1
Physical Chemistry
Lecture 11
Waves, Matrices, Operators, and
Eigenvalue Equations
Waves
λ
0

wavelength in vacuum
n(
λ
0
) 
refractive index of the medium,
which varies with wavelength (dispersion)
Wave moving through space
Dispersion relation:
Not simple, because v is a function of
݇ ൌ
2ߨ
ൌ
߱
ݒ
Waves in complex notation
Wave equation
Matrices
Matrix:
an array of
numbers or
functions.
Position in the array
is important
Labeling of matrix
elements requires
two indices
Rank
– the number
of rows or columns
9
8
7
6
5
4
3
2
1
A
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A
Special types of matrices
Square matrix
Row rank equal to column rank
Symmetric matrix
S
Offdiagonal elements across
principal diagonal are equal
Diagonal matrix
D
All offdiagonal elements are
zero
Real matrix
R
All elements are real numbers
or real functions
Complex matrix
C
Some elements are complex
numbers or functions
Transpose of a matrix
A matrix formed from another
by exchanging elements across
the principal diagonal
xy
i
iy
x
iy
x
y
x
a
a
a
a
C
B
C
a
A
B
A
a
3
6
44
2
3
25
1
)
,
(
9
8
7
6
5
4
3
2
1
0
0
0
0
0
0
33
22
11
33
22
11
C
R
D
S
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