

BUT each linear operator has eigenbits

Ω | α 〉 = ω | α 〉

eigenkets (

α

) and eigenvalues (

ω

) are specific to each

Easy example is

I | α 〉 = | α 〉

every

α

is an

I

eigenvector,

every

α

has associated eigenvalues of

A good/bad example (1.8.3) is rotation

R

2 π (^) i

) | α 〉 = ω | α 〉

has easy soln

R

2 π

i ) a | 1 〉 = a | 1 〉

other two are not so easy

Recall Characteristic Polynomials

a

b

c

d

 

x

1

x

2  

λ

x

1

x

2  

( A − λ I )

~x

For a

×

matrix the determinant is:

a

b

c

d

 

λ

λ

a

λ

b

c

d

λ

a

λ

)(

d

λ

bc

Thus characteristic eqn:

λ 2 − ( a + d ) λ

ad

bc

λ

( a + d ) ±

√ ( a + d ) 2 −

ad

bc

Eigenvalues

λ

are roots of characteristic polynomial.

Since

Tr(

A

a

d

and

det(

A

ad

bc

for real matrix elements (

a, b, c, d

λ

may be complex

An

n

×

n

matrix has polynomial equation of order

n

Extracting eigenvectors

Once we have the eigenvalues (

ω

i )

Find three sets of eigenvectors using

− ω I ) | α 〉 = | 0 〉

ω

ω

ω

a

1

a

2

a

3

 

Extracting eigenvectors example

Next: given a vector

α

α

define the

†

operator

〈 Ω α | = 〈 α | Ω †

ie. in a basis:

† ) ij

= 〈 i | Ω † | j 〉 = 〈 Ω i | j 〉 = 〈 j | Ω i 〉 ∗ = 〈 j | Ω | i 〉 ∗

ji∗

matrix representing

†

is

transpose conjugate

of

recall vector representing

α

is transpose conjugate

α

transposing conjugates

Previously seen that

−

1

−

1 Ω

−

1 , and similarly

transpose conjugate of product is

†

† Ω

†

α

α | = 〈 α |

†

α

α ) | = 〈 Λ α | Ω † = 〈 α | Λ † Ω †

proven, as reqd.

Rule for taking transpose conjugates is summarised as

reverse the product orders, take transpose conjugates.

Hermitian matrix fundamentals

1. Eigenvalues of a Hermitian matrix (operator) are

real

2. Eigenvectors of Hermitian matrices (operators) are

orthogonal

(if they correspond to distinct eigenvalues)

3. Eigenvectors of Hermitian matrices (operators)

span the space

(ie. matrix can be diagonalised)

Firstly: let

Ω | ω 〉 = ω | ω 〉

where

†

and dot away

〈 ω | Ω | ω 〉 = ω 〈 ω | ω 〉

and taking ajoint

〈 ω | Ω † | ω 〉 = ω ∗ 〈 ω | ω 〉

subtracting:

ω

ω

ω

ω ∗ 〈 ω | ω 〉

ω − ω ∗ ) 〈 ω | ω 〉

thus

ω

ω

∗

)

so

ω

ω

∗

ie. real asreqd.