Matrix Representation
Matrix Rep.
Same basics as introduced already.
Convenient method of working with vectors.
Superposition
Complete set of vectors can be used to
express any other vector.
Complete set of N vectors can form other complete sets of N vectors.
Can find set of vectors for Hermitian operator satisfying
A u  u .
Eigenvectors and eigenvalues
Matrix method
Find superposition of basis states that are
eigenstates of particular operator. Get eigenvalues.
Copyright – Michael D. Fayer, 2007
Orthonormal basis set in N dimensional vector space
e 
j
basis vectors
Any vector can be written as
N
x   xj e j
j 1
with
xj  e j x
To get this, project out
e j e j from x
sum over all e j .
xj e j  e j e j x
Copyright – Michael D. Fayer, 2007
Operator equation
y Ax
N
 yj e
N
j
j 1
 A x j e j
Substituting the series in terms of bases vectors.
j 1
N
  xj A e j
j 1
Left mult. by e i
N
yi   e i A e j x j
j 1
The N2 scalar products
ei A e j
N values of j ; N for each yi
are completely determined by
 .
A and the basis set e j
Copyright – Michael D. Fayer, 2007
Writing
aij  e i A e j
 
Matrix elements of A in the basis e j
gives for the linear transformation
N
yi   aij x j
j 1
j  1, 2,
N
 
j
Know the aij because we know A and e
In terms of the vector representatives
x   x1 , x2 ,
xN 
y   y1 , y2 ,
yN 
Q  7 xˆ  5 yˆ  4 zˆ vector
vector representative,
[7, 5, 4]
must know basis
(Set of numbers, gives you vector
when basis is known.)
The set of N linear algebraic equations can be written as
y  Ax
double underline means matrix
Copyright – Michael D. Fayer, 2007
A
array of coefficients - matrix
 a11
a
A   aij    21


aN 1
a12
a22
aN 2
a1 N 
a2 N 



a NN 
The aij are the elements of the matrix A .
The product of matrix A and vector representative x
is a new vector representative y with components
N
yi   aij x j
j 1
Copyright – Michael D. Fayer, 2007
Matrix Properties, Definitions, and Rules
Two matrices, A and B are equal
A B
if aij = bij.
The unit matrix
1 0
0 1
1   ij  


0 0
0
0  ones down

 principal diagonal

1
Gives identity transformation
N
yi    ij x j  xi
The zero matrix
0 0
0 0
0


0 0
0
0



0
0x  0
j 1
Corresponds to
y 1 x  x
Copyright – Michael D. Fayer, 2007
Matrix multiplication
Consider
y Ax
z B y
operator equations
z BA x
Using the same basis for both transformations
N
zk   bki yi
zBy
i 1
B  matrix  (bki )
z  By  B A x  C x
C BA
has elements
Example
N
ckj   bki aij
i 1
Law of matrix multiplication
 2 3  7 5   29 28 
 3 4  5 6    41 39 


 

Copyright – Michael D. Fayer, 2007
Multiplication Associative
 A B C  A BC 
Multiplication NOT Commutative
AB  B A
Matrix addition and multiplication by complex number
 A  B  C
cij   aij   bij
Inverse of a matrix A
inverse of A
A 
A 0
1
1
AA  A A1
identity matrix
A
transpose of cofactor matrix (matrix of signed minors)
A
determinant
CT
1
A
1
If A  0
A is singular
Copyright – Michael D. Fayer, 2007
Reciprocal of Product
 A B  B A
1
1
1
For matrix defined as A  ( aij )
Transpose
A   a ji 
interchange rows and columns
Complex Conjugate
 
A  aij*
*
complex conjugate of each element
Hermitian Conjugate

 
A  a *ji
complex conjugate transpose
Copyright – Michael D. Fayer, 2007
Rules
 A B  B A
transpose of product is product of transposes in reverse order
| A|  | A|
determinant of transpose is determinant
( A B )*  A B
*
*
| A |  | A |*
*

( A B )  B A

| A |  | A |*
complex conjugate of product is product of complex conjugates
determinant of complex conjugate is
complex conjugate of determinant

Hermitian conjugate of product is product of
Hermitian conjugates in reverse order
determinant of Hermitian conjugate is complex conjugate
of determinant
Copyright – Michael D. Fayer, 2007
Definitions
A A
Symmetric
A A

Hermitian
A A
*
Real
A  A
*
1
A A
Imaginary

Unitary
aij  aij  ij
Diagonal
Powers of a matrix
A 1
0
A A
e  1 A
A
1
A
A  A A 
2
2
2!
 
Copyright – Michael D. Fayer, 2007
Column vector representative
 x1 
x 
x 2
 
 
 xN 
vector representatives in particular basis
then
becomes
y  Ax
 y1   a11
 y  a
 2    21
  
  
 yN  aN 1
a12
a22
row vector
x   x1 , x2
one column matrix
aN 2
aN 




a NN 
 x1 
x 
 2
 
 
 xN 
transpose of column vector
xN 
y  Ax
y  xA
y  Ax
y  x A
transpose

Hermitian conjugate
Copyright – Michael D. Fayer, 2007
Change of Basis
orthonormal basis
e 
i
then
 i , j  1, 2,
e i e j   ij
Superposition of
N
 
ei
can form N new vectors
linearly independent
 
a new basis e i
e
i
N
  uik e k
i  1, 2,
N
k 1
complex numbers
Copyright – Michael D. Fayer, 2007
New Basis is Orthonormal
e j e i    ij
if the matrix
U   uik 
coefficients in superposition
e
i
N
  uik e k
i  1, 2,
N
k 1
meets the condition

U U 1
U
1
U

U is unitary
Important result. The new basis
  will be orthonormal
e i
if U , the transformation matrix, is unitary (see book


UU  U U 1
and Errata and Addenda ).
Copyright – Michael D. Fayer, 2007
Unitary transformation
substitutes orthonormal basis  e  for orthonormal basis  e  .
Vector x
x   xi e i
i
x   xi e i
vector – line in space (may be high dimensionality
abstract space)
written in terms of two basis sets
i
x
Same vector – different basis.
The unitary transformation U can be used to change a vector representative
of x in one orthonormal basis set to its vector representative in another
orthonormal basis set.
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 basis
Copyright – Michael D. Fayer, 2007
Example
Consider basis
 xˆ , yˆ , zˆ
y
|s
z
x
Vector s - line in real space.
In terms of basis s  7 xˆ  7 yˆ  1zˆ
Vector representative in basis  xˆ , yˆ , zˆ 
7
s  7
 
 1 
Copyright – Michael D. Fayer, 2007
Change basis by rotating axis system 45° around ẑ .
Can find the new representative of s , s'
s  U s
U is rotation matrix
x
 cos
U    sin
 0

y
sin
cos
0
z
0
0 
1 
For 45° rotation around z
 2/2

U   2 / 2
 0


2 / 2 0

2 / 2 0
0
1 

Copyright – Michael D. Fayer, 2007
Then
 2/2

s   2 / 2
 0


7 2 



s  0 
 1 


2 / 2 0 7  7 2 
  

2 / 2 0 7   0 
0
1   1   1 



vector representative of s in basis  e 
Same vector but in new basis.
Properties unchanged.
Example – length of vector

ss

1/ 2
 s s 
1/ 2
 ( s* s )1/ 2  (49  49  1)1/ 2  (99)1/ 2
 s s 
1/ 2
 ( s* s )1/ 2  (2  49  0  1)1/ 2  (99)1/ 2
Copyright – Michael D. Fayer, 2007
Can go back and forth between representatives of a vector x by
change from unprimed
to primed basis
x  U x
change from primed
to unprimed basis

x  U x
components of x in different basis
Copyright – Michael D. Fayer, 2007
Consider the linear transformation
y Ax
operator equation
In the basis  e
 can write
y  Ax
or
yi   aij x j
j
Change to new orthonormal basis
 e  using U

y  U y  U A x  U AU x
or
y  A x
with the matrix A given by

A  U AU
Because U is unitary
1

A  U AU
Copyright – Michael D. Fayer, 2007
Extremely Important
1

A  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
Copyright – Michael D. Fayer, 2007
y  Ax
AB  C
In basis  e

A B  C
Go into basis
 e 
y  A x
A B  C 
A  B  C 
Relations unchanged by change of basis.
Example
AB  C

U A BU  U C U



1
Can insert U U between A B because U U  U U  1


U AU U BU  U C U

Therefore
A B  C 
Copyright – Michael D. Fayer, 2007
Isomorphism between operators in abstract vector space
and matrix representatives.
Because of isomorphism not necessary to distinguish
abstract vectors and operators
from their matrix representatives.
The matrices (for operators) and the representatives (for vectors)
can be used in place of the real things.
Copyright – Michael D. Fayer, 2007
Hermitian Operators and Matrices
Hermitian operator
x A y  y Ax
Hermitian operator
Hermitian Matrix

A A
+ - complex conjugate transpose - Hermitian conjugate
Copyright – Michael D. Fayer, 2007
Theorem (Proof: Powell and Craseman, P. 303 – 307, or linear algebra book)
For a Hermitian operator A in a linear vector space of N dimensions,
there exists an orthonormal basis,
U1 , U 2
UN
relative to which A is represented by a diagonal matrix
1 0
0 
2
A  
0

0

0

0 
 .

N 
The vectors, U i , and the corresponding real numbers, i, are the
solutions of the Eigenvalue Equation
A U  U
and there are no others.
Copyright – Michael D. Fayer, 2007
Application of Theorem
Operator A represented by matrix A
e 
i
in some basis
. The basis is any convenient basis.
In general, the matrix will not be diagonal.
There exists some new basis
eigenvectors
U 
i
in which A represents operator and is diagonal
To get from
eigenvalues.
 e  to  U 
i
i
unitary transformation.
 U   U  e .
i
1
A  U AU
i
Similarity transformation takes matrix in arbitrary basis
into diagonal matrix with eigenvalues on the diagonal.
Copyright – Michael D. Fayer, 2007
Matrices and Q.M.
Previously represented state of system by vector
in abstract vector space.
Dynamical variables represented by linear operators.
Operators produce linear transformations.
y Ax
Real dynamical variables (observables) are represented by Hermitian operators.
Observables are eigenvalues of Hermitian operators.
A S  S
Solution of eigenvalue problem gives eigenvalues and eigenvectors.
Copyright – Michael D. Fayer, 2007
Matrix Representation
Hermitian operators replaced by Hermitian matrix representations.
A A
In proper basis, A is the diagonalized Hermitian matrix and
the diagonal matrix elements are the eigenvalues (observables).
1
A suitable transformation U AU takes A (arbitrary basis) into
A (diagonal – eigenvector basis)
1
A  U AU .
Diagonalization of matrix gives eigenvalues and eigenvectors.
Matrix formulation is another way of dealing with operators
and solving eigenvalue problems.
Copyright – Michael D. Fayer, 2007
All rules about kets, operators, etc. still apply.
Example
Two Hermitian matrices A and B
can be simultaneously diagonalized by the same unitary
transformation if and only if they commute.
All ideas about matrices also true for infinite dimensional matrices.
Copyright – Michael D. Fayer, 2007
Example – Harmonic Oscillator
Have already solved – use occupation number representation kets and bras
(already diagonal).
H



1
1 2


2
P  x  aa  a a
2
2
a n  n n 1

a n  n1 n1
matrix elements of a
0 a 0 0
0a1  1
0 a 2 0
1a 0 0
1a 1 0
1a 2  2
1a 3 0

0
1
2
3
a 

















0
1
2
0
1
0
0
0
0
0
0
0




0
0
2

3

0
0
0
0
3
0




0
4


 









 


Copyright – Michael D. Fayer, 2007




a  





0
0
0
0
1
0
0
0
0
0
0
2
0
0
3
0
0

0

0

4

0

0


aa  0
0




1
0
0
2
0
0
0
0
0
0
3
0













  
  








   0

   1
   0

4   0

   0
    
H

1
a a  a a
2
0
0
0
0
0
0
0
0
2
0
3
0
0

0

4



 1 0
 0 2
 
  0 0
 0 0
 
 


 

  








0
0
3
0

0
0
0
4


 



 
Copyright – Michael D. Fayer, 2007
H

1
a a  a a
2





a a





0
1
0
0
0
2
0
0
0

0




0 0


0 0


3 0

0 4    


    
0
0












0

0

0
0




1
0
0
2
0
0
0
0
0
0
3
0






  
 0
   
0


 
  0
4   0
 
    
  
0
1
0
0

0
0
2
0

0
0
0
3


 



 
Copyright – Michael D. Fayer, 2007
Adding the matrices a a and a a and multiplying by ½ gives H
1
0
1
H  0
2
0


0
3
0
0

0
0
5
0

0
0
0
7

  1 2 0
0
0
   0 3 2 0
0
   0
0 5 2 0
 
  0
0
0 7 2
   




 



 
The matrix is diagonal with eigenvalues on diagonal. In normal units
the matrix would be multiplied by  .
This example shows idea, but not how to diagonalize matrix when you
don’t already know the eigenvectors.
Copyright – Michael D. Fayer, 2007
Diagonalization
Eigenvalue equation
eigenvalue
Au   u
matrix representing
operator
Au   u  0
representative of
eigenvector
In terms of the components
a
N
j 1
ij
   ij  u j  0
 i  1, 2
N
This represents a system of equations
 a11    u1  a12 u2  a13 u3 
0
a21u1   a22    u2  a23 u3 
0
a31u1  a32 u2   a33    u3 
0
We know the aij.
We don't know
 - the eigenvalues
ui - the vector representatives,
one for each eigenvalue.
Copyright – Michael D. Fayer, 2007
Besides the trivial solution
u1  u2 
uN  0
A solution only exists if the determinant of the coefficients of the ui vanishes.
 a11   
a21
a31



a12
 a22   
a32
a13
a23
 a33   






know aij, don't know 's
0



Expanding the determinant gives Nth 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.
N equations for u's gives only N - 1 conditions.
Use normalization.
u1* u1  u2* u2    u*N uN  1
Copyright – Michael D. Fayer, 2007
Example - Degenerate Two State Problem
Basis - time independent kets
H   E0    
H   E0    


orthonormal.
 and  not eigenkets.
Coupling .
These equations define H.
The matrix elements are




H   E0
H  
H  
H   E0
And the Hamiltonian matrix is

  E0
H

 

 

E0 
Copyright – Michael D. Fayer, 2007
The corresponding system of equations is
( E0   )      0
   ( E0   )   0
These only have a solution if the determinant of the coefficients vanish.
E0  


E0  
0
Expanding
 E0   
2
 2  0
Dimer Splitting
2
E0
Excited State
 2  2 E0  E02   2  0
Energy Eigenvalues
  E0  
  E0  
Ground State
E = 0
Copyright – Michael D. Fayer, 2007
To obtain Eigenvectors
Use system of equations for each eigenvalue.
  a   b 
  a   b 
a , b 
Eigenvectors associated with + and -.
and  a , b  are the vector representatives of  and 
in the  ,  basis set.
We want to find these.
Copyright – Michael D. Fayer, 2007
First, for the
  E0  
eigenvalue
write system of equations.
( H 11   )a  H 12b  0
H 21a  ( H 22   )b  0
H 11  H ; H 12  H ; H 21  H  ; H 22  H 
Matrix elements of H
( E0  E0   )a   b  0
The matrix elements are
 a  ( E0  E0   )b  0
The result is




H   E0
H  
H  
H   E0
 a   b  0
 a   b  0
Copyright – Michael D. Fayer, 2007
An equivalent way to get the equations is to use a matrix form.
 E 0  
 


 a   0 
 


E0     b   0 
Substitute   E0  
 E0  E0  



 



 a   0 
 


E0  E0     b   0 
   a   0 
b   

      0 
Multiplying the matrix by the column vector representative gives equations.
 a   b  0
 a   b  0
Copyright – Michael D. Fayer, 2007
 a   b  0
 a   b  0
The two equations are identical.
a  b
Always get N – 1 conditions for the N unknown components.
Normalization condition gives necessary additional equation.
a2  b2  1
Then
a  b 
1
2
and
 
1
1
 

2
2
Eigenvector in terms of the 
basis set.

Copyright – Michael D. Fayer, 2007
For the eigenvalue
  E0  
using the matrix form to write out the equations
 E0   
 


 a   0 
 


E0      b   0 
Substituting   E0  



   a   0 
b   

     0
 a   b  0
 a   b  0
These equations give a  b
1
Using normalization a 
2
Therefore
 
1
1
 

2
2
b  
1
2
Copyright – Michael D. Fayer, 2007
Can diagonalize by transformation
1
H  U H U
diagonal
not diagonal
Transformation matrix consists of representatives of eigenvectors
in original basis.
 a
U 
 b
U
1
a   1/ 2

b   1/ 2
 1/ 2

 1/ 2

1/ 2 

1/ 2 
1/ 2 

1/ 2 
complex conjugate transpose
Copyright – Michael D. Fayer, 2007
Then
 1/ 2

H 
 1/ 2

1/ 2   E0

1/ 2   
   1/ 2

E0   1/ 2
1/ 2 

1/ 2 
Factoring out 1/ 2
1  1 1   E0

H  

2  1 1   
 1
1
E0   1 1 
after matrix multiplication
1  1 1   E0  
H  

2  1 1  E0  
E0   
 E0   
more matrix multiplication
 E0  

H 
 0
0 
E0   
diagonal with eigenvalues on diagonal
Copyright – Michael D. Fayer, 2007