Physical Chemistry Third Edition

F Some Mathematics Used in Quantum Mechanics 1277

We assume the trial solution:

ψ(x,y,z)X(x)Y(y)Z(z) (F-11)

Substitution of the trial function into the Schrödinger Eq. (F-10) and division byXYZ

completes the separation of variables:

1

X

d^2 X

dx^2

+

1

Y

d^2 Y

dy^2

+

1

Z

d^2 Z

dz^2

−

2 mE

h ̄^2

(F-12)

Sincex,y, andzare independent variables, we can keep two of these variables fixed

while allowing the other to vary. Every term must be a constant function:

1

X

d^2 X

dx^2

−

2 mEx

h ̄^2

(F-13)

1

Y

d^2 Y

dy^2

−

2 mEy

h ̄^2

(F-14)

1

Z

d^2 Z

dz^2

−

2 mEz

h ̄^2

(F-15)

whereEx,Ey, andEzare newly defined constants that obey

EEx+Ey+Ez (F-16)

We multiply Eq. (F-13) by the functionX:

d^2 X

dx^2

−

2 mEx

h ̄^2

X (F-17)

This equation is identical with Eq. (15.3-4) except for the symbols used, and has the

same boundary conditions, so that we can transcribe the solution of the one-dimensional

problem with appropriate replacement of symbols:

Xnx(x)Cxsin

(n

xπx

a

)

(F-18)

Ex

h^2

8 ma^2

n^2 x (F-19)

where we use the symbolnxfor the quantum number and whereCxis a constant.

TheYandZequations are identical except for the symbols used, so we can write

their solutions:

Yny(y)Cysin

(nyπy

b

)

(F-20)

Znz(z)Czsin

(nzπz

c

)

(F-21)

Ey

h^2

8 mb^2

n^2 y (F-22)

Ez

h^2

8 mc^2

n^2 z (F-23)

Herenyandnzare positive integers that are not necessarily equal tonx.