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c 2010 SPVM. All rights reserved. ISSN 1655-4620.http://physics.msuiit.edu.ph/spvm

The construction of the Feynman integrand of the damped harmonic oscillator asa white noise functional

Ryan John A. Cubero and Jinky B. Bornales

Department of Physics, Mindanao State University - Iligan Institute of Technology,Andres Bonifacio Avenue, Tibanga, 9200 Iligan City, Philippines

Abstract

A suitable white noise functional for the damped harmonic oscillator whose motion is described by the Caldirola-KanaiLagrangian, L = 12 m exp( t ) x 2 12 m exp( t ) 2 x 2 for a particle of mass m and frequency subjected to damping of constant , is constructed by employing a parametrization of the paths of the particle in terms of an exponentiallydecreasing Brownian motion thereby reformulating the path integral in the context of white noise analysis.c 2010 SPVM. All rights reserved.

Keywords: Feynman integrand, white noise functional approach, white noise functional, damped harmonic oscillator,parametrization

1. Introduction

One approach in quantizing systems is viaFeynmans path integral approach. [1] In Feynmansapproach, one can describe the dynamics of a quan-

tum system by obtaining its propagator. Mathemati-cally, the expression given by [2]

K (x0 , x 1 ; t0 , t 1) = x 1

x 0 N exp

ih

S D[x ] (1)

where S is the classical action of the particle with N as the normalization and D[x ] as the Lebesguemeasure, gives the quantum mechanical propagator asFeynman [1] had prescribed. Contained in this propa-gator are the wave function and energy eigenvalueswhich, in turn, describes in full detail the dynamicsof the quantum mechanical system. However, theformulation in Equation (1) has long been criticizeddue to its lack of mathematical meaning especiallywith its innite dimensional integration over a nitemeasure D[x ].

To cater this problem, Hida and Streit [3] introducedwhite noise calculus as a novel approach to innitedimensional analysis. In the context of the Hida-Streit formulation of the path integral, [3] the Feynmanintegral in Equation (1) is viewed as a weightedaverage over Brownian paths. [3,4] In this sense, the Corresponding author: cubero.ryan@yahoo.com

(R.J.A. Cubero)

propagator is being dened as an integral over theGaussian white noise measure, d (), which is anatural measure in the innite dimensional space, thatis,

K (x0 , x 1 ; t0 , t 1) = Id () (2)where I , the weighted distribution, is a suitable whitenoise functional which, for a free particle case, is

I = I 0 = N expi + 1

2 t 1

t 02d (x (t) x1) (3)

where N is an appropriate normalization. For thecase where a particle is subjected in an admissiblepotential V (x ),

I = I 0 exp ih V (x ( ))d . (4)

In this paper, we construct the Feynman integrandof the damped harmonic oscillator as a white noisefunctional or the so-called Hida distribution. [4] InSection 2, we rst outline the steps in constructingthe Feynman integrand for a free particle as a Hidadistribution. We then proceed to obtain a suitableHida distribution for a damped harmonic oscillatorby parametrizing the paths of the particle as detailedin Section 3.1 and nally, obtain the expression forthe white noise functional for a damped harmonicoscillator in Section 3.2.

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2. The Free Feynman Integrand

In this section, we briey review the construction of the Feynman integrand for a free particle, describedby the action S 0 = 12 m x2dt , as a Hida distribution.To construct the Feynman integral, the rst andthe most important step that we have to employ isto parametrize the paths, x(t ), of the particle byintroducing a Brownian uctuation. Following theworks of Hida and Streit, [3] we model the paths as

x (t) = x0 + hm B (t) (5)where B = (t)dt is the Brownian motion and is the white noise variable. With the parametrizationin Equation (5), we can now express the exponen-tial function of Feynman propagator in Equation(1) for a free particle as exp( i2 2dt ). Sincethe parametrization in Equation (5) only xes thestarting point, we affix the Donskers delta function,(x (t) x1), to pin the particles endpoint. Takingthe correspondence between the Gaussian measure,d (), and the Lebesgue measure, d x ,[3]

d x exp12 ( )2d d (), (6)

where the exponential function in d () compensatesthe Gaussian fall-off, we can nally express thepropagator for the free particle as

K = I 0d () (7)where I 0 is the white noise functional for a free particleas given in Equation (3).

3. The Feynman Integrand for the DampedHarmonic Oscillator

With the steps outlined in Section 2, we shall nowconstruct the Feynman integrand for the case of adamped harmonic oscillator. In this study, we adoptthe Caldirola-Kanai Lagrangian [5,6]

L =12

m exp( t )x2 12

exp( t ) 2x2 (8)

of a particle with a time dependent mass m (t ) =m exp( t ) where m is the real mass and frequency subjected to a damping of constant , to describe themotion of the quantum mechanical damped harmonicoscillator. ( Note that to describe damped oscillationsin quantum mechanics, one introduces the variablemass, m (t) = m exp( t ), to conserve the uncertainty relation. [7]) With the Lagrangian in Equation (8), theequation of motion, then, reads as

x = x 2x > 0, R , (9)

whose solution is shown in Figure 1.Using the Lagrangian in Equation (8), we now

proceed with the construction of the white noisefunctional for the damped harmonic oscillator.

Figure 1. The solution of the damped harmonic oscillator

for = 2 .0 rad/s and = 0 .5 /s.

3.1. The Parametrization of the Paths of theDamped Harmonic Oscillator

In Section 1, we modeled the paths of the particlein Equation (5) as

x (t) = x0 + hm B (t).This parametrization has been applied to solve thefree particle, [4] the harmonic oscillator, [3] the particlein a uniform magnetic eld [8] among others. However,

for the case of the damped harmonic oscillator, weparametrize the paths of the particle in a differentmanner. Since the kinetic energy of the dampedharmonic oscillator contains a time-dependent mass,m (t ) = m exp( t ), we parametrize the paths of theparticle, x (t), as

x (t) = x0 + hm exp( t ) B (t)= x0 + hm exp t2 B (t). (10)

Note that in Equation (10), the exponentially-decreasing Brownian uctuation manifests the dam-ping occurrences as observed in Figure 1. Thus,

Equation (10) is a sensible parametrization.3.2. The White Noise Functional of the

Damped Harmonic OscillatorWith the parametrization in Equation (10), the

exponential function in the Feynman propagatorEquation (1) yields

expih

S = expi2

t 1

02dt

i 2

t 1

0B

dBdt

dt

exp i2

2 t 1

0B 2dt (11)

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15 R.J.A. Cubero and J.B. Bornales / Proceedings of the 12th SPVM National Physics Conference

where 2 = 2 2

4 is the reduced or dampedfrequency of the particle.

The second term can be evaluated via integrationby parts [9]

B dBdt dt = ddt (B 2)dt dBdt Bdt (12)which then yields

i 2

t 1

0B

dBdt

dt =i 4

B 2(t 1)

=im 4h

exp( t 1)x2(t 1). (13)

The third term in the exponential in Equation (11)depends at most quadratically on Brownian motion.By Taylor Series expansion with x0 = 0, S V [x ] =

m exp( t ) 2 x2dt can be written as [3]S V [x ]

12

, S V . (14)

where the inner product , dt andS V [x ] =

t 1

hm exp(t )

V (0)dt, (15)

where V = m exp( t ) 2 and is some arbitrary timebetween the initial time t0 and the nal time t 1 of thepropagator.

Incorporating Eqs. (13) and (14) into Equation (11)results to

exp 12 , (i + 1) 12 ,

ih S V [x ]

exp im4h x21 exp( t 1) . (16)

Bearing in mind the correspondence between d ()and d x in Equation (6)

d x exp12 ( )2d d ()

and affixing the Donskers delta function (x (t ) x1)to the exponential in Equation (16), the Feynmanpropagator for a damped harmonic oscillator in theframework of white noise analysis is now given as

K = N S m exp( t )h exp im4h x21 exp(t 1) exp 12 , (i + 1)

12 ,

ih S V [x ]

B (t) m exp( t )h x1 d () .(17)where x0 and x1 are the initial and nal points.From Equation (17), the white noise functional fora damped harmonic oscillator is given by

I = m exp( t )h exp im4h x21 exp( t 1) exp 12 , (i + 1)

12 ,

ih S V [x ]

B (t ) m exp( t )h x1 . (18)

We note that for the case where = 0, Equation(18) reduces to a white noise functional of a simpleharmonic oscillator. [3] Furthermore, for the case where = 0 and = 0, Equation (18) reduces to the whitenoise functional for a free particle as given in Equation(3).

4. Conclusion

In this paper, we construct the Feynman integrandfor the damped harmonic oscillator as a white noisefunctional. Using the parametrization,

x (t ) = x0 + hm exp t4 B (t), (19)we cast the propagator of the damped harmonic oscil-lator in the context of white noise analysis. Now, thepropagator in Equation (17) can be evaluated usingthe T transform [3,4 ,9] of the white noise functionalfor a damped harmonic oscillator in Equation (18).This will then be the focus of our next paper.

References

1. Feynman, R. Space-time approach to non-relativistic quan-tum mechanics . Rev. Mod. Phys. 20 (1948) 367.

2. Grosche, C. and Steiner, F. Handbook of Feynman Path Integrals . Berlin : Springer, 1998.

3. Hida, T and Streit, L. Generalized Brownian Functionalsand the Feynman Integral . Stoc. Process. and their Appl.16 (1983) 55-69.

4. de Faria, M. et. al. The Feynman Integrand as a Hida distribution . J. Math Phys. 32 (1991) 2123-2127.

5. Caldirola, P. Forze non conservative nella meccanica quantistica

. Lett. Nuovo Cimento.18

(1941) 393.6. Kanai, E. On the Quantization of the Dissipative Systems .Prog. Theor. Phys. 3 (1948) 440-442.

7. Greenberger, D. M. A critique of the major approaches todamping in quantum theory . J. Math. Phys. 20 (1979) 762-770.

8. de Falco, D. and Khandekar, D. C. Application of WhiteNoise Calculus to the Computation of the Feynman Path Integral . Stoch. Process. and their Appl. 29 (1988) 257-266.

9. Bernido, C. and Carpio-Bernido, M. V. White NoiseAnalysis and the Feynman Path Integral . Bohol : Proc.of the 3rd Jagna International Workshop. Eds Bernido,Carpio-Bernido and L. Streit. (2002) 55-69.