5.61 Fall 2007 Lecture # 20 page 1
SPHERICAL HARMONICS
( ) m
Yl m θ , φ = Θ l θ Φ m φ () ()
1
( )
⎡ ⎛ 2 l + 1⎞ l − m ! ⎤
Yl θ , φ = ⎢⎜
m
(
⎥ Pl m cos θ eimφ
) 2
( )
⎟
⎢⎝ 4π ⎠ l + m ! ⎥
⎣ ⎦ ( )
l = 0, 1, 2,... m = 0, ± 1, ± 2, ± 3,... ± l
Yl m ’s are the eigenfunctions to Ĥψ = Eψ for the rigid rotor problem.
1
⎛ 5 ⎞
1
(3cos θ − 1)
2
Y00 = Y20 = ⎜ 2
( 4π )
12
⎝ 16π ⎟⎠
1 1
⎛ 3 ⎞2 ⎛ 15 ⎞ 2
Y10 = ⎜ ⎟ cos θ Y2±1 = ⎜ ⎟ sin θ cos θ e± iφ
⎝ 4π ⎠ ⎝ 8π ⎠
1 1
⎛ 3⎞ 2 ⎛ 15 ⎞ 2
Y1−1 = ⎜ ⎟ sin θ eiφ Y2±2 = ⎜ ⎟ sin 2 θ e±2iφ
⎝ 8π ⎠ ⎝ 32π ⎠
1
⎛ 3⎞ 2
Y11 = ⎜ ⎟ sin θ e− iφ
⎝ 8π ⎠
Yl m ’s are orthonormal: ∫∫ Y (θ ,φ ) Y (θ ,φ ) sin θ dθ dφ = δ δ mm′
m′∗ m
l′ l ll′
⎧1 if l = l′ ⎧1 if m = m′ normalization
Krönecker delta δ ll′ = ⎨ δ mm′ = ⎨
⎩0 if l ≠ l′ ⎩0 if m ≠ m′ orthogonality
Energies: ˆ m = E Y m)
(eigenvalues of HYl lm l
Switch l → J conventional for molecular rotational quantum #
2IE
Recall β = 2 = l l +1 ≡ J J +1
!
( ) (
J = 0, 1, 2,... )
, 5.61 Fall 2007 Lecture # 20 page 2
!2
E
∴ EJ =
2I
J J +1 ( )
6!2
J=3 E3 =
I
(
Y30 , Y3±1 , Y3±2 , Y3±3 7x degenerate )
3!2
J=2 E2 =
I
(
Y20 , Y2±1 , Y2±2 5x degenerate )
!2
J=1 E1 =
I
(
Y10 , Y10 2x degenerate )
J=0 E0 = 0 (
Y00 nondegenerate )
Degeneracy of each state (
g J = 2J + 1 )
from m = 0, ± 1, ± 2,..., ± J
Spacing between states ↑ as J ↑
!2 !2
E J +1 − E J = ( )( ) (
⎡ J +1 J + 2 − J J +1 ⎤ =
2I ⎣ ⎦ I J +1 ) ( )
Transitions between rotational states can be observed through
spectroscopy, i.e. through absorption or emission of a photon
δ+ δ+
hν Absorption
EJ EJ+1
δ- δ-
δ+ δ+
or Emission hν
EJ EJ-1
δ- δ-
SPHERICAL HARMONICS
( ) m
Yl m θ , φ = Θ l θ Φ m φ () ()
1
( )
⎡ ⎛ 2 l + 1⎞ l − m ! ⎤
Yl θ , φ = ⎢⎜
m
(
⎥ Pl m cos θ eimφ
) 2
( )
⎟
⎢⎝ 4π ⎠ l + m ! ⎥
⎣ ⎦ ( )
l = 0, 1, 2,... m = 0, ± 1, ± 2, ± 3,... ± l
Yl m ’s are the eigenfunctions to Ĥψ = Eψ for the rigid rotor problem.
1
⎛ 5 ⎞
1
(3cos θ − 1)
2
Y00 = Y20 = ⎜ 2
( 4π )
12
⎝ 16π ⎟⎠
1 1
⎛ 3 ⎞2 ⎛ 15 ⎞ 2
Y10 = ⎜ ⎟ cos θ Y2±1 = ⎜ ⎟ sin θ cos θ e± iφ
⎝ 4π ⎠ ⎝ 8π ⎠
1 1
⎛ 3⎞ 2 ⎛ 15 ⎞ 2
Y1−1 = ⎜ ⎟ sin θ eiφ Y2±2 = ⎜ ⎟ sin 2 θ e±2iφ
⎝ 8π ⎠ ⎝ 32π ⎠
1
⎛ 3⎞ 2
Y11 = ⎜ ⎟ sin θ e− iφ
⎝ 8π ⎠
Yl m ’s are orthonormal: ∫∫ Y (θ ,φ ) Y (θ ,φ ) sin θ dθ dφ = δ δ mm′
m′∗ m
l′ l ll′
⎧1 if l = l′ ⎧1 if m = m′ normalization
Krönecker delta δ ll′ = ⎨ δ mm′ = ⎨
⎩0 if l ≠ l′ ⎩0 if m ≠ m′ orthogonality
Energies: ˆ m = E Y m)
(eigenvalues of HYl lm l
Switch l → J conventional for molecular rotational quantum #
2IE
Recall β = 2 = l l +1 ≡ J J +1
!
( ) (
J = 0, 1, 2,... )
, 5.61 Fall 2007 Lecture # 20 page 2
!2
E
∴ EJ =
2I
J J +1 ( )
6!2
J=3 E3 =
I
(
Y30 , Y3±1 , Y3±2 , Y3±3 7x degenerate )
3!2
J=2 E2 =
I
(
Y20 , Y2±1 , Y2±2 5x degenerate )
!2
J=1 E1 =
I
(
Y10 , Y10 2x degenerate )
J=0 E0 = 0 (
Y00 nondegenerate )
Degeneracy of each state (
g J = 2J + 1 )
from m = 0, ± 1, ± 2,..., ± J
Spacing between states ↑ as J ↑
!2 !2
E J +1 − E J = ( )( ) (
⎡ J +1 J + 2 − J J +1 ⎤ =
2I ⎣ ⎦ I J +1 ) ( )
Transitions between rotational states can be observed through
spectroscopy, i.e. through absorption or emission of a photon
δ+ δ+
hν Absorption
EJ EJ+1
δ- δ-
δ+ δ+
or Emission hν
EJ EJ-1
δ- δ-
