Physical Chemistry Third Edition

1278 F Some Mathematics Used in Quantum Mechanics

The energy eigenfunction is

ψnxnynz(x,y,z)Csin

(nxπx

a

)

sin

(nyπy

b

)

sin

(nzπz

c

)

(F-24)

where we letCCxCyCz. The energy eigenvalue is

Enxnynz

h^2

8 m

(

n^2 x

a^2

+

n^2 y

b^2

+

n^2 z

c^2

)

(F-25)

We attach the three quantum numbersnx,ny, andnzto the symbolsψandE. A particular

energy eigenfunction is specified by giving the values of the three quantum numbers,

which we sometimes denote by writing the three quantum numbers in parentheses:

(nx,ny,nz).

F. 3 The Time-Independent Schrödinger

Equation for the Harmonic Oscillator

(the Hermite Equation)

The Schrödinger equation for the harmonic oscillator is given in Eq. (15.4-3):

d^2 ψ

dx^2

+(b−a^2 x^2 )ψ 0 (F-26)

The first step of Hermite’s solution to this equation is to find anasymptotic solution,

which is a solution that applies for very large magnitudes ofx. In this casebwill be

negligible compared witha^2 x^2 , so that

d^2 ψ

dx^2

−a^2 x^2 ψ≈ 0 (for large magnitudes ofx) (F-27)

The asymptotic solution is

ψ∞≈e±ax

(^2) / 2
(for large magnitudes ofx) (F-28)
We must choose the negative sign in the exponent to keep the solution finite as|x|
becomes large.
To represent the solution for all values ofx, we choose a trial solution of the form
ψ(x)ψ∞(x)S(x)e−ax
(^2) / 2
S(x) (F-29)
whereS(x) is a power series
S(x)c 0 +c 1 x+c 2 x^2 +c 3 x^3 +...

∑∞

n 0

cnxn (F-30)

with constant coefficientsc 1 ,c 2 ,c 3 ,.... We might have tried to represent the solution

by a power series instead of by a power series multiplied by the asymptotic solution,

but this leads to a solution that violates the condition that the wave function remains

finite for all values ofx.^9

(^9) Ira N. Levine,Quantum Chemistry, 5th ed., Prentice-Hall, Englewood Cliffs, NJ, 1991, p. 64ff.