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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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 ).
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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Hermitian Operators and Matrices
x A y y A x
Hermitian operator Hermitian Matrix
+ - complex conjugate transpose - Hermitian conjugate
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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.
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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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
Copyright Michael D. Fayer, 2009
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.
Copyright Michael D. Fayer, 2009
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.
Copyright Michael D. Fayer, 2009
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
Copyright Michael D. Fayer, 2009
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
Copyright Michael D. Fayer, 2009
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
Copyright Michael D. Fayer, 2009