BioPHYSICAL chemistry

x=rsinθcosφ

y=rsinθsinφ (9.29)

z=rcosθ

To make use of these two coordinate systems in

Schrödinger’s equation it is necessary to write out the

full expression for ∇^2. When using Cartesian coordinates

it can be written as the sum of the second partial

derivatives:

(9.30)

For spherical coordinates the expression for ∇^2 can be

shown to be:

(9.31)

Although the use of Cartesian coordinates is much easier and generally

preferred, in some circumstances spherical coordinates are used. When we

examine Schrödinger’s equation for the hydrogen atom, the potential term

for the electrostatic energy is given by the product of the charges on the

proton and electron divided by the separation distance. In this case, the

separation distance can be written in Cartesian coordinates as the square

root of the sum of the squares of x, y, and zor in spherical coordinates

simply as r. As we will see, the advantage of using spherical coordinates

to set up the problem outweighs the disadvantage of using the more com-

plicated mathematical form of the derivatives.

For the applications discussed in the following chapters, there will be no

time dependence involved. The time dependence arises when we consider

systems that change with time, for example an electron that travels near

the speed of light until it hits a target at a synchrotron (see Chapter 15).

All of the problems considered here are stationary problems that do not

change with time. For these cases, the equation simplifies because we can

write a general form for the wavefunction.

First, Planck’s relationship (eqn 9.10) is used with the definition for

angular frequency;

ω= 2 πν (9.32)

yielding

(9.33)

Eh

h

==ν ()=

π

πν ω

2

2 Z

∇=^2 + + +

2

222

2

2

211 1

sin sin

∂

∂

∂

∂θ

∂

rrrr ∂φ θ

∂∂

∂θ

θ

∂

∂θ

sin

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

∇=^2 + +

2

2

2

2

2

2

∂

∂

∂

∂

∂

xyz∂

186 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

z

z θ z

y

y

x

x

r

φ

Figure 9.6
A point in space
can be described
by Cartesian
coordinates, (x,y,z),
or by spherical
coordinates, (r,θ,φ).