100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Previously searched by you
Previously searched by you
logo
Physical Chemistry - Spherical Harmonics_lecture20 $2.60   Add to cart
Quickly navigate to
Preview
  2.  
  3. MIT Courses
  4.  
  5. MIT Courses
  6.  
  7. MIT Courses

Class notes

Physical Chemistry - Spherical Harmonics_lecture20
 1 view  0 purchase
  • Course
  • MIT Courses
  • Institution
  • MIT Courses

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 ...

[Show more]

Preview 2 out of 5  pages

Document information

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

Subjects

  • spherical harmonics

Written for

  • MIT Courses
All documents for this subject (18)

Seller

avatar-seller
tandhiwahyono

Reviews received

Content preview

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-

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller tandhiwahyono. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for $2.60. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

75323 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling

Recently viewed by you

Exam (elaborations) ·

(0)

DAT General Chemistry.

$2.60
  • (0)
  Add to cart