The quantum theory of the three-dimensional rigid rotator

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  • 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 rigid rotator is presen~d by developing a simple schematic theory of the broadening of rotational lines by collision. We consider a dilute solu- tion of l inear molecules in a simple buffer such as an inert gas under the a~sumption proposed by Gross and Lebowitz that collisions cause neglible change in the orientation---the no position change model- -and therefore contain the classical inerti'M theory as ~ limiting ca~e.

    1. - Introduction.

    I n the three-4 imens ionM case l;he t Iami l ton ian of the sys tem is g iven by

    - -21 SinO~O sinO - i -s in2 0 ,

    and t l le Sehr64inger equat ion for t i le e igenvalue prob lem by

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

    where

    ~2

    We consider ~0~,,,(0~) as spher ica l harmon ics :

    [21@ 1. 1 . . . . Pr(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 quantum theory, statistical properties of a system are given by ~ ttermit ian density matr ix o(t) and an equation describing the t ime dependence of the den- sity matr ix is

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

    Our aim is to describe the t ime behaviour of the density matr ix when the system iuteracts with the reservoir at temperature T. I f we assume that H is a t ime-constant ttamiltonian, the reservom' will drag the system towards a canonical distribution in which the density matr ix tends to @e~(t) representing the equilibrium density matr ix in accordance with Boltzmann law appropriate to the instantaneous value of the t ime-dependent ttamiltonian H(t)

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

    Then our eq. (4) takes the form

    (6) ~9 i 3~

    where the 1.h.s. is the quantum analogue of the Liouville equation and so de- scribes the variation of the density matr ix caused by the action of the ttamil- tonian. And the r.h.s, gives the effect of interaction with the reservoir col- lision term.

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

    We shall l imit our study to a special form of the collision term ~9/~t which is closely related to the strong-collision model in which the system has a Max- wellian velocity distribution after 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 Hamiltonian 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 ttamiltonian of the unperturbed system, V(t) represents the interaction with the electric fiel4 an4 # is the 4ipole moment operator of the system.

    Oar final task lies in the comput~tion of the expectation value of the 4ipole moment operator, i.e. the polarization

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

    an4 we mnst eventnally average over to the t ime of the last collision in accor- 4ance with the assumption of ran4om collisions spaced by an average time.

    We now intro4uce a quantity D(t) which obeys ~ similar equation of motion (4) an4 (6) an4 is a measure of the 4eviation of the 4ensity matr ix from the instantaneous thermal equil ibrium:

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

    On substituting of (10) in (6) an4 taking into consideration (7) we obtain the kinetic equation for the 4ensity matr ix in the form

    (11)

    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 symmetric in or4er to maintain the T[ermitian character of ~@/~t in the course of time.

    For our calculation we neglect the 3r4 term on the 1.h.s. of (11) since the potential is weak an4 is only important in cases of saturation effect.

    3. - Three-d imens iona l r ig id rotator .

    We shall now write (11) in a more explicit notation for the three-dimensional rigi4 rotator:

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

    + ~v (

  • TH~ qUAN~ THE0~Y OS T~E ~SrE-DI~ENSIONAL ~IGID ~0TAT0~ 165

    (/m]O~> on the r.h.s, and 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

    (18) exp [ - - /~ l l ] .

    Then expansion (14) becomes

    (19) ~ffsinOdOdq~ (O~[llml>: exp [--fiE,l] ,

  • 166 o. ~L,,~ o

    an4 consequently eq. (13) will be

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

    + 4= ((sin 0d0 d~0 e=p [--~E,]. 2vJJ

    From the above it is evident that

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

    exp [--flE~,] ~ ((s in0d0 d v .

    To come back into co-or4inate representation we multiply (21) above by (0' ~o' lira> to the left and 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

    , 2z+1 ~, (Z-lml)' Z -- ~ "" ,.=_, 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 fPt(eosa) Ptl(eos ~). (24) ~ U, 1/T + ~(~,,1-- ~) 4= 4=

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

  • Ttt:E QI :YANTU~ TtI:EO:RY OF THE Tt t~EE-DI :b~ElqS IONAL :RIGID :ROTATO~ 167

    where

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

    Thus eq. (2) will be transformed into the following integral equation:

    (26) =fK(cos ~) sin0d0 d~.

    ~e~ l/lint> , 9 -- ~ = ~ exp [ - -~E,~J

  • 168

    the partition function of the system becomes

    m

    and so the density matrix

    (28) ----

  • THE QUANTU'~ THEORY OF THE ~I-IREE-DIMEI~8IONAL RIGID ROTATOR 169

    We obtain an integral eqnation of the form

    ~ fz (34) = -- ~ + (0'--0)d0. ~n,ml

    The above integral equation is easily solved by Fourier analysis and :Fourier expansion of an4 :

    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 that

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

    then

    (3s) ~ D. exp [i~0']=fexp [ - in0' ] 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 better feeling for the shift of the maximum in the classical inertial region h/I < v < (2kT/I) ~ and for the region where the quan- tum and inertial domain merge v _~ h/I is developed by employing a mock distribution in place of the Boltzmann distribution. Here h = Plank constant,

  • 170 o. ~AZO

    I = moment of inert ia , k = Bo l tzmann constant , T = absolute temperature ,

    v = associate4 coll ision f requency of the buf fer gas, h/ I ~ quantum rotat iona l f requency, (2kT/ I ) 89 thermal angular f requency.

    The expectat ion va lue of the 4ipole moment operator is compute4 and

    the conclusion of the analys is as well as of a more 4eta i le4 numer ica l analysis

    is that the quantum correct ions are un impor tant at f requencies where the

    osc i l latory var ia t ion of ro ta t iona l spect rum is negligible.

    The author is great ly in4ebte4 to Prof. E .P . GRoss for suggest ing the

    prob lem an4 subsequent valui~ble discussion.

    9 R IASSUNT0 (*)

    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.

    (*) Traduzione a cura delta Redazione.

    I~BaHTOBa~I TeopHH TpeXMCpH0rO ~KeCTI~OFO pOTaTOpa.

    Pe3mMe (*). - - Ilpe)/JmraeTcn KBa~TOBOMexarmueci

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