Path integrals and wavepacket evolution for damped mechanical systemsDharmesh Jain, A. Das, and Sayan Kar

Citation: American Journal of Physics 75, 259 (2007); doi: 10.1119/1.2423040 View online: http://dx.doi.org/10.1119/1.2423040 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/75/3?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Understanding the damping of a quantum harmonic oscillator coupled to a two-level system using analogies toclassical friction Am. J. Phys. 80, 810 (2012); 10.1119/1.4735707 Low variance energy estimators for systems of quantum Drude oscillators: Treating harmonic path integrals withlarge separations of time scales J. Chem. Phys. 126, 074104 (2007); 10.1063/1.2424708 Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems J. Chem. Phys. 116, 507 (2002); 10.1063/1.1423936 Path-integral diffusion Monte Carlo: Calculation of observables of many-body systems in the ground state J. Chem. Phys. 110, 6143 (1999); 10.1063/1.478520 Evaluation of coherent-state path integrals in statistical mechanics by matrix multiplication J. Chem. Phys. 108, 1562 (1998); 10.1063/1.475527

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Path integrals and wavepacket evolution for damped mechanical systemsDharmesh Jain,a� A. Das,b� and Sayan Karc�

Department of Physics and Meteorology and CTS, Indian Institute of Technology, Kharagpur 721302, India

�Received 12 June 2006; accepted 22 November 2006�

Damped mechanical systems with various forms of damping are quantized using the path integralformalism. In particular, we obtain the path integral kernel for the linearly damped harmonicoscillator and a particle in a uniform gravitational field with linearly or quadratically dampedmotion. In each case, we study the evolution of Gaussian wave packets and discuss the characteristicfeatures that help us distinguish between different types of damping. For quadratic damping weshow the connection of the action and equation of motion to a zero-dimensional version of a scalarfield theory. We also demonstrate that the equation of motion for quadratic damping can beidentified as a geodesic equation in a fictitious two-dimensional space. © 2007 American Association ofPhysics Teachers.

�DOI: 10.1119/1.2423040�

I. INTRODUCTION

The presence of damping in a mechanical system is anatural occurrence. For example, consider a particle fallingthrough a fluid under gravity. The form of the damping forcedepends on the value of the Reynolds number Re=��v /�,where � is the fluid density, � the viscosity coefficient, � thecharacteristic length scale, and v the speed of the particle inthe fluid. For Re�1 we may assume linear damping and for103�Re�2�105 it is more natural to assume that thedamping is quadratic in the velocity.1 The assumption of anonlinear damping term makes the equation of motion non-linear and more difficult to handle in general. Several classi-cal mechanical systems with linear as well as nonlineardamping are exactly solvable.2

In contrast, the quantum mechanics of damped mechanicalsystems is not as easy to understand. Usually we write downSchrödinger’s equation for a given potential and obtain theenergy eigenvalues and eigenfunctions either exactly or byapproximation methods. This procedure does not work fordamped systems because of either the explicit time depen-dence or the complicated form of the Lagrangian and hencethe Hamiltonian.

The primary goals of this article are to show thatLagrangians can be constructed for simple damped systems,to use these Lagrangians to construct the path integral ker-nels for damped systems, and to study the wavepacket evo-lution using these kernels. Our results supplement the exist-ing literature on exact path integrals for mechanical systems.

Our examples include the damped simple harmonic oscil-lator and the freely falling particle in a uniform gravitationalfield in the presence of linear or quadratic damping. Earlierwork on the quantization of damped systems can be found inRefs. 3–5 using a variety of techniques such as variationalmethods, the Fokker-Planck equation, and canonical quanti-zation. Path integral techniques have been used by severalauthors �see for example, Refs. 6–13�. A comprehensiveanalysis of various aspects of path integrals �with associatedreferences� is given in Ref. 14.

In Sec. II we outline the path integral formalism, whichwe shall use extensively. Then in Sec. III A we consider thedamped harmonic oscillator and discuss the construction ofthe kernel for the under-damped case in detail. As a secondexample, we consider in Sec. III B a particle falling under

gravity in the presence of a linear damping force. In Sec. IV

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we focus on quadratic damping in an analogous way. In allthese systems we study wavepacket evolution and show howthe dispersion of the packet provides us with a way of dis-tinguishing between the magnitude as well as various formsof the damping force. As an aside �and a motivation for thereader who wishes to find a taste of advanced physics froman elementary standpoint� we connect the quadratic dampingscenario with a recently studied field theory. In the samespirit, we also illustrate how quadratically damped motioncan be viewed as a geodesic motion in a fictitious two-dimensional space. In Sec. V we conclude with a summaryof our results.

II. PATH INTEGRAL FORMALISM

Before we begin our discussion of the path integral treat-ment of damped mechanical systems we give some resultsthat we will use in our analyses. For readers interested in thedetails of this formalism there are several good referencesincluding Refs. 14–17.

For a particle propagating from the initial point �xi , ti� tothe final point �xf , tf� the transition amplitude is given by theintegral over all possible paths connecting the initial and thefinal points:

K�xf,tf ;xi,ti� = ��xi,ti�

�xf,tf�

eiS�L�x,x,t��/� Dx , �1�

where S and L denote, respectively, the classical action andthe Lagrangian of the particle. The transition amplitudeK�xf , tf ;xi , ti� is called the propagator. It can be shown thatfor a general quadratic Lagrangian the form of the propaga-tor reduces to16

K�xf,tf ;xi,ti� = ��tf,ti�eiS�L�xcl,xcl,t��/�, �2�

where the factor ��tf , ti� is a function of the initial and thefinal time. The subscript “cl” refers to the classical solutionof the equation of motion.

An important property of the propagator, known as tran-sitivity, is obtained by considering an instant of time t such

that tf � t� ti

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K�xf,tf ;xi,ti� =� K�xf,tf ;x,t�K�x,t;xi,ti� dx . �3�

We will make use of Eqs. �2� and �3� in our subsequentdiscussion.

III. PATH INTEGRAL FORMULATION OFLINEARLY DAMPED SYSTEMS

We now illustrate the method of path integrals outlined inSec. II by applying it to some simple damped mechanicalsystems. We will assume that the motion takes place betweenfixed initial and final points and calculate the kernel for thesystems. The kernel will then be used to study the evolutionof a Gaussian wavepacket.

A. Linearly damped harmonic oscillator

The equation of motion of a linearly damped harmonicoscillator is mx+�x+m0

2x=0, where m is the mass of theparticle, � is the damping coefficient, and 0 is the fre-quency of its oscillations when �=0. The general solution ofthe equation of motion for the over-damped �OD�, criticallydamped �CD�, and under-damped �UD� cases are shown inTable I, where =� /m, �2= ��2 /4�−0

2�=−2, and T= tf

− ti. We use the boundary conditions x�ti�=xi and x�tf�=xf toevaluate the integration constants A and B �see Table I�.

To construct the kernel we first need to know the Lagrang-ian, which is given by

L = � 12mx2 − 1

2m02x2�et. �4�

Note that the Lagrangian is explicitly time dependent. Thereare ways of choosing new coordinates so that the Lagrangianin Eq. �4� becomes time-independent.18 It is easy to checkthat this Lagrangian reproduces the correct classical equationof motion for the damped harmonic oscillator. Several au-thors have looked at the path integral kernel for thisLagrangian.19,20 A reasonably up-to-date review covering

Table I. The general solutions of the equation of motion of a damped harm

Case x�t� A

OD e−t/2�A cosh �t+B sinh �t��xie

ti/2 sinh �tf −

sinh

CD e−t/2�A+Bt� 1T �xitfe

ti/2−x

UD e−t/2�A cos t+B sin t��xie

ti/2 sin tf −

sin

Table II. The action S and the kernel K for the critically and over-damped

Case S �Action�

CDm

2T�xi

2eti +xf2etf −2xixfe

�ti+tf�/2�+m

4

ODm�

2 sinh �T��xi

2eti +xf2etf� cosh �T−2xixfe

�ti

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various aspects is available in Ref. 21. We now write downthe kernel for the under-damped case and then investigate thewavepacket evolution. The results for the other two cases aregiven in Tables II and III.

The classical action is evaluated by substituting the solu-tion for the under-damped case given in Table I into Eq. �4�and integrating it over the time interval �ti , tf�. The result is

S =m

2 sin T��xi

2eti + xf2etf� cos T − 2xixfe

�ti+tf�/2�

+m

4�xi

2eti − xf2etf� . �5�

The effect of damping appears in Eq. �5� through the pres-ence of . In particular, the second term is entirely due todamping effects. It is interesting that if we use scaled coor-dinates xi=xie

ti/2 and xf =xfetf/2, we can rewrite the first

term as a purely simple harmonic oscillator contribution.Because the Lagrangian is quadratic, the kernel is of the

form given in Eq. �2�. We make use of the transitivity of thekernel, that is, Eq. �3�, to calculate ��tf , ti�. After some alge-bra, we find

��tf,ti�

= ��tf,t���t,ti�� 2�i�

met� cos �tf − t�sin �tf − t�

+cos �t − ti�sin �t − ti�

� ,

�6�

which leads to

��tf,ti� =� me�ti+tf�/2

2�i � sin T. �7�

The complete kernel turns out to be

oscillator for different cases. �See the discussion in Sec. III A.�

B

f/2 sinh �ti� 1

sinh �T�xfe

tf/2 cosh �ti−xieti/2 cosh �tf�

/2� 1T �xfe

tf/2−xieti/2�

f/2 sin ti� 1

sin T�xfe

tf/2 cos ti−xieti/2 cos tf�

onic oscillator.

K �Kernel�

eti −xf2etf� �me�ti+tf�/2

2�i�Te�i/��S

m

4�xi

2eti −xf2etf� � m�e�ti+tf�/2

2�i� sinh �Te�i/��S

onic

xfet

�T

ftietf

xfet

T

harm

�xi

2

+tf�/2�+

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K�xf,tf ;xi,ti� =� me�ti+tf�/2

2�i � sin Te�i/��S, �8�

where S is given by Eq. �5�.To determine how a Gaussian wavepacket evolves for this

kernel, we begin with the initial �ti=0� profile of the packet:

�xi,0� = � 1

2��02�1/4

e−�xi − a�2/4�02, �9�

where �02 is the variance of the Gaussian wavepacket, which

is a measure of its width. Without any loss of generality wechoose the wavepacket to be peaked at xi=a at ti=0. Thewavepacket at a later time t is related to the wavepacket atti=0 by

�xf,t� = �−�

�

K�xf,t;xi,0� �xi,0� dxi. �10�

After some simplifications, we find

Table III. The probability distribution �xf , t�2 and the variance �t2 for the

Case �xf , t�2

CD1

�2��t2

exp�−�xf −ae−t/2�1+ �t /2���

2�t2

OD1

�2��t2

exp�−�xf −ae−t/2�cosh �t+ � /2��si

2�t2

Fig. 1. Evolution of the wavepacket in a harmonic oscillator potential w

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�xf,t�2

=1

�2��t2

exp − �xf − ae−t/2�cos t +

2sin t��2

2�t2

�11�

and

�t2 = �0

2e−t��cos t +

2sin t�2

+ � � sin t

2m�02�2� .

�12�

From Eq. �11� we see that at any time t the wavepacket ispeaked at

xf = ae−t/2�cos t +

2sin t� . �13�

The wavepacket evolution is shown in Fig. 1, and thedependence of the standard deviation �t on t and is shownin Fig. 2. From Fig. 1, we notice that the width of the wave-packet pulsates and at various times it becomes less than the

ally �CD� and over-damped �OD� harmonic oscillator.

�t2

�02e−t��1+

t

2 �2

+ � �t

2m�02 �2�

��2 � �02e−t��cosh �t+

2sinh �t�2

+ �� sinh �t

2m��02 �2�

critic

2 �nh �t

ith linear damping for the parameters =0.2, 0=0.5, and a=1.0.

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initial value �0. From Fig. 2 we see that �t shows the samebehavior and after t�11 remains less than �0 for =0.2 and0=0.5 �bold line�. We see similar behavior for �t for differ-ent values of . The oscillations are less prominent forhigher values of . This behavior is seen for the criticallyand over-damped cases. From theoretical considerations, weexpect that the under-damped case exhibits less prominentoscillations as →20. We also note that for =0, �t os-cillates between �0 and �max��0 �as is well known�, but asthe damping coefficient becomes nonzero, the standard de-viation drops below �0 at some time and goes to zero. Thisbehavior coincides with the peak of the wavepacket going tox=0. Thus, we conclude that the damping leads to localiza-tion of the particle around the minimum of the potential atx=0.

The expectation value of x is

�x� = �−�

�

x 2dx = ae−t/2�cos t +

2sin t� . �14�

This result is the same as Eq. �13� and x�t� for the UD case inTable I if A and B are evaluated using the initial conditionsx�0�=a and x�0�=0. That is, the peak of the wavepacket�corresponding to the maximum probability of finding theparticle� follows the classical trajectory as expected.

B. Uniform gravitational field with linear damping

Consider a particle of mass m in a uniform gravitationalfield with a damping force proportional to its speed. Thisdamping is an example of Stokes’ law. The equation of mo-tion of the particle is mx+�x=mg, where � is the dampingcoefficient and g is the acceleration due to gravity. Recallthat there is a terminal velocity, which the particle attainsasymptotically. The general solution of the equation of mo-

Fig. 2. Variation of �t with t and for UD case �Eq. �12�

tion is

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x�t� = A + Be−t +g

t , �15�

where =� /m and A and B are integration constants. For theinitial and final conditions x�ti�=xi and x�tf�=xf, A= ��xie

−tf −xfe−ti�+ �g /��tie

−tf − tfe−ti�� / �e−tf −e−ti�,

and B= ��xf −xi�− �g /��tf − ti�� / �e−tf −e−ti�.The equation of motion �15� can be derived from the La-

grangian

L = � 12mx2 + mgx�et. �16�

For this Lagrangian, the classical action in the time interval�ti , tf� is

S =me�ti+tf�

2�etf − eti��xf − xi −g

�tf − ti��2

+mg

�xfe

tf − xieti� +

mg2

23 �e−ti − e−tf� . �17�

The calculation of the kernel can be done in a way similar tothe damped harmonic oscillator. We obtain

K�tf,xf ;ti,xi� =� me�ti+tf�

2�i � �etf − eti�eiS/�, �18�

where S is given by Eq. �17�.We will now consider the evolution of the wavepacket

given in Eq. �9�. We make use of Eq. �10� and obtain

�xf,t�2 =1

�2��t2

�exp −�xf − a −

g

t +

g

2 �1 − e−t��2

2�t2

�19�

value of is shown just below the corresponding curve.

�. The

and

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�t2 = �0

2�1 + ���1 − e−t�2m�0

2 �2� . �20�

From Eq. �19� we see that the wavepacket is peaked at

xf = a +g

t −

g

2 �1 − e−t� . �21�

The wavepacket evolution is shown in Fig. 3, and thevariation of �t with the damping coefficient and the time t

Fig. 3. Evolution of the wavepacket for motion under gravity

Fig. 4. Variation of �t with t and for motion under gravity with linear da

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is shown in Fig. 4. From Fig. 3 we see that the width of thewavepacket increases initially and then becomes almost con-stant as the exponential part of Eq. �20� decays. From Fig. 4we see that the variance �t exhibits the same generic behav-ior for all values of . However, the time taken to reach anear-constant value of � is different for different valuesand represents the time needed to reach the terminal veloci-ties in the corresponding cases. This behavior of the wave-packet seems to be characteristic of systems involving a ter-

linear damping for the parameters =1.0, a=0.0, and g=9.8.

g �Eq. �20��. The value of is shown just above the corresponding curve.

with

mpin

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minal velocity, although, as we will show, there areinteresting differences for the case of quadratic damping. Theexpectation value of x is

�x� = a +g

t −

g

2 �1 − e−t� . �22�

This result is the same as Eqs. �21� and �15� if the constantsare evaluated using the initial conditions x�0�=a andx�0�=0.

IV. QUADRATIC DAMPING

A. Path integral kernel and wavepacket evolution

We consider a particle moving in a uniform gravitationalfield with a damping force proportional to the square of itsspeed. Much work has been done on the quantization of thisand similar systems with quadratic damping.22–27 The equa-tion of motion with this type of damping is mx+�x2=mg.Note that the equation is time-reversal invariant. The generalsolution is

x�t� =1

�ln �cosh ��gt + A��� + B , �23�

where =� /m and A and B are integration constants. Thechoice of a suitable Lagrangian for this case is interestingbecause there are nonequivalent Lagrangians that give rise tothe same equation of motion. Consider for example theforms

L = �1

2mx2 +

mg

2�e2x, �24�

L = −�1 −

gx2e−x. �25�

To quantize the system we must judiciously choose the formthat can be handled easily despite the fact that differentLagrangians can give rise to nonequivalent quantizations.The Lagrangian in Eq. �25� is not so easy to use because ofthe presence of the square root in the path integral method.Thus, we choose the Lagrangian in Eq. �24�. Despite thepresence of damping, the Lagrangians are not explicitlytime-dependent unlike the damped harmonic oscillator or aparticle in a gravitational field with linear damping. TheHamiltonian derived from Eq. �24� is a conserved quantity,but does not correspond to the energy of the system. A dis-cussion on the conserved quantities in damped systems isgiven in Ref. 32.

We note that although the Lagrangian �24� is not a qua-dratic Lagrangian, we can make it so by using the transfor-mation: X=�ex dx=ex /.33 This transformation converts itinto a Lagrangian similar to that of a simple harmonic oscil-lator with imaginary frequency whose results are known15,16

or can be deduced from those of Sec. III A by setting =0.We can write the action in terms of X and t as

S = �ti

tf m

2�X2 + gX2� dt . �26�

If we compare Eq. �26� with the harmonic oscillator action

given by

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SHO = �ti

tf m

2�x2 − 2x2� dt , �27�

we obtain = i�g= i�. We use the known results for thepropagator of the harmonic oscillator and obtain the ker-nel in terms of X and � as

K�Xf,tf ;Xi,ti� =� m�

2�i � sinh �Texp � im�

2 � sinh �T

���Xi2 + Xf

2� cosh �T − 2XiXf�� . �28�

If we transform back to x, we obtain the desired kernel:

K�xf,tf ;xi,ti� =� m�

2�i � sinh �Texp � im�

2 � 2 sinh �T

���e2xi + e2xf� cosh �T − 2e�xi+xf��� . �29�

The evolution of a Gaussian wavepacket in the X-coordinatewill be similar to Eq. �9� and �11� �after setting =0�. There-fore, in terms of the x-coordinate we can write

�xi,0� = � 1

2��02�1/4

exp �− �exi − ea�2

42�02 � , �30�

and

�xf,t�2 =1

�2��t2

exp �− �exf − ea cosh �t�2

22�t2 � , �31�

�t2 = �0

2�cosh2 �t + �� sinh �t

2m��02 �2� . �32�

If we set the exponent in Eq. �31� to zero, we see that thewavepacket is peaked at

xf = a +1

ln �cosh �t� . �33�

The wavepacket evolution is shown in Fig. 5, and thevariation of �t with time t and is shown in Fig. 6. FromFig. 5 we see that the width of the wavepacket increasesindefinitely and rapidly. From Fig. 6 we see that the standarddeviation �t exhibits similar behavior for all values of thedamping coefficient . The only difference is the rate atwhich �t grows, which can be derived from Eq. �32�. Thusfor motion under gravity, the linear and quadratic dampingcases can be distinguished from each other by following thecorresponding wavepacket evolution �see Fig. 4 for lineardamping and Fig. 6 for quadratic damping�.

As before, we would like to calculate the expectationvalue of x. We first calculate �X�, which can be determinedfrom Eq. �14� by setting =0. By using the relation X=ex / we find

�ex� = ea cosh �t . �34�

If we expand the exponential on both sides and compare thecoefficients of , we obtain

�x� = a +1

ln �cosh �t� . �35�

This result is the same as Eqs. �33� and �23�.

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B. Connection with field theory

In the following we will show that the classical equationof motion for a particle subject to damping proportional tothe square of the velocity can be obtained from a zero-dimensional version of the field theory of tachyon matter thatemerges from string theory.28 This example provides a linkbetween a field theory and a damped mechanical system.Recall other such connections such as that between thesimple harmonic oscillator and the massive Klein-Gordonfield theory.

The action for the tachyon29,30 matter field in a�p+1�-dimensional spacetime is given as31

Fig. 5. Evolution of wavepacket for motion under grav

Fig. 6. Variation of �t with t and for motion under gravity with quadratic damp

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S = −� dp+1xV�T��1 + �ij�iT� jT , �36�

where �ij is the metric for a �p+1�-dimensional flat space-time with components �ij �diag�−1,1 , . . . � for i , j=0, . . . , p,T�x� is the tachyon field, and V�T��e−�T/2 denotes thecorresponding field potential. The value of the parameter� depends on the particular type of string theory of inter-est.

We now consider the form of the action for a zero-spacedimensional case, that is, p=0. We identify the tachyon fieldwith the coordinate x �T→x� in the classical problem. Note

ith quadratic damping with =1.0, a=0.0, and g=9.8.

ity w

ing �Eq. �32��. The value of is shown just above the corresponding curve.

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that in classical mechanics the action has the form S=�L dtand the zero-dimensional action has a similar form,

S = −� dte−�x�1 − x2, �37�

where the factor of 2 in the potential V�x� has been absorbedin �. The equation of motion that results from the action inEq. �37� is

x + �x2 = � . �38�

We now scale x→bx and compare Eq. �38� with the equa-tion of motion x+x2=g. We obtain �b= and � /b=g,which can be solved to yield �=�g and b=� /g. Wesubstitute these results into Eq. �37� and recover the La-grangian in Eq. �25�. Thus, we obtain a quadraticallydamped mechanical system out of a field theory.

C. Damping as geodesic motion

Another way of looking at the problem of quadraticallydamped motion is to picture it as the motion of a particlealong a geodesic in a fictitious two-dimensional space. Con-sider the following form of a two-dimensional distance func-tion �line element�

ds2 = f�x�dx2 + h�x�dy2, �39�

where y denotes the fictitious dimension and f�x� and h�x�are unknown functions. Our aim is to show that for an ap-propriate choice of f�x� and h�x�, the equation of motion fora quadratically damped system can be identified as a geode-sic equation. For the metric g���diag�f�x� ,h�x��, we calcu-late the nonzero components of the Christoffel symbol:35

�xxx =

f�

2f, �yy

x = −h�

2f, �xy

y =h�

2h= �yx

y , �40�

where the prime denotes the derivative of the function withrespect to x. If we substitute these components into the wellknown geodesic equation given as

d2x�

d�2 + ���� dx�

d�

dx�

d�= 0 �41�

�here � is any parameter on the geodesic, which in our caseis the time t�, we obtain the equation of motion along the twodirections:

x +f�

2fx2 −

h�

2fy2 = 0, �42�

y +h�

hxy = 0. �43�

If we integrate Eq. �43� once, we have

y =C

h, �44�

where C is an integration constant. We next substitute y in

Eq. �42� and obtain

266 Am. J. Phys., Vol. 75, No. 3, March 2007

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209.213.24.107 On: Tue, 0

x +f�

2fx2 −

C2h�

2fh2 = 0, �45�

which is the same equation as the quadratically dampedequation of motion along the x direction provided thatf� /2f = and C2h� /2fh2=g. These equations can be solvedto reveal the form of the two functions:

f�x� = e2x �46�

and

h�x� = −C2

ge−2x. �47�

It is now straightforward to calculate the components ofthe the Riemann tensor, R���

� ,36 the Ricci tensor, R��=R���� ,

and the Ricci scalar, R=g��R��:34

Rxx = − 22, Ryy =2C23

ge−2x, R = − 42e−2x.

�48�

The presence of in the curvature scalar R implies that thedamping can be viewed as a curvature effect in this fictitioustwo-dimensional space. What distinguishes this case fromthe motion with linear damping discussed in Sec. III B is thatwe cannot cast the equation of motion of the linear dampingcase in the form of a geodesic equation due to the absence ofa x2 term. In this sense, quadratic damping is unique. Thusthe above connection provides us with additional geometricinsight into the nature of quadratically damped motion.

V. CONCLUDING REMARKS

We have shown how to construct kernels for severaldamped mechanical systems and studied the evolution of aGaussian wavepacket in each case. We demonstrated that forthe linearly damped harmonic oscillator and a particle in auniform gravitational field with linear and quadratic damp-ing, we can see characteristic features of the damping fromsnapshots of wavepacket evolution. To motivate our consid-eration of quadratic damping, we related the correspondingequation of motion to a field theory and to geodesic motionin a fictitious two-dimensional space.

Is a quadratically damped system damped? The Lagrang-ian is time independent and the system is Hamiltonian andconservative in the usual sense. The evolution of the wave-packet shows spreading, much like that of a free particle andunlike the linearly damped system. We also note a similaritywith the linearly damped system because in both cases, theparticle attains a terminal speed. These issues suggest that itwould be better to view the quadratically damped system asspecial and unlike the linearly damped case.

ACKNOWLEDGMENT

The work of AD is supported by the Council of Scientificand Industrial Research, Government of India.

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35The components of the Christoffel symbol depend on the metric tensorcomponents as ���

� = 12g�����g��+��g��−��g���.

36The components of Riemann-Christoffel curvature tensor are given byR���

� =������ −�����

� +���� ���

� −���� ���

� . For details of its properties, see

any text on the general theory of relativity.

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