r. Phase Problem in Quantum Oscillators

INDIAN J. CHEM., VOL. 17A, FEBRUARY 1979

SADHAN BxstrUniversity College of Science

92 Acharya Praful la Chandra Road, Calcutta 700009

Received 7 November 1978; accepted 18 December 1978

Using the ergenfuncttons of the coherent states ofharmonic oscillators it has shown that the transitionmoment is maximum when the two combining states.are in phase. Similarly the interaction energy of twocoupled oscillators is maximum when they are in1Jhase and is zero for 900 phase difference. In the case-of induced transition between two oscillator statesunder the perturbing influence of electromagnetic field,it has been established that for induced absorption the-electromagnetic field must be in phase with initialstate of the oscillator but out of phase with the finalstate. While for induced emission reverse should bethe case.

THE description of motion of a classical oscillatorrequires the specification of both amplitude

and phase as given by the relation (1)x = IAI cas (wt-rp) ... (1)where co is the circular frequency, IA I the amplitudeand rp the phase angle of the oscillator. In the caseof a quantum oscillator the phase is completelyundefined once the excitation number n is specifiedleading to the result<nlxln> = 0 '" (2)irrespective of the values of n. How can wereproduce some of the classical results which are-phase dependent with quantum oscillators? Inquantum description phase is defined in the highly-excited coherent states. A coherent state loc> isdefined as the eigen state of destruction operatorwith complex eigen value!<ocla+ = <oc:a* ... (3)and ocis given as o: = ocleiq" where rp is the phaseangle. The loc> states may be constructed out'Of the number states as follows

" n

loc> = exp(-li2IocI2) 2: (n~1/2In> ... (4)n=O

showing thereby that loc> states are highly excitednumber states. These coherent states are completebut not orthogonal. It has been shown that!l<oci~>12 = exp(-loc-~12) ... (5)Let us now consider the case of an oscillator in thecoherent state loc> making a transition to a state:~>. The transition moment square is given as

Ix,,~12= I<oc;xl~>12(expressing x in terms of creation and destructionoperators, a+ and a respectivelyj-

hlX,,~\2= -2- 1<('f.la++al~>12mw

= 2 h I ~<oc·~> +oc*<OCI~>:2mw

= ~ e-I,,-m' (~+OC*)22mw

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/

... (6)Nowloc-~? = (oc-~)(('f.*-~*)

= lC(,i2+1~12-oc~*-~oc*= [jC(,12+1~:2_2Iocl'~1cos (rp,,-rp~)J ... (7)

Hence,

I 19 ItX"~I"= 2mw exp{-:oc?-If3?+2!ocll~i cos (<Jl,,+cP~)}

x[;oc,2+i~?+21ocl ~i cas (CP,,-CPIl)] ... (8)We thus get an expression for the variation of

tra~sition moment with the phase angle of the twoOSCillator states. Let us now calculate when isIX"Il! 2 maximum?Let us put] = e21"111l1cos ('P"-'Ps)[locI2+1~12

+ 21ocll~1cos (cp,,+cP~)] ... (9)Then

~ = _e21"'~1 cos ('P,,-'PIl)[21 IIRI - ( )]rpo" . o: I'" sm <p" - rpll

[loc\2+1f3\2+2Iocll~1cos (<Jl,,+CPIl)]_e21"111l1cos ('P"-'P1l)2Iocll~1sin (<P,,+<PIl) ... (10)

Hence the condition for IX"1l12to be maximum is(locI2+1[312

) sin (<p,,-<pIl)+2Iocll~1 sin (<P,,-tfJll) cos(<P,,+<PIl)+ sin (<P,,+<PIl)= 0 ... (11)

A I?-umber of combinations of <Pocand <Jlllexist whichsatisfy the above relation<Jl,,-<Pll= 0, <P,,+<Jl1l= 0 when <Pet= tfJll= O.<P"-<Pll= 7t, <Pot+tfJll= 'It when <Jl"= 7t, <Pll= O.<P,,-<Pll= 0, <P"+<P1l= 7t when <Pet= 7t/2, <Pll= 7t/2.and so on.

The maximum value of IXocll12isit

IX"1l12= 2mwe-{I"I-IIlI)' (loci+1~J)2 '" (12)

~et us ~onsider the interaction energy of twoosc~l1ators m. st.ates lOCI> and /OC2>' The inter-~ctIon ~otentlalls .of the form AXIX, (for dipole-dipoleinteraction), A being th~ coupling constant. Nowexpressing Xl and x2 in terms of creation anddestruction operators, a+ and a respectively, we haveA Aft ( ++

XIX, = 2mw a1a2+ala2+a1a2+a!a) ... (13)

Now consider the interaction energyIiA 1 + +

2mw <OC1OC2a1a2+ala2+a!a2+a1a21 oc1oc2>Aft

= 2mw (OCIOC2+OCIOCZ+ocioc2+ocioc2)

= 2~w loc1\\oc2\[ei('PI+'P2)+ei('PI-'P')+e-i('PI-'P2)

+e-i('PI-'P')]Ali

= mw \oc111oc21[COS(<Jl1+<Jl2)+COS (<P1-<Jl2)]

2Ait= mw \oc11Ioc2\cos <Jl1COS'P2 ... (14)

NOTES

This gives us the variation of interaction energywith phase angles. The energy is maximum whenCPI = CP2 = 0, i.e. the oscillators are in phase. Itis zero when CPI = 0, CP2 = ~!2, i.e. when the oscil-lators are 90° out of phae.

If we start with two coupled oscillators at t = 0,then it is well known that with progress of timethe energy will oscillate back and forth betweenthe two oscillators. The exchange frequency <.UA

is given as"

s.; (15)<.UA=T ...

and after a time t = E~n the energy is transferredin!

from oscillator 1 to oscillator 2. We thus get for tfrom (14)

t= ~m<.U ••• (16)2'-1 alii a21 cos 'PI COS CP2

i.e. t is lowest when the oscillators are in phase. Asthe phase difference increases the time for energytransfer from oscillator 1 to oscillator 2 increasesand becomes infinite for 90° phase difference, i.e.no transfer.

Finally let us consider the probability of inducedtransition between the coherent states Ia> andI~> by a time dependent perturbing potential ofthe form xF(t). If this purturbation is caused byelectromagentic field of the light beam, then we have,

F(t) = eEo (e-i(wt-cp) +ei(wt-cp)) ... (17)2

where <.Uand 'P are the circular frequency and phaseof the electric oscillation. The probability ofinduced transition may be written as

alX/3= IJe-iwlXt<Ct.lxF(t)I~>e;wptI2 ••• (18)

where WIX and Wf3 are the average oscillation fre-quencies in the states la> and I~> respectively.Writing x in terms of creation and destructionoperators and replacing F(t) by the relation (17)we have

a = [eEo1i <Ct.la++al~> J[ <ei(wlX/3+w)te-iCPrt./3 4mw

+ei(wlXll-w)tei'P)]dtl2 '" (19)

(where WIX/3= lOll - wlX)

alXo = [eEl/, <Ct.I~>(ICt.[e-icp'+I~leiCP.)" 4mw

Jei(WIXi3+W)te-iCPci(WIXIl-W)tei<PdW •.. (20)

where CPI and 'P2 are the phase of the states la> andI~> respectively.

a = leEl/, < al~> [Jei(wrt./3+cu)t dt ([I)([ e-i(<p.-<p)1X/3 4mw

+1~lei(<P.-'P)+Jei(wlXll-w)t dt (II)(I e-i(<p.-<p)

+1~lei(<p.H)JI2 ... (21)

Since we are interested in the phase relations bet-ween the electric field and the two oscillator states, wekeep the phase dependence of the non-orthogonalityintegral out of our consideration. Now the second

v•f

(

term in the square bracket in (21) gives us inducedabsorptions. It is evident that for induced ab-sorption the probability is maximum when 'P = CPl =-CP2' i.e. the electric field must be in phase withthe oscillation in state II)(> but out of phase witht hat in state I~>. For induced emission thereverse will be the case as is evident from the firstterm in (21).

It may. be observed that standard quantummechanical expression deduced for orthogonal basishas been used in the calculation of coherent stateswhich are not orthogonal. It has been pointedout by Lamb that when la-~I is greater than unitythe coherent states are nearly orthogonal, so thestandard procedure may be followed as a firstapproximation. This is what has been done inpresent work which is, in any way, approximate.

Sincere thanks are due to Prof. Chanchal Majumderfor helpful suggestions and constructive criticisms.References1. CARRUTHERS, P. & NIETO, M. M., Am. J. Phys., 33 (1965),

537; Rev. mod. Phys., 40 (1968),411.2. DAVYDOV, A. S., Quantum mechanics (Pergamon Press,

New York), 1973.3. FORSTER TH., Ann. Phys., 2 (1948), 55.4. LANDAU, L. D. & LIFSHITZ, E. M., Quantum mechanics,

(Pergamon Press, New York), 1958.5. SARGENT,M., SCULLEY, M. O. & LAMB, W. E., Laser

physics (Addison-Wesley, Reading, Mass), 1974, 248.

Spectral & Partition Coefficient Studieson the Ability of Small Oligomers of

Amylose to Form Complexes withPotassium Iodide & Iodine

BRA]ESH KUMAR & A. K. BANERJEE*

Department of Chemistry, University of DelhiDelhi 110007

Received 15 May 1978; accepted 7 August 1978

The partition coefficients of iodine (KI+I2) parti-tioned between eel. and amylose solution (0·75 g/litre)and its degradation products obtained by a-amylasehave been determined at 35° and different time inter-vals. The values are: 5·15 (0 rrrln), 7·90 (30 rntn),12·28 (75 rntn), 14·41 (100 rrrin), 17·70 (infinite time).The value of the partition coefficient of I.(KI+I2}

partitioned between eel. and water and containing alarge amount of a-amylase is 22·38. These results.clearly reveal that as more and more time is allowedfor the action of a-amylase (when smaller and smallerchains of amylose are formed). the capacity of amyloseto incorporate iodine and form complex also decreases.

DDRING the course of our investigation on the. blue-coloured c?mplex obtained by the inter-

action of amylose WIth potassium iodide and iodine,we ~anted to obtain data on the ability of smallercham molecules of amylose to incorporate iodineand to form complex with it. In order to obtainsmaller chains of amylose we used I)(-amylase, anenzyme well-known for its capacity to breakdownamylose chams to give oligomers .of shorter chain

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