Physical Chemistry , 1st ed.

Now we distribute the product XYZto all three derivatives in the Hamiltonian

operator:



2 m

^2

(^) 




x

2

2 XYZ (^) 



y

2

2 XYZ (^) 



z

2

    2 XYZE XYZ

Next we take advantage of a property of partial derivatives: they act on the

stated variable only and assume that any other variables are constant. In the

first derivative term, the partial derivative is taken with respect to x, meaning

that yand zare held constant. As we defined it above, only the Xfunction de-

pends on the variable x; the functions Yand Zdo not. Thus, the entire func-

tion Yand the entire function Z—whatever they are—are constants and can

be removed to outside the derivative. The first term then looks like this:

YZ

d

d

x

2

    2 X

The same analysis can be applied to the second and third derivatives, which

deal with yand z, respectively. The Schrödinger equation can therefore be

rewritten as



2 m

^2

(^) YZ
d
d
x
2
2 XXZ (^) d
d
y
2
2 YXY (^) d
d
z
2
2 ZE XYZ
Finally, let us divide both sides of this expression by XYZand bring ^2 /2m
to the other side. Some of the functions will cancel from each term on the left
side, leaving us with
^ X

1

d

d

x

2

2 X (^) Y

1

d

d

y

2

2 Y (^) Z

1

d

d

z

2

    2 Z

2



m

2

E

Each term on the left side depends on a single variable: either x,or y,or z.Every

term on the right side is a constant: 2,m,E, and . In such a case, every term

on the left side must also be a constant—this being the only way that the three

terms, each dependent on a different variable, could sum up to a constant

value. Let us define the first term as (2mEx)/^2 :

X

1

d

d

x

2

2 X 

2 m

^2

Ex

where Exis the energy of the particle that derives from the Xpart of the over-

all wavefunction. Similarly, for the second and third terms:

Y

1

d

d

y

2

    2 Y 

2 m

^2

Ey

Z

(^1)
d
d
z
2
2 Z 
2 m
^2
Ez
where Eyand Ezare the energies derived from the Yand Zparts of the overall
wavefunction. These three expressions can be rewritten as

2



m

2

d

d

x

2

    2 XExX



2



m

2

d

d

y

2

    2 YEyY (10.18)



2



m

2

d

d

z

2

    2 ZEzZ

300 CHAPTER 10 Introduction to Quantum Mechanics