Physical Chemistry II (Chapter 4 1) Rigid Rotor Models and Angular Momentum Eigenstates

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1 Physical Chemisty II (Chapte 4 ) Rigid Roto Models and Angula Momentum Eigenstates Tae Kyu Kim Depatment of Chemisty Rm. 30 (tkkim@pusan.ac.k) SUMMAR CHAPTER 3 A simple QM example: Paticle in D Box d E mdx [Bounday Condition] n x n h sin, En whee n,,3 n a a 8ma Enegy level: Quantiation and eo point enegy Wavefunction & pobability: Aveage value and othogonality Coesponding pinciple: the pedictions of QM become to CM

2 SUMMAR CHAPTER 3 Paticle in D & 3D Box: Sepaation of Vaiables 4 nx x ny y ( xy, ) X sin sin ab a b h n n E x y 8m a b x y FEMO Model: Paticle in D Box (Conjugated hydocabon) h h hc n n E E E n n n 8ma 8ma Quantum Mechanical Tunnelling: Pinciples fo STM 3 IN THE CHAPTER 3 A molecule has tanslational, vibational, and otational degees of feedom. Each of these can be sepaately descibed by its own enegy spectum and enegy eigenfunctions. As shown in Chapte 3, the paticle in a box is useful model fo exploing the tanslational degee of feedom. In this chapte, quantum mechanics is used to study the otation of a diatomic molecule. We fomulate and solve a quantum mechanical model fo otational motion(chapter 4). This model povides a basis fo undestanding the obital motion of electons aound the nucleus of an atom as well as the otation of a molecule aound its pincipal axes. In the next chapte, we will conside the vibational degee of feedom, modeled by the hamonic oscillato(chapter 5). 4

3 MOTIONS OF A DIATOMIC MOLECULE Sepaation of Vaiable fo Diatomic Molecule cente of mass (C.M) m a a b m b H m a () b V a mb Intoduction of cente of mass and elative coodinate educe the twopaticle poblem into two sepaate one paticle poblems cente of mass and elative coodinate m m a a b b R a b ma mb ma m M ma mb mm a b b H R V() M MOTIONS OF A DIATOMIC MOLECULE Sepaation of Vaiable fo Diatomic Molecule Hˆ Hˆ ( ) Hˆ ( ) Hˆ (, ) total tans cm vib int ot cm cm E E ( ) E ( ) E (, ) total tans cm vib int ot cm cm ( ) ( ) (, ) total tans cm vib int ot cm cm mm m m

4 RIGID ROTOR IN D CLASSICAL MECHANICS A paticle otating aound a fixed axis (angula momentum and kinetic enegy) T p m m v T m m m I Angula Momentum: L I m p T L I I RIGID ROTOR IN D CLASSICAL MECHANICS A Model of Rigid Diatomic Molecule m m R m m R R mm mm T m m m m I m m mm I m m m R m R R R mm mm mm T L L I R

5 RIGID ROTOR IN D QUANTUM MECHANICS H R V() M y xcos y sin ` x y x H oto Rigid oto V() (, ) E(, ) RIGID ROTOR IN D QUANTUM MECHANICS H oto E m e e im im m 0,,,... ( ) ( ) Ae im Be im im Ae ( m 0,,,...)

6 RIGID ROTOR IN D QUANTUM MECHANICS im Ae ( m 0,,,...) E m m I two-fold degeneate Physical Chemisty II (Chapte 4 ) Rigid Roto Models and Angula Momentum Eigenstates Tae Kyu Kim Depatment of Chemisty Rm. 30 (tkkim@pusan.ac.k)

7 SUMMAR CHAPTER 4 Rotation of Diatomic Molecule (Tansfoming to C.M. Coodinate) Cente of Mass M M μ H () a b V m a m H R V() b M 3 SUMMAR CHAPTER 4 QM Model fo D Rigid Roto H V() V() (, ) E(, ) ( ) E( ) m ( ) m ( ) E im Ae ( m 0,,,...) E m m I 4

8 SUMMAR CHAPTER 4 QM Model fo D Rigid Roto (Bounday Conditions) 5 Ψ n and E n fo D Rigid Roto m m E I im e m 0,,,... 6

9 Schödinge Equation x y 3D Rigid Roto sin sin sin sin sin sin sin (, ) E (, ) I sin sin (, ) ( ) ( ) I sin sin E sin 7 I 3D Rigid Roto sin sin E sin I sin sin E sin constant I sin (, ) E (, ) I sin sin im ( ) e ( m0,,...) m I sin sin sin E 8

10 Final Solution 3D Rigid Roto m sin sin E I sin /! m l l m m ( ) Pl cos m0 l m! and l l m! l m! P m l / cos m0 im e These constaints ae imposed by the foms of the polynomials which must be acceptable wavefunction and eigenfunctions of the Schodinge equation l 0,,... m l 9 Spheical Hamonics, l,m (θ,φ) Spheical hamonics: eigenfunctions of the igid oto lm im e S lm ( ) cos 4 *. 0 0 lm l ' m' sin dd ll ' mm' (othonomal) ll ˆ l l. Hlm lm Elm Ej I I lm 3. m lm 0 3 sin e cos i 5 sin cos e 8 5 sin 3 i e i 0

11 Spheical Hamonics, l,m (θ,φ) 3 px sin cos cos 4 3 py sin sin i 4 p 0 3 cos sin e cos i 5 sin cos e 8 5 sin 3 i e i d cos 5 d x sin cos cos 4 5 d y sin cos sin i 4 5 sin cos d x y d xy i 6 5 sin sin 6 Spheical Hamonics, l,m (θ,φ)

12 Spheical Hamonics, l,m (θ,φ) 3 Angula Momentum (L) Angula Momentum lˆ ˆ l l Hlm lm Elm I I Z component of angula momentum (L ) l p l yp p l p xp x y y x l xp yp lˆ lˆ lˆ lˆ y x x y l ˆ lm ll lm l ll ( ) lˆ x y i y x i ˆ m im l A P ( ) e m i i lm lm lm l lm Spheical hamonics ae simultaneous eigenfunctions of both L and L. 4

13 Hˆ lm lm l l I ˆ llm ll lm l 0,,, Angula Momentum l ˆ m ml, l,, l, l lm lm l l ˆ 6 l 6 m m m l ˆ l l ˆ l l ˆ 0 l l ˆ l l ˆ l l? l? x y l l l l, 5,6,5, x y 5 Angula Momentum 4 The possible magnitude of angula momentum is quantied. Spatial quantiation: the vecto can only have cetain oientation in space. 6

14 Raising & Loweing Angula Momentum Opeato Raising and loweing angula momentum opeatos lˆ lˆ [ lˆ ˆ ˆ ˆ, l ] l x ily [ lˆ, lˆ ] lˆ lˆˆ, ˆ ˆ, [ ˆ, ˆ l l m ll l m l l] l, m mlˆ lm, lˆ lm, ( ) lˆ m lm, lˆlˆ l, m ( m ) lˆ l, m Popotionality constants ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y x y lm l l lm lm l il l il lm lm l l l lm ( ) ( ) ll mm lˆ l, m lm, lˆ l, m cn l, m lˆ lm, d l, m Raising and loweing (quantum numbe) n 7 Raising & Loweing Angula Momentum Opeato * ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ lm, lxily l lm, d lx ily lm, llm, d ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x y x y lm l l lm lm l il l il lm lm l l l lm ( ) ( ) ll mm * * l l d c d c lm, lm, lm, lm, lm, lm, * lm, lm, c l( l) m( m) d l( l) m( m) ˆl, ll ( ) mm l m ( ) l, m 8