18_12afig_pchem.jpg rotational motion center of mass translational motion r1r1 r2r2 motion of two...

18_12afig_PChem.jpg

Rotational Motion

Center of Mass

Translational Motion

r1

r2

2

ˆ ˆ ˆ2

Hm

2

ˆ ˆ ˆ2

HI

L L

22ˆ ˆ ˆ ( )

2 2 eq

kH

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

k

2 22 2

1 2 1 21 2

ˆ ˆ ˆ ˆ ( , )2 2

H Vm m

r r

RcInternal coordinates

Cartesian

Internal motion (w.r.t CM)

Motion of the C.M.

2r

1r

Origin

rVibrational Motion

Centre of Mass

1 1 2 21 1 1 1 2 2 2 2

1 2

, , , , , ,m m

X Y Z x y z x y zm m

r r

R 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 zZ

m m

2 22 2ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )

2 2H H H V

M R rR r r

Motion of Two Bodies

Internal Coordinates:

1 1 2 2& r r R r r R1 2 r r r

1 2 1 2 r r R r R r r

In C.M. Coordinates:

( , , )x y zr

Kinetic Energy Terms

2 22 2ˆ ˆ ˆ ˆ ˆ( ) ( )

2 2K K K

M R rR r

? ?

2 2 2 2 22

2 2 2ˆ ˆ( )

2 2

d d dK

M M dX dY dZ

RR

2 2 2 2 22

2 2 2ˆ ˆ( )

2 2

d d dK

dx dy dz

rr

? ? ?

? ? ?

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


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

Centre of Mass Coordinates2

1 12

1

m md d d d d

dx M dX dx M dX dx

2 2 21 1 12 2 2

m m md d d d d d

M dX M dX dx dx M dX dx

22 2 22 2

2 2 2 22

2m md d d d

dx M dX M dXdx dx

Similarly

2 2 21 12 2 2

2m md d d

M dX M dXdx dx


2 2 21

2 2 2 21 1 1

1 2 1md d d d

m dx M dX M dXdx m dx

2 22 2 2 21 2

2 21 2 1 1 2 2

1 1ˆ2 2

x xx

d dK

m m m dx m dx

2 2 22

2 2 2 22 2 2

1 2 1md d d d

m dx M dX M dXdx m dx

2 2 2 2 2 21 2

2 2 2 2 2 2 2 21 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


2 2 2 21 2

2 2 2 2 2 21 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 mass

2 2 2

2 2

1 1ˆ2x

d dK

M dX dx

22 2

internal

1 1ˆ ˆ ˆ2 CMK 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

Hamiltonian2 2

2 2ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )2 2

H H H VM

R rR r r

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

ˆ ( ) ( ) ( )H E RR R R

22ˆ ˆ( ) ( ) ( )

2V E

r rr r r

22ˆ ( ) ( )

2E

M R RR R

ˆ ( ) ( ) ( )H E rr r r

C.M. Motion3-D P.I.B

Internal MotionRotationVibration

Separable!

Rotational Motion and Angular Momentum2

2ˆ ( ) ( )2

K

rr r

We rotational motion to internal coordinates

i ii

p m vLinear momentum of a rotating Body

i ii

p m r

i i

dv r

dt

s

is r i

dsv

dt

i iv r

d

dt

Angular Velocity

Parallel to moving body

p(t1)

p(t2)

Always changing direction with time???

Always perpendicular to r

Angular Momentum

L r p

vm

r

p

L

L r p

sinr p r p

Perpendicular to R and pOrientation remains constant with time

Rotational Motion and Angular Momentum

i ii

L r p

IL

2

i ii

m r

2

i ii

m r

2

i ii

I m r

Moment of inertia

As p is always perpendicular to r

2 r r r r r

r

Center of mass

R1r

2r

Rotational Motion and Angular Momentum

1 2 1 2 r r r r r

2 2

1 1 2 2I m m r r

r

Center of mass

R1r

2r

2 2

1 1 2 2m m r R r R

2 21 2

m 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

221 2

1 2

1 1 m m

m m M

r

2 221

r r

1 12 2 1

m m

M M r R r r r

2 22 22 1

1 2

m mI m m

M M

r r

Rotational Motion and Angular MomentumClassical Kinetic Energy

2

. .2

i

i i

pK E

m

22

2 2i ii i

i i

m rm v

222

2 2 2

LII

I I

2 22 2

2 2i ii

m r

r

r

Center of mass

R1r

2r

Rotational Motion and Angular Momentum2

. .2

LK E

I

21ˆ ˆ2

K LI

22ˆ

2K

r

2ˆˆ ( ) ( )

2

LK

I r r

22ˆ ( ) ( )

2K

rr r

L̂ i r r

2 r rr r Since r and p are perpendicular

r

Center of mass

R1r

2r

2 2

2 2 2 2ˆ IL

r r r r

r rr r

Momentum Summary

21

2K r p

I

22ˆ

2K

L̂ i r

Linear

Classical QM

Rotational (Angular)

Momentum

Energy

Momentum

Energy 2

2ˆ2

K rI

p̂ i

L r p

drp mv m

dt

2 2

22 2

p m d rK

m dt

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

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

Angular Momentum in QM

ˆ ˆ ˆ ˆˆ( ) ( )

ˆ ˆ ˆˆˆ( ) ( )

ˆ ˆˆ ˆ ˆ( ) ( )

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

ˆ

ˆ

ˆ

x

y

z

d dL i y z

dz dy

d dL i z x

dx dz

d dL i x y

dy dx

ˆ ˆ ˆ ˆx y zL L L L i j k

Angular Momentum

ˆ ˆ ˆ ˆi L r p r

d d dx y z i

dx dy dz

L

Angular Momentum

x y zL L L L i j k

ˆ

ˆ

ˆ

x

y

z

d dL i y z

dz dy

d dL i z x

dx dz

d dL i x y

dy dx

Two-Dimensional Rotational Motion

cos( )x r

x

y

r

sin( )y r

d d

dx dy

i j

How to we get:

( , ) ( , )d d

r rdx dy

i j

Polar Coordinates

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

dr dr dx dr dy dx dy

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

r rd d dx d dy dx dy

2 22

2 2

d d

dx dy

2 22

2 2( , ) ( , )

d dr r

dx dy


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

dr dr dx dr dy dx dy

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 dr x y

r 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


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

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

2

2

d d d d dy x y x

d dx dy dx dy

2 22 2

2 2

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

dx dx dy dx dy dx dy dx dy

d d d d d d d dy 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 dy

2 22 2

2 2

d d d d d dy y yx x xy x

dx dy dxdy dx dydx dy

product rule

Two-Dimensional Rotational Motion2 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 dr

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 dy

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 dr

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

2 2 2 2 2 2

2 2 2 2 2 2

x y d x y d

r r dx r r dy

2 22

2 2

d d

dx dy


22

2 2

1 1d d dr

dr r dr r d

2 2 2

2 2 2 2

1 1d d d d dr

dr r dr r d dx dy

2 2 22

2 2

1 1ˆ2 2

d d dH r

dr r dr r d

Two-Dimensional Rigid Rotor

22ˆ ( , ) ( , ) ( , )

2H r r E r

Assume r is rigid, ie. it is constant

2 2 22

2 2

1 1ˆ2 2

d d dH r

dr r dr r d

2 2 22

2 2

1ˆ2 2r

dH

r d

2ˆˆ

2zL

HI

ˆz r

dL i i

d

As the system is rotating about the z-axis

18_05fig_PChem.jpg


2 2

2( ) 0

2

dE

I d

2

2 2

2( ) 0

d IE

d

22

2( ) 0

dk

d

r

2 22

2

2

2

I kk E

I

2 2

2ˆ ( ) ( ) ( )

2

dH E

I d

18_05fig_PChem.jpg


22

2( ) 0

dk

d

( ) 0d d

ik ikd d

( ) 0 & ( ) 0d d

ik ikd d

( ) ikA e ( ) ikA e

18_05fig_PChem.jpg


( ) ikA e Periodic

2 2 2 2

2 2 m

k mE E

I I

2( 2 ) ( )ik ikA e A e

2 1 2 2ike k m k m k

m = quantum number

( ) imm A e ( ) im

m A e

18_05fig_PChem.jpg


( ) imm A e

2 2 22

2ˆ

2 2r

dH

I d

ˆz r

dL i i

d

2 2

ˆ ( ) ( )2m m

mH

I

2 2

2m

mE

I

ˆ ( ) ( )z m mL m

22 2

2ˆ dL

d

2 2 2ˆ ( ) ( )m mL m

zmL m

2 2 2mL m


( ) imm A e

ˆ ( ) ( )m mH E 2 2

2m

mE

I

ˆ ( ) ( )z m mL m

zmL m

E

mzmLmEm

6

5

4

3

2

1

2

I

18.0

12.5

8.0

4.5

2.00.5

6

5

4

3 2

6

5

4

321

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

Normalization

( ) imm A e

2 2

* *

0 0

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

** *( ) im imm A e A e

( ) imm A e

*A A

*( ) ( )m m

Normalization

2 2

*

0 0

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

2 2 22 2

0 0 0

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

1

2A

1( )

2im

m e

1( )

2im

m e

2 2

*

0 0

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

18_06fig_PChem.jpg

1( )

2im

m e

Orthogonality

2

*

,

0

( ) ( )m m m md

*2

0

1 1

2 2im ime e d

2

0

1

2im ime e d

m = m’ 2

0

1 21 1

2 2d

m ≠ m’

2 2

0 0

1 1cos( ) sin( )

2 2i m me d m m i m m d

2 2

0 0

1cos( ) sin( )

2 2

im m d m m d

1

0 0 02 2

i

14_01fig_PChem.jpg

x y z r i j k

sin cos sin sin cosr r r r i j k

Spherical Polar Coordinates

d d d

dx dy dz i j k

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

r r rdx dy dz

i j k

( , , ) ( , , )

( , , )

x r y r

z r

r i j

k

?

14_01fig_PChem.jpg

sin cos sin sin cosx r y r z r

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

cos cos sin cos sind d d d

r r rd dx dy dz

sin sin cos sind d d

r rd dx dy

.... & ....d d

d d

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 dxd d

r r rd dy

r rd d

d dz

.SP Cart WGradient in Spherical Polar coordinates expressed in Cartesian Coordinates

14_01fig_PChem.jpg


cos cos sincos sin

sinsin cos cos

sin sinsin

sincos 0

Cart

d d

dx r r drd d

dy r r ddd

r ddz

1.Cart SP

WGradient in Cartesian coordinates expressed in Spherical Polar Coordinates

14_01fig_PChem.jpg


1.

cos cos sincos sin

sin

sin cos cossin sin

sin

sincos

Cart SP

d d dddr r d r ddx

d d d d

dy dr r d r d

d dd

dr r ddz

W

.ˆ

Carti L r

1ˆSPi L r W

14_01fig_PChem.jpg


cos cos sincos sin

sincos( )sin( )

sin cos cossin( )sin( ) sin sin

sincos( )

sincos

d d d

dr r d r dr

d d di r

dr r d r dr

d d

dr r d

ˆ ˆ ˆ ˆx y zL L L L i j k

cosˆ sin cossin

cosˆ cos sinsin

ˆ

x

y

z

d dL i

d d

d dL i

d d

dL i

d

ˆ i L r

14_01fig_PChem.jpg

The Laplacian in Spherical Polar Coordinates

22 2

2 2 2 2 2

1 1 1sin

sin sin

d d d d dr

r dr dr r d d r d

22

2

1.....

dr

r dr OR OR

22

2

2....

d d

dr r dr

Radial Term Angular Terms

2 1 1Cart SP SP

W W

Three-Dimensional Rigid Rotor

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

2 2 2 2 22

. 2 2 2ˆ

2 2Cart

d d dH

dx dy dz

2 2 22

2 2 2 2

1 1 1ˆ sin2 2 sin sinSP

d d d d dH r

r dr dr r d d r d

2 2

2 2 2

1 1ˆ sin2 sin sin

d d dH

r d d d

2

2I

L

22 2

2 2

1 1ˆ sinsin sin

d d dL

d d d

18_05fig_PChem.jpg


2 2

2 2

1 1ˆ ( , ) sin ( , ) ( , )2 sin sin

d d dH E

I d d d

2 2

2 2

1 1sin ( ) ( ) ( ) ( )

2 sin sin

d d dE

I d d d

22

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

d d d

d d d

2

2

2

2

IEE

I

Separable?


22

2

1( )

( )

dk

d

2 21sin sin ( ) sin

( )

d dk

d d

22

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

d d d

d d d

22

2

1 1sin sin ( ) sin ( )

( ) ( )

d d d

d d d

Two separate independent equations

k2= separation Constant

18_05fig_PChem.jpg


22

2( ) 0

dk

d

1( )

2im

m e

1( )

2im

m e

k m k

22

2

1( )

( )

dk

d

Recall 2D Rigid Rotor

18_05fig_PChem.jpg


2 21sin sin ( ) sin

( )

d dm

d d

2 2sin sin sin ( ) ( )d d

md d

, ( ) cos( )mml m l lC 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


2 ( 1)

2l

l lE

I

2 2

2 2

1 1sin ( , ) ( , )

2 sin sin

d d dE

I d d d

1( )

2im

m e

, ,ˆ ( ) ( ) ( ) ( )l m m l l m mH E , ( ) (cos( ))mm

l m l lC P

21 ( 1) 1ˆ (cos( )) (cos( ))22 2

m mm im m iml l l l

l lH C P e C P e

I

,

1( , ) (cos( ))

2mm im

l m l lY C P e

Spherical Harmonics

, ,ˆ ( , ) ( , )l m l l mHY E Y

The Spherical Harmonics

,

1( , ) (cos( ))

2mm im

l m l lY C P e

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

1/2(2 1)( )!

for 02( )!

ml

l l mC m

l m

1/2(2 1)( )!

( 1) for 02( )!

m ml

l l mC m

l m

For l=0, m=0

21(cos ) (cos 1)

2 ! (cos )

l

ll l

dP

l d

(cos ) sin (cos )(cos )

m

m ml l

dP P

d

0

2 00

1(cos ) (cos 1) 1

2 0! (cos )l

dP

d

0

0 00 (cos ) sin 1 1

(cos )

dP

d


,

1( , ) (cos( ))

2mm im

l m l lY C P e

00 (cos( )) 1P

00,0

1 1 1( , ) (1)

2 2 2iY e

For l=0, m=0

1/2

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

2(0)! 2mlC

0,0

1, (0, ), (0,2 )

2oY r r

Everywhere on the surface of the sphere has value 1

2

what is ro ? r = (ro,


r = (1,

Normalization:

*

0,0 0,0

S

Y Y dV

2 sin( ) sin( )o

dV dxdydz

r d d d d

*( ) ( , )

( ) ( , )

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 1sin( )

2 2d d

In Spherical Polar Coordinates

2

0 0

1sin( )

4d d

2

0 0

1sin

4d d

2

0 0

1 1cos 2(2 ) 1

4 4

1or

r is fixed at ro.

The wavefunction is an angular function which has a constant value

over the entire unit circle.

X

Y

Z

1

2


01,0

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

22 2iY e

1,0

3( ) cos( )

2(0, ), (0,2 )

Y

r = (1,

X

Y

Z

The wavefunction is an angular function which has a value varying as on the entire unit circle.

3cos( )

2

The spherical Harmonics are often plotted as a vector strating from the origin with orientation and and its length is Y()

Along z-axis

2 11 1

1(cos ) (cos 1) cos

2 1! (cos )

dP

d

0

0 01 (cos ) sin cos cos

(cos )

dP

d

For l=1, m=0

18_05fig_PChem.jpg


1, 1

3( , ) sin( )

2 2iY e

Complex Valued??

1, 1 1, 1

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

2 4 2 2 2i iY Y e e

Along x-axis

1, 1 1, 1

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

2 4 2 2 2i iY Y e e

i i

Along y-axis

1

1 11 1(cos ) sin (cos ) sin cos sin

(cos ) (cos )

d dP P

d d

For l=1, m =±1

18_05fig_PChem.jpg


0 22

1(cos( )) 3cos ( ) 1

2P

12 (cos( )) 3sin( )cos ( )P 2 2

2 (cos( )) 3sin ( )P

22,0

5( , ) (3cos ( ) 1)

4Y

2, 1

15( , ) sin cos

2 2iY e

2, 1 2, 1 15sin cos cos

2 8

Y Y

2, 1 2, 1 15sin sin cos

2 8

Y Y

i

YZXZ

2 22, 2

15( , ) sin

4 2iY e

The Spherical Harmonics Are Orthonormal

2*, , , , ,

0 0

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

2

0 0

1 1(cos ) (cos ) sin

2 2m mm im m im

l l l lC P e C P e d d

2

0 0

1(cos ) (cos ) sin

2m mim im m m

l l l le e d C P C P d

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 00 (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

Yl,m are Eigenfuncions of H, L2, Lz

,

1( , ) (cos( ))

2mm im

l m l lY C P e

2 2

2 2

1 1ˆ sin2 sin sin

d d dH

I d d d

22 2

2 2

1 1ˆ sinsin sin

d d dL

d d d

ˆz

dL i

d

2

, ,ˆ ( , ) ( 1) ( , )

2l m l mHY l l YI

2 2, ,

ˆ ( , ) ( 1) ( , )l m l mL Y l l Y

, ,ˆ ( , ) ( , )z l m l mL Y m Y

2

( 1)2lE l l

I

2 2 ( 1)lL l l

( 1)lL l l

zL m

Dirac Notation

*,i j i j

S

ds

*,i j i j v v

,

N

i i m mm

c m is complete*,m m m m

S

* * * *,1 ,2 , 1 ,i i i i N i Nc c c c v

,1

,2

, 1

,

j

j

j

j N

j N

c

c

c

c

v

Continuous Functions

Vectors

*,|i j i j i j

S

ds i i vDirac

j j v

*ˆ ˆ ˆ ˆi j i j i j

S

ds O O O O

Bra

Ket

ˆ | j j jH E

Dirac Notation

2*

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

0 0

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

1,1 1,1

ˆ|Y HY

2 2 2

1,1 1,1 1,1 1,1 1,1ˆ 1(1 1) |

2HY Y Y Y Y

I I I

2 2

1,1 1,1|Y YI I

2

, ,ˆ ( , ) ( 1) ( , )

2l m l mHY l l YI

2

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

I

Degenerate

18_16fig_PChem.jpg

3-D Rotational motion & The Angular Momentum Vector

zL m

( 1)l l L

m indicates the orientation of the angular momentum with respect to z-axis

L

l determines the length of the angular momentum 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

E

l zmLlE,..,lm mY

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

22,1,0, 1, 2Y

11,0, 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 -1 00Y

1 0-1 -2

2

1 0-1 -2

2

-3

3

0

2

0

2

32

0

22

19_01tbl_PChem.jpg

Rotational Spectroscopy

19_13fig_PChem.jpg

Rotational Spectroscopy2

2( 1)

2Jo

E J Jr

1J JE E E

2

1E JI

J : Rotational quantum number

2

2( 1)( 2) ( 1)

2 o

J J J Jr

2

( 1)2JE J J

I


hcE h hc

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

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

Frequency (v)

vv

Line spacing


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

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

2726 27

1 1 11.774 10

2.656 10 1.673 10kg

kg kg

2 28 o

hB

r c

34 2

122 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

Rotational SpectroscopyThe line spacing for 1H35Cl 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

2726 27

1 1 11.626 10

5.807 10 1.673 10kg

kg kg

2 2 28 8oo

h hB r

r c B c

34 2

10

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 / )

1059.52 2

v cm cm mB m

ˆ

ˆ

ˆ

x

y

z

d dL i y z

dz dy

d dL i z x

dx dz

d dL i x y

dy dx

zL m

( 1)l l L

L

2 2 2 2ˆ ˆ ˆ ˆL x y zL L L

?

?

, ( , )l mY

zL m

2 ( 1)L l l

ˆ ˆ ˆ ˆx y zi L L L L r i j k

The Transverse Components of Angular Momentum

Ylm are eigenfunctions of L2 and Lz but not of Lx and Ly

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

Commutation of Angular Momentum Components

ˆ ˆ ˆ ˆ ˆ ˆ,x y x y y xL L L L L L

d d d d d d d d d d d dy 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 dy yz yx z zx

dx dzdx dz dydx dydz

2 d d d d d d d dy z z x z x y z

dz dy dx dz dx dz dz dy

dz d d d d d d d d dy z yx zz zx

dz dx dz dx dz dz dy dx dy dz

product rule


d d d d d d d d d d d dz 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 dzy z xy x xz

dxdz dxdy dz dy dzdy

d d d d d d dz d d dzy zz xy x z

dx dz dx dy dz dz dz dy dz dy

product rule


22

22

22

2

ˆ ˆ,x y

d d d d dy yz yx z zx

dx dzdx dz dydx dydzL L

d d d d dzy z xy x xz

dxdz dxdy dz dy dzdy

ˆz

d di ih x y i L

dy dx

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

2 d d d dy x i ih y x

dx dy dx dy


2 ˆx

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

dx dz dz dy dz dy dx dz

ˆ ˆ ˆ ˆ ˆ ˆ,x z x z z xL L L L L L

2 ˆy

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

dz dy dy dx dy dx dz dy

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

ˆ ˆ ˆ ˆ ˆ ˆ,y z y z z yL L L L L L

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

Cyclic Commutation of Angular Momentum

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

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

ˆ ˆ ˆ,x y zL L i L

ˆ ˆ ˆ,y x zL L i L

ˆxL

ˆyLˆ

zLi

ˆxL

ˆyLˆ

zLi

Commutation with Total Angular Momentum

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

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , ,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

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

ˆ ˆ ˆ ˆ ˆ ˆx x z z x xL L L L L L

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

ˆ ˆ ˆ ˆx y y xi L L i L L


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

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

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

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , ,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

ˆ ˆ ˆ ˆy x x yi L L i L L

ˆ ˆ ˆ ˆ ˆ ˆy y z z y yL L L L L L


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

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

ˆ ˆ ˆ ˆ ˆ ˆ 0z z z z z zL L L L L L

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

2ˆ ˆ, 0xL L 2ˆ ˆ, 0yL L

This means that only any one component of angular momentum can be determined at one time.

Ladder Operators

ˆ ˆ ˆx yL L iL ˆ ˆ ˆ

x yL L iL

2ˆ ˆy xi L i L ˆ ˆ

y xi L L

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

ˆ ˆ ˆx yL iL L

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

ˆ ˆ ˆx yL iL L

ˆ ˆy xi L L ˆ ˆ( )y xi L i i L

Ladder OperatorsWhat do these ladder operators actually do???

, , ,ˆ ˆ ˆ

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

Recall That: ˆ ˆ ˆ,zL L L ˆ ˆ ˆ ˆ ˆ

z zL L L L L

, ,ˆ ˆ ˆ ˆ ˆ

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

, , , ,ˆ ˆ ˆ ˆ ˆ ˆ ˆ

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

, , ,ˆ ˆ ˆ ˆ

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

, ,ˆ ˆ ˆ( 1)z l m l mL L Y m L Y

, 1 , 1 , , 1ˆ ˆ( 1)z l m l m l m l mL Y m Y L Y Y Raising Operator

Lowering Operator, , 1

ˆl m l mL Y Y Similarly

Therefore is an eigenfunction of with eigen values l and m+1

Ladder Operators

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

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

2 2 2, ,

ˆ ˆ ˆ ˆ ˆ ˆ0 , l m l mL L Y L L L L Y

2 2, ,

ˆ ˆ ˆ ˆl m l mL L Y L L Y

2 2, ,

ˆ ˆ ˆ( 1)l m l mL L Y l l L Y

2 2, ,

ˆ ˆ ˆ( 1)l m l mL L Y l l L Y

,ˆ

l mL Y2ˆ ˆ& zL L

Which implies that , , , 1ˆ

l m l m l mL Y C Y

Ladder Operators

, , , 1ˆ

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

ˆl m l m l mL Y C Y

This is not an eigen relationship!!!! ,l mCis not an normalization constant!!!

These relationships indicates that a change in state, by m=+/-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 of L, and therefore must be smaller than L. ie.

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

2,0 2,1L̂ Y Y

Why is lL

2,1 2,2L̂ Y Y

2,2ˆ 0L Y Not allowed

?

,ˆ ?

n

l mL Y ,

ˆ ?n

l mL Y

2,0 2, 1L̂ Y Y

2, 1 2, 2L̂ Y Y

2, 2ˆ 0L Y

More Useful Properties of Ladder Operators

2 2 2 2ˆ ˆ ˆ ˆx y zL L L L 2 2 2 2ˆ ˆ ˆ ˆ

x y zL L L L

2 2 2 2, ,

ˆ ˆ ˆ ˆx y l m z l mL L Y L L Y

2 2, ,

ˆ ˆl 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 of L 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 by ±1

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

max,ˆ 0l mL Y


min,ˆ 0l mL Y

max,ˆ ˆ 0l mL L Y min,

ˆ ˆ 0l mL L Y

Lets show that mmin and mmax are l and -l resp.

ˆ ˆ ˆ ˆ&L L L L Have to be determined in terms of 2ˆ ˆ& zL L

2 2ˆ ˆ ˆ ˆ ˆ ˆ2 2 zL L L L L L

2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆx y x y x x y y x yL L L iL L iL L iL L iL L L

2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆx y x y x x y y x yL L L iL L iL L iL L iL L L

2 21 ˆ ˆ ˆ ˆ ˆ ˆ

2 x yL L L L L L


ˆ ˆ ˆ ˆ ˆ ˆ, , ,x yL L L L i L L

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , , ( ) ,x x y x x y y yL L i L L i L L i i L L ˆ ˆ ˆ0 ( ) ( ) 0 2z z zi i L i i L L

Also note that:

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

2 2ˆ ˆ ˆ ˆ ˆz zL L L L L

2 2ˆ ˆ ˆ ˆ ˆz zL L L L L

2 2ˆ ˆ ˆ ˆ ˆ ˆ2 2 zL L L L L L

2 2ˆ ˆ ˆ ˆ ˆ2 2 2 2z zL L L L L

Similarly

Ladder Operators

2 2,max ,max

ˆ ˆ ˆ ˆ ˆ0 l z z lL L Y L L L Y

max max max

2 2, , ,

ˆ ˆ ˆl m z l m z l mL Y L Y L Y

max max max

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

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

max max1 ( 1) 0l l m m

maxm l

Ladder Operators

min min

2 2, ,

ˆ ˆ ˆ ˆ ˆ0 l m z z l mL L Y L L L Y

min min min

2 2, , ,

ˆ ˆ ˆl 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

2min 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 .

minm l

18_12afig_pchem.jpg rotational motion center of mass translational motion r1r1 r2r2 motion of two...

Documents