Chemistry 482 Homework Set #5 Due Mar 13, 2013

1. Consider the wavefunction )1cos3(165),( 2

2/1

a. Show that this function is an eigenfunction of the total energy operator for a rigid rotor b. Show that this function is normalized over the interval 0 and 20 2. The moment of inertia of 1H35Cl is I=2.644 x 10-47 kg m2. a. Calculate Erot/kBT for J =0,5,10, and 20 at 298 K b. For which of these values of J is Erot/kBT ~ 1? 3. A rigid rotor is described by the wavefunction

1,1 2,0 3,01( , ) , 2 ( , 3 ( ,14

Y Y i Y , where , ,mY are the spherical harmonics with

the total angular momentum quantum number and the magnetic quantum number m. a. If Lz is measured for this system, what are the possible values that will be observed and what is the probability for each value? b. If the magnitude of angular momentum, L , is measured for this rigid rotor, what are the possible values that will be observed and what is the probability for each value? 4. Consider a rigid rotor which has 2 as one of its quantum numbers.

a. What are the possible angles between L

and the z axis for this rigid rotor? b. What are the possible eignevalues of the operators (if any) for 2 2 2 2, , and x y x yL L L L ?

5. Find the most probable angles and for a particle described by the rigid rotor wavefunction

3, sin8

iY e

#5

b. S: sillll de \~; Jf f·f (o!OBcSTL, 0 ~o/s; 2fC )

_ 21' I~Sine d fj (k) ( 3CVS6 -/ t = :11t r:(-rSr ) ( qCOS~s/ nfI - 6 COlO s!'r1l7 +sinfl)d

= t J: (9C:eS1ne - 6~sin9 + sine )Je= -f[ q. + -6-+ + 2) = 1 (norma,{jied)

.2. n:1a . £rcrl; = ..2I J(J + I )

.t,..a IErot (KaT::: ~ I<sT J (J +t)

_ (1.055)(Jo-34- J·S)~ J (J+{)- 2.2.f;44-X 10- 4-i~.m:l.· ,.3gX(O-~3J' k-I . 2q~I<'

- 0 .oS f J (J + I ):. 0

::::1·5~

(J=o)(J=5)

( .1=10)

(:r=2.o)= !;.63= .21.5D

b.

b. L~ N/A

LS N/A2- ~ ;;. L2Lx t L'J ::: L - c

.") 2

= 61T~ - LlOm:=. 0 1 ~ ~_~_' :x-:.....~-tJ-~ -_

/ -h 2 f -h <- m ::::±} .:=: i 6 -t; '2. ( Tn ==0 -f7fJ. 5--n2 m-=±)4-~2 m =1-2 2tfz; til::- ± 2

\((9, ~)-o,g <;lnB e--J'~

p -= f~ r* r sfne elL ta ji'rki the most prohoJoJe (J / ~

dp _ dp _ dsm~ _ \ :2 _. 2-

\ ae -0 ale - dfl - 3smE! vose - 3(f- C{J)e) case =-0

I~ -= 0 CQ;6= ±/ 0/ w;(J =0 e = 0, !} ,7rdcp ~ (r'n depeJ)cie;vt ) ~ == 0 ~ 2n

3.a.

L i :::mtr ( P-1Qa - l~q4-)

(,,1 £=/ m=/ Ll= n

)2,0 l=2 m=- 0 Ll= D

13,0 t=-6 m= 0 L=I= 0

b· ILl::: n('1 lUtl)

Y; J 1 l =-/ J LI = 12tY2,V ,{:= z. I L } = fbt

Y3/v ~ =-3 ILl == ~ TI-

t:=:2 m= 0/ ±/, ± ~

a. Ll:= m Pi =- 0 I ±yf, ± 211/L I =-nJU~+J) ::: JTn

4·

Ip= 1t1~+?

t -IiJ p:::- 14-

I~Tt

!p= Ifp~tP=#

m= f

m=-j

~.~ . ~=: .±2j)lin .

£iL L e := 35: 30

2~~ (J-/4+.7-'

m=2

m.:::-2