Physical Chemistry , 1st ed.

In comparing these three equations with the original Schrödinger equation for
this system, it is not too difficult to see that

EExEyEz (10.19)
We have seen expressions of the form given in equations 10.18:they have
the same form as the Schrödinger equation for the one-dimensional particle-in-a-
box.Rather than having to re-derive solutions for the three-dimensional case,
we can simply use the same functions, but with the appropriate labels for a
three-dimensional system. Therefore, the solution for the xdimension is

X(x)       2

a

sin^ nx

a

x

where nxis 1, 2, 3, 4... and is a quantum number (note the xsubscript on the
quantum number). The quantized energy,Ex, can also be taken from the one-
dimensional box system:

Ex

8

n

m

x^2 h

a

2

(^2)
This analysis can be repeated for the other two dimensions. It should be
readily apparent that the answers are similar:
Y(y) 

2

b

sin^

ny

b

y

Z(z)       2

c

sin^ nz

c

z

Yand Zdepend only on the coordinates yand zrespectively. The constants b
and crepresent the lengths of the box in the yand zdirection, and the quan-
tum numbers nyand nzrefer to the corresponding dimension only. Remember,
however, that the wavefunction is the product ofX,Y, and Z, so that the
complete 3-D wavefunction is

(x,y,z) 

a

8

bc

sin^ nx

a

x sin ny

b

y sin nz

c

z (10.20)

where the constants have all been grouped together. The total energy for a par-
ticle in this three-dimensional box is

E

8

n

m

x^2 h

a

2

2  (^8)

n
m
y^2 h
b
2
2 
n
8 m
z^2 h
c^2
2

8
h
m
2
(^) 
n
a^2
x^2  n
b^2
y^2  n
c^2
z^2
 (10.21)
Although the wavefunctions of the 3-D particle-in-a-box are qualitatively
similar to those for the 1-D particle-in-a-box, there are some differences. First,
every observable has three parts: an xpart, a ypart, and a zpart. (See the
expression for Ein equation 10.21.) For example, the momentum of a particle
in a 3-D box is more properly referred to as a momentum in the xdirection,
denoted px; a momentum in the ydirection,py; and a momentum in the z
direction,pz. Each observable has a corresponding operator, which in the
momentum example is either ˆpx,ˆpy,or ˆpz:
ˆpxi




x

ˆpy i





y

(10.22)

ˆpz i





z

10.11 The Three-Dimensional Particle-in-a-Box 301