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Quantum Mechanics for
Scientists and Engineers
David Miller
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Background mathematics 9
Matrix notation
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Matrix notation
A matrix is, first of all, a rectangulararray of numbers
AnMNmatrix has
Mrows (here 2)
Rows are horizontalNcolumns (here 3)
Columns are vertical
The array is enclosed in square brackets
2 1 36 5 4
A
This is arectangular
matrix
2 3
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Symbol for a matrix
As a symbol for a matrix
we could just use a capital letter, likeA
Here, we need to distinguish matrices
and other linear operators
from numbers and simple variablesso we put a hat over a symbol
representing a matrix
which distinguishes a matrixsymbol when we write it byhand
A
2 1 36 5 4
A
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Rectangular and square matrices
Because all matrices are, bydefinition, rectangular
when we say a matrix isrectangular
we almost always mean it is nota square matrix
one with equal numbers ofrows and columns
This is asquarematrix
2 2
1.5 0.5
0.5 1.5
iB
i
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Rectangular and square matrices
The numbers or elements in a matrixcan be
real, imaginary, or complex
The elements are indexed in row-
column orderB12 is the element (value -0.5i) in the
first row and second column
We often use the same letter, hereB, forthe matrix and for its elements
or the lower case version, e.g., b12
This is asquarematrix
2 2
1.5 0.5
0.5 1.5
iB
i
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Diagonal elements
The leading diagonal of a matrix
or just the diagonal
is the diagonal from top left tobottom right
Elements on the diagonalhere those with value 1.5
are called diagonal elements
Elements not on the diagonalhere those with value 0.5i and -0.5i
are called off-diagonal elements
This is asquarematrix
2 2
1.5 0.5
0.5 1.5
iB
i
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Vectors
In the matrix algebra version of vectors
which are matrices of size 1 in one oftheir directions
we must specify whether a vector is a
row vectora matrix with one row
or a column vector
a matrix with one column
2 3
5 2
4
7 6
i
i
d i
i
4, 2,5, 7c
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Transpose
An important manipulation for matricesand vectors is
the transpose
denoted by a superscript T
a reflection about a diagonal linefrom top left to bottom right fora matrix
Algebraically
2 1 36 5 4
A
2 6 1 5
3 4
TA
T nmmn
A A
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1.5 0.50.5 1.5
iBi
Transpose
An important manipulation for matricesand vectors is
the transpose
denoted by a superscript T
a reflection about a diagonal linefrom top left to bottom right fora matrix
Algebraically
1.5 0.50.5 1.5
T i
Bi
T nmmn
B B
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Transpose
An important manipulation for matricesand vectors is
the transpose
denoted by a superscript T
a reflection about a diagonal linefrom top left to bottom right fora matrix
or at 45 for a vector
4, 2,5,7c
42
5
7
Tc
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Transpose
An important manipulation for matricesand vectors is
the transpose
denoted by a superscript T
a reflection about a diagonal linefrom top left to bottom right fora matrix
or at 45 for a vector
2 3
5 2
4
7 6
i
i
d i
i
[2 3 5 2 4 7 6 ]Td i i i i
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Hermitian transpose or adjoint
Another common manipulation is the
Hermitian adjoint, Hermitiantranspose, or conjugate transpose
denoted by a superscript
pronounced daggera reflection about a diagonal line
from top left to bottom right for amatrix or at 45 for a vector
and taking the complexconjugate of all the elements
1.5 0.50.3 1.5
iDi
1.5 0.3
0.5 1.5
iD
i
nmmnD D
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Hermitian transpose or adjoint
Another common manipulation is the
Hermitian adjoint, Hermitiantranspose, or conjugate transpose
denoted by a superscript
pronounced daggera reflection about a diagonal line
from top left to bottom right for amatrix or at 45 for a vector
and taking the complexconjugate of all the elements
2 3
5 2
4
7 6
i
i
d i
i
2 3 5 2 4 7 6d i i i i
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Hermitian matrix
A matrix is said to be
Hermitian
if it is equal to its own Hermitianadjoint
i.e.,
or, element by element
B B
nm nmB B
1.5 0.50.5 1.5
iBi
1.5 0.5 0.5 1.5
iB B
i
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Background mathematics 9
Matrix algebra
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Adding and subtracting matrices
If two matrices are the same
sizei.e., the same numbers ofrows and columns
we can add or subtractthem by
adding or subtractingthe individual matrix
elementsone by one
1
2 1 3
iF
i
5 4
6 7 8
iG
i
K F G
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Adding and subtracting matrices
If two matrices are the same
sizei.e., the same numbers ofrows and columns
we can add or subtractthem by
adding or subtractingthe individual matrix
elementsone by one
1
2 1 3
iF
i
5 4
6 7 8
iG
i
1 5 42 6 1 7 3 8
K F G
i ii i
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Adding and subtracting matrices
If two matrices are the same
sizei.e., the same numbers ofrows and columns
we can add or subtractthem by
adding or subtractingthe individual matrix
elementsone by one
1
2 1 3
iF
i
5 4
6 7 8
iG
i
1 5 42 6 1 7 3 8
6 5
4 8 5
K F G
i i
i i
i
i
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Multiplying a vector by a matrix
Suppose we want to multiply a column
vector by a matrixThe number of rows in the vector
must match the number ofcolumns in the matrix
This is generally true for matrix-matrixmultiplication
The number of rows in the matrix onthe right
must match the number ofcolumns in the matrix on the left
1 2 3
4 5 6
matrix
7
8
9
vector
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Multiplying a vector by a matrix
First we put the vector sideways on
top of the matrixthen multiply element by element
and add to get the first element
of the resulting vectorMove down
and repeat for the next row
1 2 3
4 5 6
7
8
9
matrix vector
1 2 3
4 5 6
7 8 91 7
2 8
3 950
50
1 2 3
4 5 67 8 9
4 7
5 86 9
122
50122
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Multiplying a vector by a matrix
1 2 3
4 5 6
7
8
9
matrix vector
7
8
9
1 2 3
4 5 6
50
122
=
m mn n
n
d A c
d A c
First we put the vector sideways on
top of the matrixthen multiply element by element
and add to get the first elementof the resulting vector
Move down
and repeat for the next row
We can also write this multiplication
with a sum over the repeatedindex
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Multiplying a matrix by a matrix
To multiply a matrix by a matrix
repeat this operation for eachcolumn of the matrix on theright
working from left to right
Write down the resultingcolumns in the resulting matrix
also working from left to right
Summation notationsums over the repeated index
7 150 14 1 2 3
8 2122 32 4 5 6
9 3
R B A
mp mn np
nR B A
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Vector vector products
An inner product
of a row and a column vectorcollapses two vectors to a
number
analogous to geometricalvector dot product
An outer product
of a column and a row vector
generates a square matrix
4
32 1 2 3 5
6
4 8 12 4
5 10 15 5 1 2 3
6 12 18 6
n n
n
f c d c df
mp m pF d c
F d c
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Matrix algebra properties
Matrix algebra, like normal algebra
is associative
and has distributive properties
but matrix multiplication is
not in general commutativeas is easily proved byexample
CB A C BA A B C AB AC
1 2 5 6 19 22
3 4 7 8 43 50
5 6 1 2 23 347 8 3 4 31 46
BA AB in general
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Multiplying a matrix by a number
Multiplying a matrix by a number
means we multiply every element ofthe matrix by that number
Also, we can take out a common factor
from every elementmultiplying the matrix by that factor
Such results are easily proved insummation notation
e.g., for matrix vector multiplicationwhereBmn= Amn
1 2 2 42
3 4 6 8
2 4 1 2
26 8 3 4
m mn n mn n
n n
mn n mn n
n n
d A c A c
A c B c
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Multiplying a matrix by a number
Since number multiplication is commutative
we can move simple factors aroundarbitrarily in matrix products
e.g., for a number and a vector c
This result is also easily proved usingsummation notation
ABc A Bc AB c
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Hermitian adjoint of a product
The Hermitian adjoint of a product
is the reversed product of the Hermitian adjoints
We can prove this using summation notation
Suppose so that and so
AB B A
R AB mp mn npn
R A B
*
( ) ( )
( ) ( ) ( ) ( )
pm mp mn np mn npn npm
nm pn pn nmn n pm
AB R R A B A B
A B B A B A
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Inverse of a matrix
For ordinary algebra, the reciprocal or
inverse of a number or variablex is
which has the obvious property
For a matrix, if it has an inverseit has the property
where is called the identity matrix
which is the diagonal matrix with1 for all diagonal elements
and zeros for all other elements
1 1x
x xx
1/x 1xor
1
1 A A I I
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Identity matrix
For example, the 3x3 identity matrix is
The identity matrix in a givenmultiplication has to be the right size
so we do not typically bother to state
the size of the identity matrixFor any matrix
we can write
Like the number 1 in ordinary algebra
the identity matrix is almost trivial
but is very important
1 0 0
0 1 0
0 0 1
I
AI IA A
A
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