Volume77A, number1 PHYSICSLETTERS 28 April 1980

ThE QUANTIZATION OF QUADRATIC FRICTION

F. NEGROand A. TARTAGLIA1stituto di Fisica SperimentaledelPolitecnico, Turin, Italy

Received8 January1980Revisedmanuscriptreceived21 February1980

A generallagrangianis found for a pointobject moving througha viscousfield quadraticallydependingon thevelocity.A simplespecific caseis workedout, equivalentto afreevarying massobject.Thesystemis canonicallyquantizedandanexplicit expressionfor the wave function is found.

The quantizationof systemswith friction forces al andis describedclassicallyby theequationof mo-hasso far beenperformedfor somesimplelinearcases, tion:i.e. for point particlessubjectto friction forcespro-

G(xx)=x+7x =0. (1)portional to their speed.Two different approacheshavebeenproposed:let uscall thefirst thecanonical Any externalforcehasbeenexcluded,7 representsaquantizationmethod(CQM) [1,2] andthe secondthe constantfriction coefficientand dotsindicatetimenonlinearSchrodingerequationmethod (NLSEM) derivatives.We caneasilyverify that the Helmholtz[3—8].Both methodsgive rise to problemswith the criterioninterpretationof theresults: the first producescorn-

aG/ax = (d/dt)aG/axmutatorsof the particlepositionandkmetic momen-tumthat vanish with time; the secondoneleadssimul- is notsatisfiedandconsequentlyeq.(1) cannotbetaneouslyto dampedandundampedsolutions.Recent- derivedfrom a lagrangianfunction. It is howeverpos-ly an interpretationhasbeenproposedthat endows sible,applying a methodalreadyusedfor the lineartheCQM with a physicalmeaningrelating its results case [Ill, to multiply eq.(1) by a suitableintegratingto thedescriptionof varyingmassobjects [9—11]; factorf(~,x, t) is sucha way that the newequation,this factand a more formal completenessmakeuspre- classicallyequivalentto the former one,can indeedfer theCQM. be derivedfrom a lagrangian.The conditionf must

When consideringfriction at the quantumlevel for satisfy in orderto obtain the desiredresult is in thispracticalpurposes,for instancein nuclearphysics,in case [121generalwe haveto do with systemsin which the dissi-pativeprocesseshavenot simply a linear dependence yx a_i-~-_f+ 2’yi~= + - (2)on the velocity. So it is reasonableto studysituations ax X

somewhatmore complicatedthan thoseconsideredso The solutionof eq. (2) yields:far. The simplestof thesemore complexcasesis thatof a friction forcedependingquadraticallyon the fX

2~’7 exp(~c/7x+ bx + Ct), (3)speedof the particle.This casecould also be interesting whereb andc arearbitraryconstants.Thenewequa-in the light of the realisticint~rpretationproposedin tion of motion isnowrefs. [9—11]becauseit could haverelativistic implica- b .

/ 2Y’IYexp(—c/yx+ bx + ct)(x + ‘yx ) = 0. (4)The systemwe aregoingto studyis one dimension- Eq. (4) is entirely equivalentto eq.(1) only when b

Volume 77A, number 1 PHYSICS LETTERS 28 April 1980

= 2y, c = 0; otherwisethe situationfor whichi = 0 The resulting Schrodinger equation ismust beexcluded.We want eq.(4) to coincidewith h2e~2Yx(a2~/ax2— 2ya~/ax+ 2y2~i)

(d/dt) aL/ai~— 8L/3x = 0, (5) (11)= ih a~/at,

L being the lagrangianfunction looked for. From thecoincidenceof eq. (4) with eq.(5) we obtain whoseunnormalizablesolution correspondingto an

x eigenvalueof thehamiltonianisL = exp(bx+ Ct) f dz r E(b2Y)I~exp(_c/y~)dE,(6) ~ = e~~J

1(~~/’yh) eYX) + aJ ~

which representsa family of possiblelagrangiansfor x e~c4/7~)t. (12)the problemwe arestudyingandhasessentiallythesameshapeasfor the linear case [11]. From (6) we A is an eigenvalueof thehamiltonianand a is a con-canderivethegeneralexpressionsfor the conjugate stant; .)‘~ is a Bessel functionof imaginaryorder i.

momentump andfor thehamiltonianHofthe system: The Ehrenfesttheoremis indeedsatisfied;weseefor instancethat:

p = exp(bx+ Ct) fz(b_2~)/Yexp(~c/yz)dz, = (i/~)[~ ~ = 2

7H,

(7)x just as in the classicalcase.The wave function (12) is

H= exp(bx+ ct)f z(~7)I7exp(_c/yz)dz. everywheresinglevaluedand finite except forx —~ +°°;

that meansthat the particle,as for theclassicalcase,The lagrangianscorrespondingto differentchoicesof neverstops,whereasthe probability densityof findingthe parametersb andc are indeedequivalentandto it for x —~ ~~oo tendsto zero [13] -

display the generalfeaturesof the behaviourof oursystemwe canselecta simple specialcase,namelythe Referencesone for which b = 2’y and c = 0. With thesevaluesfortheparameters,eqs.(6) and(7) become: [1] P. Caldirola,NuovoCimento 18 (1941) 393.

[21E. Kanai, Prog. Theor. Phys. 3 (1948)440.L = 4~2e

2~, p = ~ e2~, H = ~p2e_2YX. (8) [3] M.D. Kostin,J. Q~em.Phys.57 (1972)3589.

H is evidentlya constantof the motion andthe classi. [41K.K. Kanand J.J. Griffin, Phys. Lett. SOB (1974) 241.[5] K. Albrecht,Phys.Lett. 56B (1975)127.cal coordinateof the particleis givenby: [6] B.S.K. Skagerstam,J. Math. Phys. 18 (1977) 308.

x = ‘y~lg(y\/~7~t). (9) [71M. Razavy,Can. J. Phys. 56 (1978) 311.[8] W. Stocker and K. Albrecht, Ann. Phys. 117 (1979)436.

The systemcannow be canonically quantized just [9] J.R. Ray, Lett. Nuovo Cimento 25 (1979)47;26(1979)64.

by substitutingx —~x and—iFi a/ax—~p; however,the [10] V.V. Dodonovand V.1. Man’ko, Phys.Rev. 20A (1979)problemarisesof the order of the factorsin the 550

hamiltonian,since~ and~3do not commute.We can [11] A. Tartaglia, to be published.go aroundthis difficulty by introducingthesymme- [12] P. Havas,NuovoCimento 5 (1957) Suppi. 363.

trized Hamiltonoperator [13] E. Lommel, Math. Ann. III (1871)481—486.

JJ ~(~2~2’YX +e2’Y~2)

(10)= ..!J~2e_27X(&2/8x2— 2y a/ax+ 272).

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