242 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
The values of the constant mlare fixed by the physical constraint that, if the object is fully
rotated by 2π, then the solution must be the same. This gives:
Φ(φ+ 2 π) =Φ(φ) (db12.22)
Aeimlφ=Aeiml(φ+^2 π)=Aeimlφeiml^2 π (db12.23)
For this to be true then:
1 =eiml^2 π=cos(2πml) +isin(2πml) (db12.24)
with ml=0, ±1, ±2, ±3,....
The Θterm is called the Legendre equation and it has a series of solutions of the form
commonly denoted as Pl(cosΘ) for which there are restrictions on the separation constants:
(db12.25)
Here, l=0, 1, 2, 3, 4,... and ml=−l, −l+1, −l+2,..., l−1, l. The combined angular
terms are called spherical harmonics, with some solutions given in Table 12.1.
Some of the solutions can be easily seen. If we substitute:
Θ(θ) =A (db12.26)
where Ais a constant, then the equation is simply:
0 +[l(l+1) sin^2 θ−ml^2 ]A= 0 (db12.27)
sin sin
()
θ ()sin
θ
θ
θ
θ
θ
d
d
d
d
⎡ Θ
⎣
⎢
⎢
⎤
⎦
⎥
⎥
++⎣⎡ll 1 2 −−m^2 l⎦⎤Θθ()= 0
Table 12.1
Some solutions to the angular part of Schrodinger’s equation.
lml Y(ΘΘ,ΦΦ)
00
11
10
20
15
16
312
π
(cos )θ−
−
3
4 π
cosθ
−
3
8 π
sinθeiφ
1
4 π
