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Physical Chemistry - Spherical Harmonics_lecture20
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This course presents an introduction to quantum mechanics. It begins with an examination of the historical development of quantum theory, properties of particles and waves, wave mechanics and applications to simple systems — the particle in a box, the harmonic oscillator, the rigid rotor and the ...

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Document information

  • April 25, 2023
  • 5
  • 2007/2008
  • Class notes
  • Prof. robert guy griffin
  • All classes

Subjects

  • spherical harmonics

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  • MIT Courses
All documents for this subject (18)

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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-

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