Physical Chemistry Third Edition

732 17 The Electronic States of Atoms. I. The Hydrogen Atom

This function is an eigenfunction of̂Lzwith eigenvaluehm ̄ ifBis chosen to equal

zero, or an eigenfunction of̂Lzwith eigenvalue−hm ̄ ifAis chosen to equal zero. If

we setBequal to zero theΦfunction is

ΦΦmAeimφ

1

√

2 π

eimφ (17.2-16)

We will refer to theΦmfunctions as thecomplexΦfunctions. The constant 1/(

√

2 π)

is chosen for normalization. The eigenvalue equation for̂Lzis

̂LzΦhm ̄ Φ (17.2-17)

Thezcomponent of the angular momentum is quantized, with eigenvalues 0,±h ̄,± 2 h ̄,

and so on.

The second set ofΦfunctions contains two types of functions:

Φmx

1

√

π

cos(mφ)(m0) (17.2-18)

and

Φmy

1

√

π

sin(mφ)(m0) (17.2-19)

where the constants are chosen for normalization. We callΦmxandΦmytherealΦ

functions. The functionΦ 0 is real and belongs to both sets ofΦfunctions. We use the

subscriptmxbecauseΦmxhas its maximum value at thexaxis and the subscriptmy

becauseΦmyhas its maximum value at theyaxis for odd values ofm. The realΦ

functions form0 are not eigenfunctions of̂Lz.

The realΦfunctions and the complexΦfunctions are related by the identities

cos(α)

1

2

(eiα+e−iα) (17.2-20)

and

sin(α)

1

2 i

(eiα+e−iα) (17.2-21)

Either the complexΦfunctions or the realΦfunctions can be factors in the energy

eigenfunctions. The complexΦmfunctions are more useful when we discuss angular

momentum, and the realΦmxandΦmyfunctions are more useful in discussions of

chemical bonding.

Exercise 17.3

a.Show thatΦmx(φ)

1

√

2

(

Φm(φ)+Φ−m(φ)

)

form0.

b.RelateΦmy(φ)toΦm(φ) andΦ−m(φ).

c.Show thatΦmxandΦmyare not eigenfunctions of̂Lzform0.