Physical Chemistry Third Edition

1280 F Some Mathematics Used in Quantum Mechanics

or the formula^10

Hn(y)n!

[∑n/2]

m 0

(−1)m(2y)n−^2 m

m!(n− 2 m)!

(F-41)

The sum contains only even powers ofyifnis even or only odd powers ofyifnis

odd. The symbol [n/2] stands forn/2ifnis even and for (n−1)/2ifnis odd.

There are a number of identities obeyed by Hermite polynomials.^11 One useful

identity is

yHn(y)nHn− 1 (y)+

1

2

Hn+ 1 (y) (F-42)

An important fact is that ifnis even, thenHn(y) is an even function ofy, and ifnis

odd, thenHn(y) is an odd function ofy:

Hn(−y)Hn(y)(neven) (F-43)

Hn(−y)−Hn(y)(nodd) (F-44)

F. 4 The Hydrogen Atom Energy Eigenfunctions

The energy eigenfunctions are written as products of three factors

ψnlmRnl(r)Ylm(θ,φ)Rnl(r)Θlm(θ)Φm(φ) (F-45)

TheΦfunctions are discussed in Chapter 17. TheΘlmfunctions obey Eq. (17.2-22).

With a change of variables,ycos(θ),P(y)Θ(θ), the equation becomes, after some

manipulation

(1−y^2 )

d^2 P

dy^2

− 2 y

dP

dy

−

m^2

1 −y^2

P+KP 0 (F-46)

Equation (F-46) is the same as theassociated Legendre equationifKl(l+1), where

lis an integer that must be at least as large as|m|. The set of solutions is known as the

associated Legendre functions, given for non-negative values ofmby^12

Pml(y)(1−y^2 )m/^2

dmPl(y)

dym

(F-47)

wherePl(y) is the Legendre polynomial

Pl(y)

1

2 ll!

dl

dyl

(y^2 −1)l (F-48)

With suitable normalization,

ΘΘlm

(

(2l+1)(l−m)!

2(l+m)!

) 1 / 2

Plm(cos(θ)) (F-49)

(^10) A. Erdelyiet al., eds.,Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953,
p. 192ff.
(^11) A. Erdelyiet al.,op. cit.(note 3).
(^12) J. C. Davis, Jr.,Advanced Physical Chemistry, Ronald Press, New York, 1965, p. 596ff.