Physical Chemistry Third Edition

B Some Useful Mathematics 1247

where we use the common symbolrfor the magnitude ofz. If a complex number is

written in the form of Eq. (B-61),

r^2 |z|^2 z∗z(x−iy)(x+iy)x^2 +iyx−ixy+y^2

x^2 +y^2 (B-65)

Complex numbers are sometimes represented by a point in theArgand plane, in which

the real part is plotted on the horizontal axis, and the imaginary part is plotted on the

vertical axis, as shown in Figure B.8. By the theorem of Pythagoras, the magnitude

of a complex number is the length of the directed line segment from the origin to the

point representing the number in the Argand plane. The angleφis the angle between

the positivexaxis and this directed line segment:

φarctan(y/x) (B-66)

By using the identity in Eq. (B-63), we can show that a complex number can be

x

y

r

f

z 5 x 1 iy





Figure B.8 The Argand Plane.
A point in this diagram represents
a complex number in the form
x+iyorreiφ.

represented in terms ofrandφ:

zx+iyreiφ (B-67)

wherer

√

x^2 +y^2. From this equation, we obtain

|z|^2 z∗z

(

re−iφreiφ

)

r^2 (B-68)

B.6 Some Properties of Hermitian Operators

In Chapter 16 we asserted several properties of hermitian operators. We provide proofs

of two of these properties here.

Property 4: The eigenvalues of a hermitian operator are real.

To establish this property, we take the eigenvalue equation for an arbitrary hermitian

operator

̂Afj(q)ajfj(q) (B-69)

We multiply both sides by the complex conjugate of the eigenfunctionfjand integrate

over all values of the coordinates, factoring the constant eigenvalue out of the integral:

∫

fj∗̂Afjdqaj

∫

fj∗fjdq (B-70)

We now apply the definition of a hermitian operator, Eq. (16.3-28) to the left-hand side

of this equation:

∫

fj∗̂Afjdq

∫

(̂A∗fj∗)fjdq (B-71)

From the complex conjugate of the eigenvalue equation, Eq. (B-69), we can replace

̂A∗fj∗byaj∗fj∗, and by using Eq. (B-70) we obtain

∫

fj∗̂Afjdqaj

∫

fj∗fjdq

∫

(̂A∗fj∗)fjdqaj∗

∫

fj∗fjdq (B-72)