rotational motion 2009 7

7/30/2019 Rotational Motion 2009 7

1/85

18_12afig_PChem.jpg

Rotational Motion

Center of Mass

Translational Motion

r1

r2

2

2H

m

2

2H

I

L L

22

( )2 2 eqk

H

r r

Motion of Two Bodies

Each type of motion is best represented

in its own coordinate system best suited

to solving the equations involved

w

k

2 22 2

1 2 1 2

1 2

( , )2 2H Vm m r r

RcInternalcoordinates

Cartesian

Internal motion (w.r.t CM)

Motion of the C.M.

2r

1r

Origin

r

Vibrational Motion


2/85

Centre of Mass

1 1 2 2 1 1 1 1 2 2 2 21 2

, , , , , ,m m

X Y Z x y z x y zm m

r rR r r

Weighted average of all positions

1 1 2 2

1 2

m x m xX

m m

1 1 2 2

1 2

m y m yY

m m

1 1 2 2

1 2

m z m z Z

m m

2 22 2

( ) ( ) ( )2 2H H H VM R rR r r

Motion of Two Bodies

Internal Coordinates:

1 1 2 2& r r R r r R 1 2 r r r

1 2 1 2 r r R r R r r

In C.M. Coordinates:

( , , )x y zr


3/85

Kinetic Energy Terms

2 22 2

( ) ( ) 2 2K K K M R rR r

? ?

2 2 2 2 2

22 2 2

( )2 2

d d dKM M dX dY dZ

RR

2 2 2 2 22

2 2 2 ( )

2 2

d d dK

dx dy dz

rr

? ? ?

? ? ?


4/85

Centre of Mass Coordinates

1 1 1

d dX d dx d

dx dx dX dx dx

1 1 2 2 1 1

1 1 1 2 1 2

m x m x m mdX d

dx dx m m m m M

1 21 1

1dx d

x xdx dx

1

1

md d d

dx M dX dx


5/85


2 2 2

d dX d dx d

dx dx dX dx dx

1 1 2 2 2 2

2 2 1 2 1 2

m x m x m mdX d

dx dx m m m m M

1 22 2

1dx d

x xdx dx

2

2

md d d

dx M dX dx


6/85


2

1 12

1

m md d d d d

dx M dX dx M dX dx

2 2 2

1 1 1

2 2 2

m m md d d d d d

M dX M dX dx dx M dX dx

22 2 2

2 2

2 2 2 2

2

2m md d d d

dx M dX M dXdx dx

Similarly

2 2 2

1 1

2 2 2

2m md d d

M dX M dXdx dx


7/85


2 2 2

1

2 2 2 2

1 1 1

1 2 1md d d d

m dx M dX M dXdx m dx

2 22 2 2 2

1 2 2 2

1 2 1 1 2 2

1 1

2 2x xx

d dK m m m dx m dx

2 2 2

2

2 2 2 2

2 2 2

1 2 1md d d d

m dx M dX M dXdx m dx

2 2 2 2 2 2

1 2

2 2 2 2 2 2 2 2

1 1 2 2 2 1

1 1 1 1m md d d d d d

m dx m dx M dX M dX m dx m dx


8/85


2 2 2 2

1 2

2 2 2 2 2 2

1 1 2 2 1 2

1 1 1 1m md d d d

m dx m dx M M dX m m dx

2 2

2 2

1 1d d

M dX dx 1 2

1 2 1 2

1 1 1 m m

m m m m

Reduced mass2 2 2

2 2

1 1

2x

d dK

M dX dx

2

2 2internal1 1

2CMK K K

M R r

2 2 22

2 2 2

d d d

dX dY dZ

R

2 2 22

2 2 2

d d d

dx dy dz

r


9/85

Hamiltonian2 2

2 2

( ) ( ) ( )2 2H H H VM R rR r r

( , ) ( ) ( ) ( ) ( ) ( ) ( )H H H E E R rR r R r R r R r

( ) ( ) ( )H E RR R R

22 ( ) ( ) ( )

2

V E

r rr r r

22 ( ) ( )

2E

M R RR R

( ) ( ) ( )H E rr r r

C.M. Motion

3-D P.I.B

Internal Motion

Rotation

Vibration

Separable!


10/85

Rotational Motion and Angular Momentum2

2 ( ) ( )

2

K

rr r We rotational motion to internal coordinates

i i

i

p m v

Linear momentum of a rotating Body

i ii

p m rw

i i

dv r

dt

Ds

is r D Didsvdt

i iv rw

d

dt

w

f

Angular Velocity

Parallel to moving body

p(t1)

p(t2)

Always changing direction with time???

Always perpendicular to r


11/85

Angular Momentum

L r p

v

m

r

p

L

L r p

f

sinf r p r p

Perpendicular to R and p

Orientation remains constant with time


12/85

Rotational Motion and Angular Momentum

i ii

L r p

IwL

2

i i

i

m w r

2i i

i

m w

r

2

i i

i

I m rMoment of inertia

As p is always perpendicular to r

2 r r r r r

r

Center

of mass

R

1r

2r


13/85


1 2 1 2 r r r r r

2 2

1 1 2 2I m m r r

r

Center

of mass

R

1r

2r

2 2

1 1 2 2m m r R r R

2 21 2m m

M M r r r

1 1 2 21 1 1 1 2 1 1 1 2 2

1m mMm m m m

M M M

r rr R r r r r r

2

21 2

1 2

1 1 m mm m M

r

2 221

r r

1 12 2 1m m

M M r R r r r

2 22 22 1

1 2

m mI m m

M M

r r


14/85


Classical Kinetic Energy

2

. .2

i

i i

pK E

m

22

2 2

i ii i

i i

m rm v w

222

2 2 2

LII

I I

ww

2 22 2

2 2i i

i

m rw w

r

r

Centerof mass

R

1

r

2r


15/85


2

. . 2

L

K E I

21 2

K LI

22

2K

r

2

( ) ( )2

LK

I r r

22 ( ) ( )

2K

rr r

L i r r

2 r rr r Sincer and p areperpendicular

r

Center

of mass

R

1r

2r

2 2

2 2 2 2 IL

r r r rr r r r


16/85

Momentum Summary

21

2K r p

I

22

2K

L i r

Linear

Classical QM

Rotational

(Angular)

Momentum

Energy

Momentum

Energy 2

2

2K r

I

p i

L r p

drp mv m

dt

2 2

22 2

p m d r

K m dt


17/85

Angular Momentum

L r p

x y zx y z p p p L

x y z

x y z

p p p

i j k

L


18/85

Angular Momentum

x y zL L L L i j k

x z y

y x z

z y x

L yp zp

L zp xp

L xp yp


19/85

Angular Momentum in QM

( ) ( ) ( ) ( )

( ) ( )

x x z y

y y x z

z z y x

L L yp zp

L L zp xp

L L xp yp

r r

r r

r r

x

y

z

d dL i y z

dz dy

d dL i z xdx dz

d dL i x y

dy dx

x y zL L L

L i j k


20/85

Angular Momentum

i L r p r

d d d

x y z i dx dy dz

L

i x y z

d d d

dx dy dz

i j k

L


21/85

Angular Momentum

x y zL L L L i j k

x

y

z

d dL i y zdz dy

d dL i z x

dx dz

d dL i x y

dy dx


22/85

Two-Dimensional Rotational Motion

cos( )x r f

x

y

fr

sin( )y r f

d d

dx dy

i j

How to we get:

( , ) ( , )d d

r rdx dy

f f

i j

Polar Coordinates

cos( ) sin( )d dx d dy d d d dr dr dx dr dy dx dy

f f

sin( ) cos( )d dx d dy d d d

r r

d d dx d dy dx dy

f f

f f f

2 22

2 2

d d

dx dy

2 22

2 2( , ) ( , )

d dr r

dx dyf f


23/85


cos( ) sin( )d dx d dy d d d

dr dr dx dr dy dx dyf f

d d dr x y

dr dx dy

2 2

1 d x d y d

r dr r dx r dy

2 2

1 d d x d y d d d

r x yr dr dr r dx r dy dx dy

2 2 2 2

x d d x d d y d d y d dx y x y

r dx dx r dx dy r dy dx r dy dy

2 2 2 2

2 2 2 2 2 2 2 2

x d x d xy d yx d y d y d

r dx r dx r dxdy r dydx r dy r dy

2 2 2 2

xx d d xy d d yx d d y dy d d dy

r dx dx r dx dy r dy dx r dy dy dy dy

product rule


24/85


sin( ) cos( )d dx d dy d d d d d

r r y xd d dx d dy dx dy dx dy

f ff f f

2

2

d d d d d y x y x

d dx dy dx dyf

2 22 2

2 2

d dx d d d d d d d y y yx x xy x

dx dx dy dx dy dx dy dx dy

d d d d d d d d y y y x x y x x

dx dx dx dy dy dx dy dy

2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

1 d y d y d yx d x d xy d x d

r d r dx r dy r dxdy r dx r dydx r dyf

2 22 2

2 2

d d d d d d y y yx x xy x

dx dy dxdy dx dydx dy

product rule


25/85


2 2 2 2

2 2 2 2 2 2 2 2

1 d d x d x d xy d yx d y d y d r

r dr dr r dx r dx r dxdy r dydx r dy r dy

2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

1 d y d x d yx d xy d y d x d

r d r dx r dx r dxdy r dydx r dy r dyf

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

1 1d d d x d y d x d y d r

r dr dr r d r dx r dx r dy r dyf

2 2 2 2 2 2

2 2 2 2 2 2x y d x y dr r dx r r dy

2 22

2 2

d d

dx dy


26/85


2

2

2 21 1d d dr

dr r dr r d f

2 2 2

2 2 2 2

1 1d d d d d r

dr r dr r d dx dyf

2 2 2

22 2

1 1

2 2

d d dH rdr r dr r d f


27/85

Two-Dimensional Rigid Rotor

22

( , ) ( , ) ( , )2H r r E r f f f

Assume ris rigid, ie. it is constant

2 2 22

2 2

1 1

2 2

d d dH r

dr r dr r d f

2 2 22

2 2

1

2 2r

dH

r d f

2

2

zLHI

z rd

L i idf

As the system is rotating about the z-axis


28/85

18_05fig_PChem.jpg


2 2

2( ) 0

2

dE

I d f

f

2

2 2

2( ) 0

d IE

d f

f

2

22 ( ) 0d kd

f

r

2 22

2

2

2

I kk E

I

2 2

2 ( ) ( ) ( )

2

dH E

I d

f f f

f


29/85

18_05fig_PChem.jpg


22

2 ( ) 0

d

kd ff

( ) 0d d

ik ik

d d

f

f f

( ) 0 & ( ) 0d d

ik ik d d

f f f f

( ) ikA e f f ( )ikA e f f


30/85

18_05fig_PChem.jpg


( ) ikA e f f

Periodic

2 2 2 2

2 2 m

k m

E EI I

2( 2 ) ( )

ik ikA e A ef f f f

2 1 2 2ik

e k m k m k

m = quantum number

( ) imm A ef f ( )

im

m A ef f


31/85

18_05fig_PChem.jpg


( ) imm A ef f

2 2 22

2

2 2r

dH

I d f

z r

dL i i

df

2 2 ( ) ( )

2m m

mHI

f f

2 2

2

m

mE

I

( ) ( )z m mL m f f

22 2

2

dL

df

2 2 2 ( ) ( )m mL m f f

z mL m

2 2 2

mL m


32/85


( ) imm A ef f ( ) ( )m mH E f f

2 2

2m

mE

I

( ) ( )z m mL m f f

z mL m

E

mz mLmEm

6

5

4

3

21

2

I

18.0

12.5

8.0

4.5

2.00.5

6

5

4

3

2

6

5

4

3

21

Only 1 quantum number is require to determine the state of the system.


33/85

Normalization

( ) imm A ef f

2 2* *

0 0( ) ( ) 1 & ( ) ( ) 1m m m m

f f f f

** *( ) im imm A e A e

f f f

( ) imm A ef f

*A A

*

( ) ( )m m f f


34/85

Normalization

2 2*

0 0

1 ( ) ( ) ( ) ( )m m m m

d d

f f f f f f

2 2 2

2 2

0 0 0

1 1 [2 ]im im im imA A e e d A A e e d A d A

f f f f f f f 1

2A

1( )

2

im

m ef f

1( )

2

im

m ef f

2 2

*

0 0

( ) ( ) ( ) ( ) 1m m m md

f f f f f

O h li


35/85

18_06fig_PChem.jpg

1( )

2

im

m ef f

Orthogonality

2

*

,

0

( ) ( )m m m md

f f f *2

0

1 1

2 2

im ime e d

f f f

2

0

1

2

im ime e d

f f f

m = m 2

0

1 21 1

2 2d

f

mm

2 2

0 0

1 1

cos( ) sin( )2 2

i m m

e d m m i m m d

f

f f f f

2 2

0 0

1cos( ) sin( )

2 2

im m d m m d

f f f f

1

0 0 02 2

i

S h i l P l C di t


36/85

14_01fig_PChem.jpg

x y z r i j k

sin cos sin sin cosr r r f f r i j k

Spherical Polar Coordinates

d d d

dx dy dz i j k

( , , ) ( , , ) ( , , )d d d

r r rdx dy dz

f f f

i j k

( , , ) ( , , )

( , , )

x r y r

z r

f f

f

r i j

k

?


37/85

14_01fig_PChem.jpg

sin cos sin sin cosx r y r z r f f

Spherical Polar Coordinates

d dx d dy d dz d

dr dr dx dr dy dr dz

cos sin sin sin cosd d d d

dr dx dy dz f f

cos cos sin cos sind d d d

r r rd dx dy dz

f f

sin sin cos sind d d

r rd dx dy

f f f

.... & ....d d

d d f

Th G di i S h i l P l C di


38/85

14_01fig_PChem.jpg

The Gradient in Spherical Polar Coordinates

cos sin sin sin cos

cos cos sin cos sin

sin sin cos sin 0

d d

dr dx

d dr r r

d dyr rd d

d dz

f f

f f

f f

f

.SP Cart WGradient in Spherical Polar

coordinates expressed in

Cartesian Coordinates

Th G di t i S h i l P l C di t


39/85

14_01fig_PChem.jpg


cos cos sincos sin sin

sin cos cossin sin

sin

sincos 0

Cart

d d

dx r r dr

d d

dy r r d

ddr ddz

f ff

f ff

f

1

.Cart SP

WGradient in Cartesian

coordinates expressed inSpherical Polar Coordinates



40/85

14_01fig_PChem.jpg


1

.

cos cos sincos sin

sin

sin cos cossin sinsin

sincos

Cart SP

d d dd

dr r d r d dx

d d d d

dy dr r d r d

d dd

dr r d dz

f ff

f

f ff f

W

.

Carti

L r

1SPi

L r W



41/85

14_01fig_PChem.jpg


cos cos sincos sin

sincos( )sin( )sin cos cos

sin( ) sin( ) sin sinsin

cos( )sin

cos

d d d

dr r d r d rd d d

i rdr r d r d

rd d

dr r d

f ff

ff f f

f f f

x y zL L L L i j k

cos sin cos

sin

cos

cos sinsin

x

y

z

d dL i

d d

d d

L i d d

dL i

d

f f

f

f f f

f

i L r

Th L l i i S h i l P l C di t


42/85

14_01fig_PChem.jpg

The Laplacian in Spherical Polar Coordinates

22 2

2 2 2 2 2

1 1 1

sinsin sin

d d d d d

rr dr dr r d d r d f

22

2

1 .....d rr dr

OR OR2

2

2

2....

d d

dr r dr

Radial Term Angular Terms

2 1 1

Cart SP SP

W W

Th Di i l Ri id R t


43/85

Three-Dimensional Rigid Rotor

Assume ris rigid, ie. it is constant. Then all energy is from rotational motion only.

2 2 2 2 22

. 2 2 2

2 2Cart

d d dH dx dy dz

2 2 22

2 2 2 2

1 1 1 sin

2 2 sin sin

SP

d d d d dH r

r dr dr r d d r d

f

2 2

2 2 2

1 1

sin2 sin sin

d d d

H r d d d f

2

2IL

22 2

2 2

1 1 sin

sin sin

d d dL

d d d

f

Th Di i l Ri id R t


44/85

18_05fig_PChem.jpg


2 2

2 2

1 1 ( , ) sin ( , ) ( , )

2 sin sin

d d dH E

I d d d f f f

f

2 2

2 2

1 1sin ( ) ( ) ( ) ( )

2 sin sin

d d dE

I d d d f f

f

22

2sin sin ( ) ( ) ( ) ( ) sin ( ) ( )

d d d

d d d

f f f

f

2

2

2

2

IEE

I

Separable?

Th Di i l Ri id R t


45/85


22

2

1( )

( )

dk

df

f f

2 21 sin sin ( ) sin( )

d dk

d d

22

2( )sin sin ( ) ( ) ( ) sin ( ) ( )

d d d

d d df f f f

22

2

1 1sin sin ( ) sin ( )

( ) ( )

d d d

d d d

f

f f

Two separateindependent

equations

k2= separation

Constant

Th Di i l Ri id R t


46/85

18_05fig_PChem.jpg


22

2 ( ) 0d

kd ff

1( )

2

im

m eff

1( )

2

im

m eff

k m k

22

2

1

( )( )

d

kd ff f

Recall 2D Rigid Rotor

Th Di i l Ri id R t


47/85

18_05fig_PChem.jpg


2 21 sin sin ( ) sin

( )

d dm

d d

2 2sin sin sin ( ) ( )d d

md d

, ( ) cos( )mm

l m l l C P

This equation can be solving using a series expansion, using a Fourier Series:

Where ( 1)l l 2

2 ( 1)

2 2

l

l lE E

I I

Legendre polynomials

2 2sin sin ( ) sin ( ) ( )d d

md d

Th Di i l Ri id R t


48/85


2 ( 1)

2l

l lE

I

2 2

2 2

1 1sin ( , ) ( , )

2 sin sin

d d dE

I d d d

f f

f

1( )

2

im

m eff

, , ( ) ( ) ( ) ( )l m m l l m mH E f f , ( ) (cos( ))

mm

l m l l C P

21 ( 1) 1 (cos( )) (cos( ))22 2

m mm im m im

l l l l

l lH C P e C P e

I

f f

,

1( , ) (cos( ))

2

mm im

l m l l Y C P ef f

Spherical Harmonics

, , ( , ) ( , )l m l l mHY E Y f f

Th S h i l H i


49/85

The Spherical Harmonics

,

1( , ) (cos( ))

2

mm im

l m l l Y C P ef f

, 1,..., 1,m l l l l

1/2

(2 1)( )!for 0

2( )!

m

l

l l mC m

l m

1/2

(2 1)( )!( 1) for 0

2( )!

m m

l

l l mC m

l m

For l=0, m=0

21(cos ) (cos 1)2 ! (cos )

l

l

l l

dP

l d

(cos ) sin (cos )(cos )

m

m ml l

dP Pd

0

2 0

0

1(cos ) (cos 1) 1

2 0! (cos )l

dP

d

0

0 0

0 (cos ) sin 1 1(cos )

d

P d



50/85


,

1( , ) (cos( ))

2

mm im

l m l l Y C P ef f

0

0 (cos( )) 1P

0

0,0

1 1 1( , ) (1)

2 2 2

iY e f

For l=0, m=0

1/2

0 (2* 0 1)(0 0)! 1( 1)2(0)! 2

mlC

0,0

1

, (0, ), (0,2 )2 oY r r f

Everywhere on the surface of the

sphere has value 1

2

what is ro ? r = (ro, , f



51/85


r = (1, , f

Normalization:

*

0,0 0,0

S

Y Y dV

2 sin( ) sin( )o

dV dxdydz

r d d d d f f

*( ) ( , )

( ) ( , )

1 11

2 2

f o f j f f f

i o i i i i i

x r y f x z g x y

x r y f x y g x y

dxdydz

2 2 2 2Where ox y z r

*2

0 0

1 1 sin( )2 2

d d

f

In Spherical Polar Coordinates

2

0 0

1sin( )

4d d

f

2

0 0

1sin

4d d

f

2

0 0

1 1cos 2(2 ) 1

4 4

f

1or

r is fixed at ro.

The wavefunction is an angular

function which has a constant valueover the entire unit circle.

X

Y

Z

1

2



52/85


01,01 3 3

( , ) 1 cos( ) cos( )22 2

iY e f f

1,0

3( ) cos( )

2

(0, ), (0, 2 )

Y

f

r = (1, , f

X

Y

Z

The wavefunction is an angular

function which has a value varying ason the entire unit circle.

3cos( )

2

The spherical Harmonics

are often plotted as a

vector strating from the

origin with orientation

and f and its length is

Y(,f)

Along z-axis

2 1

1 1

1(cos ) (cos 1) cos

2 1! (cos )

dP

d

0

0 0

1 (cos ) sin cos cos(cos )

dPd

For l=1, m=0



53/85

18_05fig_PChem.jpg


1, 1

3( , ) sin( )

2 2

iY e f f

Complex Valued??

1, 1 1, 1

1 3 3( , ) ( , ) sin( ) sin( ) cos( )

2 4 2 2 2

i iY Y e ef f f f f

Along x-axis

1, 1 1, 1

1 3 3( , ) ( , ) sin( ) sin( )sin( )

2 4 2 2 2

i iY Y e ei i

f f f f f

Along y-axis

1

1 1

1 1(cos ) sin (cos ) sin cos sin(cos ) (cos )

d dP P

d d

For l=1, m =1



54/85

18_05fig_PChem.jpg


0 221

(cos( )) 3cos ( ) 12

P

1

2 (cos( )) 3sin( )cos ( )P 2 22 (cos( )) 3sin ( )P

2

2,0

5( , ) (3cos ( ) 1)

4Y f

2, 1

15( , ) sin cos

2 2

iY e f f

2, 1 2, 1 15sin cos cos

2 8

Y Y f

2, 1 2, 115

sin sin cos2 8

Y Y

i f

YZXZ

2 2

2, 2

15( , ) sin

4 2

iY e f f

The Spherical Harmonics Are Orthonormal


55/85

The Spherical Harmonics Are Orthonormal

2

*

, , , , ,

0 0

sin( )l m l m l m l mY Y d d

f 2

0 0

1 1(cos ) (cos ) sin

2 2

m mm im m im

l l l l C P e C P e d d

f f f

2

0 0

1(cos ) (cos ) sin

2

m mim im m m

l l l l e e d C P C P d

f f f

2

,

0

(cos ) (cos ) sin2

m mm ml l

m m l l

C CP P d

2

,

0

When (cos ) (cos ) sin2

m mm ml l

l l l l

C Cm m P P d

Example0

1 (cos( )) cos( )P 0

0 (cos( )) 1P

2 20 10 10 0 0 11 1

0 1

0 0

(cos ) (cos ) sin cos sin 02 2

C CC CP P d d

Y are Eigenfuncions of H L2 L


56/85

Yl,m are Eigenfuncions of H, L , Lz

,

1( , ) (cos( ))

2

mm im

l m l l Y C P ef f

2 2

2 2

1 1 sin

2 sin sin

d d dH

I d d d

f

22 2

2 2

1 1 sin

sin sin

d d dL

d d d

f

z

dL i

df

2

, , ( , ) ( 1) ( , )

2l m l mHY l l Y

I f f

2 2

, , ( , ) ( 1) ( , )l m l mL Y l l Y f f

, , ( , ) ( , )z l m l mL Y m Y f f

2

( 1)2

lE l lI

2 2 ( 1)lL l l

( 1)lL l l

zL m

Dirac Notation


57/85

Dirac Notation

*

,i j i j

Sds

*

,i j i j v v

,

N

i i m m

mc f

mfis complete*

,m m m m

Sf f

* * * *

,1 ,2 , 1 ,i i i i N i N c c c c v

,1

,2

, 1

,

j

j

j

j N

j N

c

c

cc

v

Continuous Functions

Vectors

*

,|i j i j i jS

ds i i vDirac

j j v

* i j i j i j

S

ds O O O O

Bra

Ket

|j j j

H E

Dirac Notation


58/85

Dirac Notation

2

*

1,1 1,1 1,1 1,1

0 0

( , ) ( , )sin | |H Y HY d d Y H Y

f f f 1,1 1,1

|Y HY

2 2 2

1,1 1,1 1,1 1,1 1,1 1(1 1) |

2HY Y Y Y Y

I I I

2 2

1,1 1,1|Y YI I

2

, , ( , ) ( 1) ( , )

2

l m l mHY l l Y

I

f f

2

1,1 1,1 1,0 1,0 1, 1 1, 1 | | | | | |Y H Y Y H Y Y H Y

I

Degenerate

Dirac Notation


61/85

18_16fig_PChem.jpg

3 D Rotational motion & The Angular Momentum Vector

zL m

( 1)l l L

m indicates the orientation ofthe angular momentum with

respect to z-axis

L

l determines the length

of the angularmomentum vector

Rotational motion is quantized not continuous. Only certain

states of motion are allowed that are determined by quantum

numbers l and m.

Three Dimensional Rigid Rotor States


62/85

Three-Dimensional Rigid Rotor States

lz mLlE,..,

l

m mY

33,2,1,0, 1, 2, 3Y

2

2,1,0, 1, 2Y

1

1,0, 1Y

2

I

6.0

3.0

1.0

0.5

0

3

2

10

Only 2 quantum numbers are require to determine the state of the system.

2

( 1)

2lE l l

I

( 1)lL l l zL m

12

6

2

Lm

0

1 0 -10

0Y

1 0-1 -2

2

1 0-1 -22

-3

3

0

2

0

2

32

0

22

Rotational Spectroscopy


63/85

19_01tbl_PChem.jpg




64/85

19_13fig_PChem.jpg


2

2( 1)

2J

o

E J J

r

1J JE E ED

2

1E JI

D

J : Rotational quantum number

2

2( 1)( 2) ( 1)

2o

J J J Jr

2

( 1)2J

E J JI



65/85


hcE h hc

D

2

( 1)

4

h J

Ic

2 ( 1)B J

2 28 o

hB

r c

Wavenumber (cm-1)

Rotational Constant

1J Jv c c D D

2 ( 1 1) 2 ( 1) 2c B J B J cB

Frequency (v)

Dvv

Line spacing



66/85


8 1 122 2 (2.99 10 / ) (1890 ) 1.13 10v cB m s m Hz D

Predict the linespacing for the 16O1H radical.

mO = 15.994 amu = 2.656 x 10-26 kg

mH = 1.008 amu = 1.673 x 10-26 kg

r = 0.97 A = 9.7 x 10-11 m

1 amu = 1 g/mol = (0.001 kg/mol)/6.022 x 10-23 mol-1

= 1.661 x 10-23 kg

27

26 27

1 1 11.774 10

2.656 10 1.673 10kg

kg kg

2 28o

hB

r c

34 21

22 27 11 8

6.626 10 /1890

8 1.774 10 9.7 10 (2.99 10 / )

kgm sm

kg m m s

1 1 12 2 (1890 ) 3780 37.8v B m m cm D



67/85

Rotational SpectroscopyThe line spacing for1H35Cl is 21.19 cm-1,

determine its bond length .

mCl = 34.698 amu = 5.807 x 10-26 kg

mH = 1.008 amu = 1.673 x 10-26 kg

27

26 27

1 1 11.626 10

5.807 10 1.673 10

kg

kg kg

2 2 28 8o

o

h hB rr c B c

34 210

1 2 27 8

6.626 10 /1.257 10 1.257

(1059.5 )8 1.626 10 (2.99 10 / )

kgm sm A

m kg m s

11(21.19 )(100 / ) 1059.5

2 2

v cm cm mB m

D

The Transverse Components of Angular Momentum


68/85

x

y

z

d dL i y z

dz dy

d dL i z x

dx dz

d dL i x ydy dx

zL m

( 1)l l L

L

2 2 2 2 L x y zL L L

?

?

, ( , )l mY f

zL m

2 ( 1)L l l

x y zi L L L L r i j k

The Transverse Components of Angular Momentum

Ylm are eigenfunctions ofL2 and Lz but not ofLx and Ly

Therefore Lx and Ly do not commute with either L2 or Lz!!!

Commutation of Angular Momentum Components


69/85

g p

,x y x y y xL L L L L L

d d d d d d d d d d d d y z z x y z y x z z z x

dz dy dx dz dz dx dz dz dy dx dy dz

22

2

d d d d d y yz yx z zx

dx dzdx dz dydx dydz

2 d d d d d d d d y z z x z x y zdz dy dx dz dx dz dz dy

dz d d d d d d d d d y z yx zz zx

dz dx dz dx dz dz dy dx dy dz

product rule



70/85

g p

d d d d d d d d d d d d z x y z z y z z x y x z

dx dz dz dy dx dz dx dy dz dz dz dy

22

2

d d d d d zy z xy x xz

dxdz dxdy dz dy dzdy

d d d d d d dz d d d

zy zz xy x zdx dz dx dy dz dz dz dy dz dy

product rule



71/85

g p

22

2

22

2

2

,x y

d d d d d y yz yx z zx

dx dzdx dz dydx dydz L L

d d d d d zy z xy x xz

dxdz dxdy dz dy dzdy

z

d di ih x y i L

dy dx

,y x y x x y zL L L L L L i L

2 d d d d y x i ih y xdx dy dx dy



72/85


2 x

d d d d d d d d

z x y z y z z x i Ldx dz dz dy dz dy dx dz

,x z x z z xL L L L L L

2 y

d d d d d d d d y z x y x y y z i L

dz dy dy dx dy dx dz dy

,z x z x x z yL L L L L L i L

,y z y z z yL L L L L L

,z y z y y z xL L L L L L i L

Cyclic Commutation of Angular Momentum


73/85

Cyclic Commutation of Angular Momentum

,z x yL L i L

,y z xL L i L

,x y zL L i L

,z y x

L L i L ,

x z yL L i L ,y x zL L i L

,x y zL L i L

,y x zL L i L

xL

yL

zL

i

xL

yL

zL

i

Commutation with Total Angular Momentum


74/85


2 2 2 2 , , , ,z x z y z z zL L L L L L L L

2 2 2 ,x z x z z xL L L L L L

, , ,x z x z z x x z x z z x z x x z x zL L L L L L L L L L L L L L L L L L

, ,x x z z x x z x z xL L L L L L L L L L

x x z z x xL L L L L L

, ,x x z x z x x z x x z xL L L L L L L L L L L L

x y y xi L L i L L



75/85

g

2 2 2 ,y z y z z yL L L L L L

, ,y y z y z y y z y y z yL L L L L L L L L L L L

, ,y y z z y y z y z yL L L L L L L L L L

, , ,y z y z z y y z x z z y z y z y x zL L L L L L L L L L L L L L L L L L

y x x yi L L i L L

y y z z y yL L L L L L



76/85

g

2 2 2 2 , , , ,z x z y z z zL L L L L L L L

2 2 2 ,z z z z z zL L L L L L

0z z z z z zL L L L L L

0 0x y y x y x x yi L L i L L i L L i L L

2 , 0xL L 2 , 0yL L

This means that only

any one component of

angular momentum

can be determined at

one time.

Ladder Operators


77/85

p

x yL L iL

x yL L iL

2 y xi L i L y xi L L

, , ,z z x z y

L L L L i L L

x yL iL L

, , ,z z x z yL L L L i L L

x yL iL L

y xi L L

( )y xi L i i L

Ladder Operators


78/85

p

What do these ladder operators actually do???

, , ,

l m x l m y l mL Y L Y iL Y

? ?Recall That: ,zL L L

z zL L L L L

, ,

l m z z l mL Y L L L L Y

, , , ,

z l m z l m z l m l mL L Y L L Y L L Y m L Y

, , ,

l m z l m l mL Y L L Y m L Y

, ,

( 1)z l m l mL L Y m L Y

, 1 , 1 , , 1 ( 1)

z l m l m l m l mL Y m Y L Y Y Raising Operator

Lowering Operator, , 1

l m l mL Y Y Similarly

Ladder Operators


79/85

Therefore is an eigenfunction of with eigen values land m+1

p

2 2 2 , , , 0 0 0x yL L L L i L L i 2 2 2 , , , 0x yL L L L i L L

2 2 2

, , 0 , l m l mL L Y L L L L Y

2 2

, ,

l m l mL L Y L L Y

2 2

, , ( 1)l m l mL L Y l l L Y

2 2

, ,

( 1)l m l mL L Y l l L Y

,

l mL Y2 & zL L

Which implies that, , , 1

l m l m l mL Y C Y

Ladder Operators


80/85

p

, , , 1

l m l m l mL Y C Y

, , , 1 l m l m l mL Y C Y

This is not an eigen relationship!!!! ,l mC

is not an normalization constant!!!These relationships indicates that a change in state, by Dm=+/-1, is caused by L+and L-

Can these operators be applied indefinitely??

Remember that there is a max and min value for m, as

it represents a component ofL, and therefore must be

smaller than L. ie.

( 1) ( 1) ( 1)m l l l l m l l

2,0 2,1L Y Y

Why is lL

2,1 2,2L Y Y

2,2 0L Y Not allowed

?

, ?

n

l mL Y ,

?n

l mL Y

2,0 2, 1L Y Y

2, 1 2, 2L Y Y

2, 2 0L Y

More Useful Properties of Ladder Operators


81/85

p p

2 2 2 2 x y zL L L L

2 2 2 2 x y zL L L L

2 2 2 2, , x y l m z l mL L Y L L Y

2 2

, ,

l m z l mL Y L Y

2

,( 1) l ml l m Y

This is an eigen equation of a physical observable that is always greater than zero,

as it represents the difference between the magnitude ofL and the square of its

smaller z-component, which are both positive.

2( 1) 0 ( 1) ( 1)l l m l l m l l

This means that m is constrained by l, and since m can be changed by1

, 1, 2,...., 2, 1, .m l l l l l l



82/85

max, 0l mL Y

p p

min, 0l mL Y

max, 0l mL L Y min,

0l mL L Y

Lets show that mmin and mmax are land -lresp.

&L L L L Have to be determined in terms of2

& zL L

2 2 2 2 zL L L L L L

2 2 x y x y x x y y x yL L L iL L iL L iL L iL L L

2 2 x y x y x x y y x yL L L iL L iL L iL L iL L L

2 21 2

x yL L L L L L



83/85

, , ,x yL L L L i L L

, , , ( ) ,x x y x x y y yL L i L L i L L i i L L

0 ( ) ( ) 0 2z z zi i L i i L L

Also note that:

2 2 , 2 2 zL L L L L L L L

2 2 z zL L L L L

2 2 z zL L L L L

2 2 2 2 zL L L L L L

2 2 2 2 2 2z zL L L L L

Similarly

Ladder Operators


84/85

2 2

,max ,max 0 l z z l L L Y L L L Y

max max max

2 2

, , ,

l m z l m z l mL Y L Y L Y

max max max2 2 2

, max , max ,1 l m l m l ml l Y m Y m Y

2 max max ,1 ( 1) l ml l m m Y

max max1 ( 1) 0l l m m

maxm l

Ladder Operators


85/85

min min

2 2

, , 0 l m z z l mL L Y L L L Y

min min min

2 2

, , ,

l m z l m z l mL Y L Y L Y

min min min

2 2 2

, min , min ,1 l m l m l ml l Y m Y m Y

min

2

min min ,1 ( 1) l ml l m m Y

min min1 ( 1) 0l l m m

2min min min1 0 1,&m m l l m l l

Since the minimum value cannot be larger than the maximum value, therefore .

rotational motion 2009 7

Documents