Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

250 WAVE MECHANICS OF SIMPLE SYSTEMS

It is also a member of the class ofeigenvalue equations, in which an operatorOˆ

operates on theeigenfunctionφ(x)togiveaneigenvalue ktimes the same function:

Oˆφ(x)=kφ(x)

In this case, the operator isd^2 /dx^2 , and the eigenvalue isk=− 4 π^2 /λ^2.

The eigenfunction may be represented as a vector:

φ(x)=

(

ξ 1 (x)

ξ 2 (x)

)

The effect of the operator is to stretch or contract theeigenvectorby an amount equal

to the eigenvalue, or to change its direction, or both.

Waves in two dimensions (x, y) such as those of a vibrating plate or membrane are

described by

∂^2 φ(x,y)

∂x^2

+

∂^2 φ(x,y)

∂y^2

=−

4 π^2

λ^2

φ(x,y)

and the wave equation in 3-space is

∂^2 φ(x,y,z)

∂x^2

+

∂^2 φ(x,y,z)

∂y^2

+

∂^2 φ(x,y,z)

∂z^2

=−

4 π^2

λ^2

φ(x,y,z)

16.3 THE SCHRODINGER EQUATION ̈

By the de Broglie equationp=h/λ,it follows thatλ^2 =h^2 /p^2. Substituting forλ^2

in a three-dimensional wave equation, we obtain

∂^2 φ(x,y,z)

∂x^2

+

∂^2 φ(x,y,z)

∂y^2

+

∂^2 φ(x,y,z)

∂z^2

=−

4 π^2

λ^2

φ(x,y,z)=−

4 π^2 p^2

h^2

φ(x,y,z)

The momentum of a moving particle is its mass times its velocity,p=mv. This leads

top^2 =m^2 v^2 = 2 m(^12 mv^2 )= 2 mT, whereTis the classical kinetic energy^12 mv^2.

Now, with a slight change in symbol for notational simplicity, we obtain

∂^2 

∂x^2

+

∂^2 

∂y^2

+

∂^2 

∂z^2

=

(

∂^2

∂x^2

+

∂^2

∂y^2

+

∂^2

∂z^2

)

=−

8 π^2 mT

h^2



The bracketed operator

(

∂^2 /∂x^2 +∂^2 /∂y^2 +∂^2 /∂z^2

)

operates on the wave function

(x,y,z). It is given the shorthand notation∇^2. Because its eigenvalue contains a