The Chemistry Maths Book, Second Edition

14 Partial differential equations

14.1 Concepts

An equation that contains partial derivatives is a partial differential equation. For

example, iffis a function of the independent variables xand ythen an equation that

contains one or more of∂f 2 ∂x,∂f 2 ∂yand higher partial derivatives, as well asf,x

and y, is a partial differential equation. Examples of such equations that are important

in the physical sciences are

1. 1-dimensional wave equation


2. 1-dimensional diffusion equation


3. 3-dimensional Laplace equation


4. ∇

2

f 1 = 1 g(x, y, z) 3-dimensional Poisson equation

5. time-independent Schrödinger equation


6. time-dependent Schrödinger equation

These are all second-order linear equations. Equation 4 is inhomogeneous, the

others are homogeneous equations. In equations 1 and 2 , the unknown functionfis

a function of the coordinate xand of the time t;f 1 = 1 f(x, 1 t). In 3 , 4 and 5 , the function

depends on the three coordinates of a point in ordinary space, and in 6 it is also a

function of the time.

As for ordinary differential equations, there are several important standard types

of partial differential equation that frequently occur in mathematical models of

physical systems, and whose solutions can be expressed in terms of known functions.

The equations discussed in this chapter are equation 5 above for the particle in a

rectangular box (Section 14.4) and in a circular box (Section 14.5), and for the

hydrogen atom (Section 14.6), and equation 1 as applied to the vibrations of an

elastic string, such as a guitar string (Section 14.7). These examples demonstrate a

number of important principles in the solution of partial differential equations. They

show that different boundary and initial conditions can lead to very different types

of particular solution, that the symmetry properties of the system being modelled

can lead to the phenomenon of ‘degeneracy’, and that, in some cases, the solutions

are expressed in the form of the ‘orthogonal expansions’ discussed in Chapter 15.

−∇+ ,, =

∂

∂





2

2

2 m

Vxyzt i

t

ψψ

ψ

(,)

−∇+ ,, =



2

2

2 m

ψψψVxyz E()

∂

∂

• 
  

∂

∂

• 
  

∂

∂

=∇ =

2

2

2

2

2

2

2

0

f

x

f

y

f

z

f

∂

∂

=

∂

∂

2

2

f 1

x

D

f

t

∂

∂

=

∂

∂

2

22

2

2

f 1

x

f

v t