Page 70 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.8 Particle in a three-dimensional box 61

cated and is discussed in Chapter 8. Such systems include atoms or molecules
with more than one electron, and molecules with two or more identical nuclei.

2.8 Particle in a three-dimensional box
A simple example of a three-dimensional system is a particle con®ned to a
rectangular container with sides of lengths a, b, and c. Within the box there is
no force acting on the particle, so that the potential V(r)isgiven by
V(r) ˆ 0, 0 < x < a,0 < y < b,0 < z < c

ˆ1, x , 0, x . a; y , 0, y . b; z , 0, z . c
The wave function ø(r) outside the box vanishes because the potential is
in®nite there. Inside the box, the wave function obeys the Schrodinger equation
È
(2.70) with the potential energy set equal to zero
!
2
2
2
ÿ" 2 @ ø(r) @ ø(r) @ ø(r)
‡ ‡ ˆ Eø(r) (2:75)
2m @x 2 @ y 2 @z 2
The standard procedure for solving a partial differential equation of this type is
to assume that the function ø(r) may be written as the product of three
functions, one for each of the three variables
ø(r) ˆ ø(x, y, z) ˆ X(x)Y(y)Z(z) (2:76)
Thus, X(x) is a function only of the variable x, Y(y) only of y, and Z(z) only of
z. Substitution of equation (2.76) into (2.75) and division by the product XYZ
give
2
2
2
2
2
2
ÿ" d X ÿ" d Y ÿ" d Z
‡ ‡ ˆ E (2:77)
2mX dx 2 2mY dy 2 2mZ dz 2
The ®rst term on the left-hand side of equation (2.77) depends only on the
variable x, the second only on y, and the third only on z. No matter what the
values of x,or y,or z, the sum of these three terms is always equal to the same
constant E. The only way that this condition can be met is for each of the three
terms to equal some constant, say E x , E y , and E z , respectively. The partial
differential equation (2.77) can then be separated into three equations, one for
each variable
2
2
2
d X 2m d Y 2m d Z 2m
‡ E x X ˆ 0, ‡ E y Y ˆ 0, ‡ E z Z ˆ 0
dx 2 " 2 dy 2 " 2 dz 2 " 2
(2:78)
where
E x ‡ E y ‡ E z ˆ E (2:79)