rotations vs. translations translations rotations

Rotations vs. Translations

Translations Rotations

mass m kg 2 2moment of inertia I=mR kgm

-1velocity v ms -1vangular velocity ω= s

R

-1

momentum

p=mv kgms 2 -1

angular momentum

L=pR=Iω kgm s

2 2

KineticEnergy

mv pK= =

2 2m

J 2 2

KineticEnergy

Iω LK= =

2 2I

J

Quantized Planar Rigid Rotor

• Schroedinger’s Wave Equation

• General Solution:

• Continuity Condition

22

2

d ψ φ- =Eψ φ2I dφ

22

2IEk =ik ikA e A e

2 2 2z

k z

ψ φ =ψ φ+2π 0, 1, 2, 3

L kE = = andL = k

2I 2I

k

Quantized Planar Rigid Rotor(cont.)

• Wave Function:

• Orthonormality Condition

ikAe

2 2 22

0 0 0

1

1 1

2 2

ik ik

ik

d Ae Ae d A d

A e

Quantized Rigid Rotor

• Schroedinger’s Wave Equation:

• Separation of Variables:

• Results in two equations:

2 2

2 2

1 1sin , ,

2 sin sinY EY

I

,Y

2

2 22

2

1 1sin sin sin

2where = and m is a constant

m

IE

The Phi Equation

• This equation is the same as the plane rigid rotor, so it has the same solution:

2 22 2

2 2

10

1

2im

m

m m

e

The Theta Equation

• The theta equation can be put into the form of a standard (a.k.a. “already solved”) equation.

2 2

2 22

2 2

sin sin sin 0

cos

After some busy work:

1 2 01

d dm

d d

Let x

d x d x mx x x

dx dx x

Legendre’s equation

The theta equation has the form of a famous differential equation called Legendre’s equation:

an equation that was solved by Adrien Legendre about 180 years ago

2 2

22 2

1 2 1 01

d x d x mx x J J x

dx dx x

2

2 2

2If 1

then 1 for 0,1,2,3...2 2J

IEJ J

LE J J J

I I

Visualizing Complex Wave Functions

•Problems involving the quantization of angular momentum produce wave functions that are complex.•We encounter complex wave functions in:

–Planar Rigid Rotor–Rigid Rotor–Hydrogen Atom

Complex Wave Functions• Planar Rigid Rotor (a.k.a particle-on-a-

ring):

• Rigid Rotor:

• Hydrogen Atomic Orbital

1; 0, 1, 2

2ik

k e k

, ,, ;

0,1, 2,3 and 0, 1, 2,J M J M MY

J M J

, , , ,, , , ;

1, 2,3 ; 0,1,2,3 1;and 0, 1, 2,n m n m n m mr R r Y R r

n n m

Spherical Harmonics are Complex

l m

0 0

1 0

1 ±1

2 0

2 ±1

2 ±2

,l m m ,m m mY

22

32 cos34 sin

258 3cos 1

12

12

12

/ 2ie

154 sin cos / 2ie

21516 sin 2 / 2ie

14

34 cos

38 sin ie

2516 3cos 1

158 sin cos ie

2 21532 sin ie

Visualizing the Imaginary

• Note that spherical harmonics are real if m=0 and complex otherwise.

• A graphical representation of the real function functions is given below. Surfaces of (e.g. Y00, Y10, Y20) the function will only appear green and/or red, depending upon whether the function is positive or negative for those values of

• If the function is complex (e.g. Y11, Y21, etc. ) other colors represent

complex values. For example, if the function is proportional to +i or –i on a surface that can be displayed by yellow/blue.

,

Complex and Real Spherical Harmonics

310 4

31 1 8

2520 16

2 21522 32

, cos

, sin

, 3cos 1

, sin

i

i

Y

Y e

Y

Y e

Getting Rid of the Imaginary

• In most chemistry texts, atomic orbital wave functions are displayed as real functions. This is done by taking linear combinations of complex functions. Using the

complex functions… we define the normalized REAL wave functions:

1, 1

3, sin

8iY e

11 1 1

11 1 1

10

1 3 3, sin cos

4 42

3 3, sin sin

4 42

3 3, cos

4 4

X

Y

Z

xY Y

r

i yY Y

r

zY

r

Summary of Rigid Rotor Properties

• Energy:

• Angular Momentum:

2

2

1

1 ; 0,1,2,32

2 22

J

J J

E J J JI

E E E JI

2 2 1 or 1

; 0, 1, 2, 3Z

L J J L J J

L M M J

Statistical Thermodynamics of Rotations

• Partition Function (assumes EJ<<kBT)

• Probability of being in J energy level

2

2 2 22

1 /2/

0 0

1 /2 /2( 1)2 (2 1)

0 0

2

2 2

1 12 1 2 1

1 22 1

2 8whereσ=1for heteronuclear;σ=2for homonuclear;

BJ B

B B

J J Ik TE k Trot

J J

J J Ik T x Ik Tx J Jxdx J dJ

B B

q J e J e

J e dJ xe dx

Ik T Ik T

h

22/

( 1)/22

2 12 1

8

J B

B

E k TJ J Ik T

Jrot B

h JeP J e

q Ik T