Physical Chemistry Third Edition

734 17 The Electronic States of Atoms. I. The Hydrogen Atom

Table 17.1 Normalized Spherical Harmonic FunctionsYlm(θ,φ)

Θlm(θ)Φm(φ)

ComplexΦfunctions, eigenfunctions of̂Lz.

Φm(φ)

1

√

2 π

eimφ

RealΦfunctions, not necessarily eigenfunctions of̂Lz.

Φmx(φ)

1

√

π

cos(mφ)

Φmy(φ)

1

√

π

sin(mφ)

Θfunctions:

Θ 00 (θ)

√

2

2

Θ 10 (θ)

√

6

2

cos(θ); Θ 11 (θ)Θ1,− 1 (θ)

√

3

2

sin(θ)

Θ 20 (θ)

√

10

4

(3cos^2 (θ)−1)

Θ 21 (θ)Θ2,− 1 (θ)

√

15

2

sin(θ)cos(θ)

Θ 22 (θ)Θ2,− 2 (θ)

√

15

4

sin^2 (θ)

Θ 30 (θ)

3

√

14

4

(

5

3

cos^3 (θ)−cos(θ)

)

Θ 31 (θ)Θ3,− 1 (θ)

√

42

8

sin(θ)(5cos^2 (θ)−1)

Θ 32 (θ)Θ3,− 2 (θ)

√

105

4

sin^2 (θ)cos(θ)

Θ 33 (θ)Θ3,− 3 (θ)

√

70

8

sin^3 (θ)

AdditionalΘfunctions can be obtained from formulas in Appendix F.

The values in Eq. (17.2-29) differ from the assumed valuesh ̄,2h ̄,3h ̄,..., in the Bohr

theory of the hydrogen atom. The Bohr theory also did not provide for states of zero

angular momentum or for different values ofLzcorresponding to a given value of|L|.

The functionΦmis an eigenfunction of̂Lzwith eigenvaluehm ̄ , so thatYlmis an

eigenfunction of̂Lz:

̂LzYlmΘlm̂LzΦmΘlmhm ̄ ΦmhmY ̄ lm (m0,±1,...,±l) (17.2-30)

The possible values ofLzare:

Lzmh ̄0, ±h ̄, ± 2 h ̄, ± 3 h ̄,...±lh ̄ (17.2-31)

For every value oflthere are 2l+1 values ofm(lpositive values,lnegative values,

plus zero). The degeneracy of a value of|L|equal toh ̄

√

l(l+1) is

gl 2 l+ 1 (17.2-32)