Download pdf - The quantum theory of the three-dimensional rigid rotator

IL NUOV0 C1-M'ENT0 VOL. 21 B, N. 1 11 Ma~gio 1974

The Quantum Theory of the Three-Dimensional Rigid Rotator.

O. ~ALO

University o/ Zrai,robi - ~'ai,robi

(ricevuto il 12 Giugno 1973)

Summary . - - A quaaatttm-mechanical theory of a three-dimensional r igid ro ta tor is p r e sen~d by developing a simple schematic theory of the broadening of ro ta t ional lines by collision. We consider a dilute solution of l inear molecules in a simple buffer such as an inert gas under the a~sumption proposed by Gross and Lebowitz tha t collisions cause neglible change in the orientat ion--- the no position change mode l - - and therefore contain the classical inerti'M theory as ~ l imit ing ca~e.

1. - Introduction.

I n t h e t h r e e - 4 i m e n s i o n M case l;he t I a m i l t o n i a n of t h e s y s t e m is g i v e n b y

- - 2 1 SinO~O sinO - i - s in2 0 ,

a n d t l l e S e h r 6 4 i n g e r e q u a t i o n for t i l e e i g e n v a l u e p r o b l e m b y

(2) H ~ - - - - Ez~ ~m ,

w h e r e

~2

W e c o n s i d e r ~0~,,,(0~) as s p h e r i c a l h a r m o n i c s :

[21@ 1. 1 . . . . P r ( eos O) exp [imp] (3) Y, . , O_ qo) \ 4:~ l +

162

TItE QUANTUI%f T}I~0RY OF TIIE THI~EE-DI~ENSIONAL I~I~ID I~OT'AT01~ 1 ~

where _P~(cos 0), 1 ~- O, 1, 2 ... c~, m ~- O, ~ 1 . . . • l, are the associated Le- gendre polynomials.

In quan tum theory, s tat is t ical propert ies of a sys tem are given b y ~ t t e rmi t ian dens i ty ma t r ix o(t) and an equat ion describing the t ime dependence of the densi ty m a t r i x is

(4) ~e i ~-/+~[ el= o.

Our a im is to describe the t ime behaviour of the dens i ty ma t r ix when the system iuteracts with the reservoir a t t empera tu re T. I f we assume th a t H is a t ime-constant t tami l tonian , the reservom' will drag the system towards a canonical dis tr ibut ion in which the densi ty ma t r ix tends to @e~(t) represent ing the equil ibrium densi ty ma t r ix in accordance with Bol tzmann law appropr ia te to the ins tantaneous value of the t ime-dependent t t ami l ton ian H ( t )

(5) p~.(t) = cxp [ - - f iH( t ) J /Tr exp [--f iH(t)] .

Then our eq. (4) takes the form

(6) ~9 i 3~

where the 1.h.s. is the quan tum analogue of the Liouville equat ion and so de- scribes the var ia t ion of the densi ty mat r ix caused b y the act ion of the t tami l - tonian. And the r.h.s, gives the effect of in teract ion with the reservoir collision term.

2. - Density matrix in external field. Gross-Lebowitz model.

We shall l imit our s tudy to a special form of the collision t e rm ~9/~t which is closely related to the strong-collision model in which the system has a Max- wellian veloci ty distr ibution af ter impact (1).

__ ~o.(00') [ e(O0) e(0'0') ] (7) ~o (oo't) - ~(oo') + + St ~ aT [~o~(00) ~o~ '

where T is the t ime between collisions. Our Hami l ton ian is

(8) H(t ) : Ho ~- V(t) = Ho -- At cos 0 Eo cos o~t,

(1) E . P . GROSS and J. L. L~BOWITZ: _Phys. Rev., 104, 6 (1956).

164 o. ~Ano

where He is the t t ami l ton ian of the unper tu rbed system, V(t) represents the in teract ion with the electric fiel4 an4 # is the 4ipole moment operator of the

system. Oar final task lies in the comput~t ion of the expectat ion value of the 4ipole

momen t operator , i.e. t he polar izat ion

(9) P(t) = Tr {@(t), #o~},

an4 we mnst eventnal ly average over to the t ime of the last collision in accor- 4ance with the assumption of r an4om collisions spaced b y an average time.

We now intro4uce a quan t i ty D(t) which obeys ~ similar equat ion of mot ion (4) an4 (6) an4 is a measure of the 4eviat ion of the 4ensi ty ma t r ix f rom t h e

ins tantaneous thermal equi l ibr ium:

(10) D(t) = @(t) -- @~(t).

On subst i tu t ing of (10) in (6) an4 taking into consideration (7) we obtain the kinet ic equat ion for the 4ensi ty ma t r ix in the form

( 1 1 )

where

8D i i -~ + ~ [HOD] -I- ~ [V(t)D] -- eeo~ D eo(OO') [D(O0) D(0'0')] ~t T + ~ [ ~ + e,(0'0')]

@o ~- exp [-- flHo]/Tr exp [-- flHo].

The [ ] bracket is symmetr ic in or4er to main ta in the T[ermitian character of ~@/~t in the course of t ime.

For our calculation we neglect the 3r4 t e rm on the 1.h.s. of (11) since the poten t ia l is weak an4 is only impor t an t in cases of saturat ion effect.

3 . - T h r e e - d i m e n s i o n a l r ig id r o t a t o r .

We shall now write (11) in a more explicit nota t ion for the three-dimensional rigi4 ro ta tor :

(12) 8Ds__t + ~i [HD] -- 8q~St D~ +

+ ~v (<Oq~[DIOq~ (Oq~[O~ + (O~[ff"lO'q~') (O'T'IDIO'~')}"

The above equat ion can be han41e4 in a nnmber el ways. We t r y to form, say,

TH~ q U A N ~ THE0~Y OS T~E ~SrE-DI~ENSIONAL ~IGID ~0TAT0~ 165

(/m]O~> on the r.h.s, and <O'qY/llml> on the 1.h.s., t he reby move into tile energy represen ta t ion :

2~ ~ 2~

f f f ~-2-vv sinOdO clq~ sinO'dO' d~'-

0 0 0 0

where %z,= (E~- E~,)/ti is t he angular f requency associated with ~he t ran- sition l -+ l~ of the nnperturbecl molecules.

The difficult th ing we have is how to t r ea t the t e rm in the curly brackets. Le t us consider the following:

(14) =-- /sinOdOd~sinO dO dq0 <~mlOq~><O~lDlO~o><Oq~l~olO'q~><O'~o'lgim~> z v d

an(t first look at the piece

(~5) fsin O' dO ~ d~'(O~]Oo]O'~'> <O'~'lZxm~>.

Leaving O~ free, we in tegra te over O'~v'

(16) sin O'dO'd~v' ~ ~ (O~vlr/~ > exp [--fiE~](/~rlO'qY>(O'~'II, m~>, r

and applying the completion theorem of spherical harmonics, i.e.

(17) fsin OdOd~ * ,

our (16) above takes the form

( 1 8 ) < O ~ l l l ~ 1 > e x p [ - - / ~ l l ] .

Then expansion (14) becomes

(19) ~ffsinOdOdq~<lmlOq~> <Oq~lDl@~> (O~[llml>: exp [--fiE,l] ,

1 6 6 o . ~L,,~ o

an4 consequently eq. (13) will be

(20) (io9 + ioo~,-~-1) (lm[D[~m~>--~ --(~m I ~@'--~ [/lm~> +

+ 4= ((sin 0d0 d~0 <~1o~><o~1OlO~> <O~olt~> e=p [--~E,] . 2vJJ

From the above it is evident that

(21) (lmlDlllm~> = 1/~ + i(w~--w) +

exp [--flE~,] ~ ( ( s in0d0 d v <~mlOv><OvlDIOv><Ovlqm~>.

To come back into co-or4inate representation we multiply (21) above by (0' ~o' lira> to the left and <l~m~]O' ~} to the right, and form (0 ~ ~]D]0' ~0'> thereby obtaining an integral equation in co-ordinate space.

For the moment let us leave alone the inhomogeneons term and t ry to re- duce the other term involving D.

We obtain

4= (O'q/[Im>exp[--flEq](l~m~lO'q/>(O~lltm~> sinOdOdq~(Oq~lDlOcf>. (22) ~ ~ air + i(o),,,-- ~)

r 1

The above can only be expan4e4 when we sum over m~ an4 m2 an4 thus, applying the a44ition theorem for spherical harmonics, we obtain

, 2 z + 1 ~, (Z-lml) ' Z <o'~'lZm> <lmlOq~> -- ~ "" ,.=_, 4= (l + m) !

21+ 1 = , , �9 P'~(cosO)P~(cosO')exp[im(q~--qY)]-- ~ Y4cos~),

(23) ( l l - - Im~[)~

211-~- 1 �9 P~(eos0)P~]'(cos0') exp [ - - im(~--~ ' ) ] : 4= P~(eosa) ,

where

cos a --~ cos 0 cos 0 '+ sin 0 sin 0' cos (~ -- ? ' ) .

Consequently expression (22) will take the form

4= 3" exp [--fiEq] 2lq- 1 211-~ 1 fP t (eosa) Ptl(eos ~). (24) ~ U, 1/T + ~(~,,1-- ~) 4= 4=

�9 sin0d0 d~ (Oq~]D[O~) ~--fK(cos ~) sin0d0 d~0 <O~[D[O?>,

Ttt:E QI :YANTU~ TtI:EO:RY OF T H E T t t ~ E E - D I : b ~ E l q S I O N A L :RIGID : R O T A T O ~ 167

where

(25) 4~r ~ exp [-- fiE~,] 21 + 1 211§ 1 Pdcos ~)Pz~(cos ~).

Thus eq. (2) will be t ransformed into the following integral equat ion:

(26) <0'~'ID[0'~'> = f K ( c o s ~) s in0d0 d~.

~e~ l/lint> <llmllO,qJ,>, �9 <0~ ID 10~> - - ~ <O'q~' I~m> <Zm I -~/- ~m

which is to be solved exact ly. The exact solution of the above integral equat ion (26) is presented in a paper

by G~oss and the au thor (5). Below we shall proceed to check the above cal- culations by considering a two-dimensiona] case whose integral equat ion is exact ly solvable.

4 . - T w o - d i m e n s i o n a l c a s e .

Here the eigenvalue problem is

Where

~ 1 E~ = ~ m s , y~,,~(0) ----- ~ exp [imO] = <ml0> �9

~V27i

Therefore applying Dirac 's quan tum relat ion (a) for the off-diagonal demen t s

(27) <0lexp [--#R]]0'> = ~ <0[/> exp [ - - ~ E , ~ J < m l O ' > =

1 = - - ~ exp [--fiE,~] exp [im(O--O')]

and the diagonal elements given b y

1 co

<Ol exp [-flH]10> = ~ , Z e~p=_ [-fiE,.],

(2) E . P . GI~OSS and g. O. 1VIA:Lo: Journ. Chem. Phys., 57, 6 (1972). (3) P . A . !VI. DIRAe: The Principles of Quantum Mechanics, fourth edition (Oxford, 1967).

168

the partition function of the system becomes

m

and so the density matrix

(28) <01eo]0> ---- <01exp [--flH~ Z

Then eq. (20) takes the form

(29) / 1 § i(o~,~ - -~) <mtDIm,> (-e

from which it follows

(30)

1 2~

O. "~I'ALO'

4= exp [-#E.j faO<miO><OlDlO> <mlDlm*) = 2 ; 11~ § i(eo . . . . --co) <0ira1> +

§ inhomogeneous t e rm.

lqow in order to form <O']D[O'>, i.e. to move inr eo-or4inate representation, we multiply (30) to the left and right by

~, <O'[m> ... <m,10'>,

an4 obtain

(31) 4z 2"-~ ~ , 1/'r § i ( r - - co) J

exp [i(m -- m~)0] exp [-- i (m - - m,)0'].

<O[DIO> (2~)�89 § inhomogeneous te rm.

Let us now form K m , ( O - 0') by carrying o~t the summation over m, i.e.

(32) ~exp[--im(O'--O)](1) �89 Kml(O--O') = ~ T ~ _ ~ ) ~ '

then the snmmation over m,:

(33) 4Z ~ exp [-- fl.E,~,]Km,(O' - - O) exp [iml(O' - - 0)] ---- Z(O' -- O).

= - < m I ~ i~nl>'l"

4. f § ~ exp [--flEml] dO<re]O> ~OIDIO> <O[m,>

THE QUANTU'~ THEORY OF THE ~I-IREE-DIMEI~8IONAL RIGID ROTATOR 1 6 9

We obtain an integral eqnation of the form

~ f z (34) <OlD[O> = -- ~ <0'lm><~I ~ { Imp> <m~10'> + (0'--0)<0lDl0>d0. ~ n , m l

The above integral equation is easily solved by Fourier analysis and :Fourier expansion of <OlD[O> an4 <O']D[O'>:

D = ~ D, exp [@0]. (35)

We have

(36) ~ D . exp [inO'] : f z ( o'-- 0)d0 ~D~ exp [ ipO] d- ,~ , n 2o

where ~. is the Fourier transform of the inhomogeneons term. We know tha t

(37) fz(o'- 0)d0 = ~Zo exp [iq(O'-- 0)], q

then

(3s) ~ D. exp [i~0']=fexp [ - i n0 ' ] d0' ~ Z~ exp [i~(O'-O)] ~ D. exp [ /p0]d-2.=

= Z go exp [ - ~qOJfexp [~(n-- q) O'] d0' Z D~ exp [ipO] @ 2~. q

We can only say something about the integral above on condition that n = q. So we come to

(39) ~D ~ exp [inO'] = ~ Z~ exp [-- iqO] ~ D~ exp [ipO] d- ~ ,

n q

2~rD~ = ~ Z~D~ exp [i(q--p)O] d- ~ = (2:~)~Z~D~d - 2~, q~

from which we obtain the following result:

2~(1 -- 2:r

In the appendix of (2) a bet ter feeling for the shift of the maximum in the classical inertial region h/I < v < (2kT/I) ~ and for the region where the quantum and inertial domain merge v _~ h/I is developed by employing a mock distribution in place of the Boltzmann distribution. Here h = Plank constant,

1 7 0 o . ~AZO

I = m o m e n t of i ne r t i a , k = B o l t z m a n n c o n s t a n t , T = a b s o l u t e t e m p e r a t u r e ,

v = a s soc i a t e4 col l is ion f r e q u e n c y of t h e b u f f e r gas, h / I ~ q u a n t u m r o t a t i o n a l

f r e q u e n c y , (2kT / I ) �89 t h e r m a l a n g u l a r f r e q u e n c y .

T h e e x p e c t a t i o n v a l u e of t h e 4 ipo le m o m e n t o p e r a t o r is c o m p u t e 4 a n d

t h e conc lus ion of t h e ana ly s i s as wel l as of a m o r e 4 e t a i l e 4 n u m e r i c a l ana lys i s

is t h a t t h e q u a n t u m cor rec t ions a re u n i m p o r t a n t a t f r e q u e n c i e s whe re t h e

o sc i l l a t o ry v a r i a t i o n of r o t a t i o n a l s p e c t r u m is negl ig ib le .

T h e a u t h o r is g r e a t l y i n 4 e b t e 4 to Prof . E . P . GRoss for sugges t i ng t h e

p r o b l e m a n 4 s u b s e q u e n t valui~ble discussion.

�9 R I A S S U N T 0 (*)

8i presen~a la ~eoria quan~o-meeeaniea di un ro~atore rigido ~ridimensionale sviluppando una sempliee teoria sehematiea dell 'allargamento delle linen rotazionali per eollisiono. Si eonsidera nna soluzione diluita di moIeeole lineari in un sempliee tampone quale un gas inerte nell 'ipotesi proposta da Gross e Lebowi~z ohe le eollisioni eausino nn ~rasenrabile eambiamento di orientamento - - il modello del eambio di posizione nullo - - e quindi eon~eng~ la teoria inerziale el~ssiea come easo limi~e.

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