8/8/2019 On the White Noise Path Integration Approach to the Damped Harmonic Oscillator

1/3

Proceedings of the 11th SPVM National Physics Conference and Workshop

c 2009 SPVM. All rights reserved. ISSN 1655-4620. http://physics.msuiit.edu.ph/spvm

13

On the White Noise Path Integration Approach to the

Damped Harmonic Oscillator

Ryan John A. Cuberoa and Jinky B. Bornalesa.

aPhysics Department, Mindanao State University - Iligan Institute of Technology,A. Bonifacio Avenue, Tibanga, 9200 Iligan City, Philippines

The propagator for the damped harmonic oscillator whose motion is described by the Caldirola-KanaiLagrangian, L(x ,x,t) = 1

2(exp(t)x2exp(t)2x2), is calculated by reformulating the path integral in terms

of the Brownian functionals and employing the S- and T- transforms in white noise analysis to evaluate thepath integral.

Dissipative systems occur frequently in na-ture. In the Newtonian formulation, dissipativeforces are not so difficult to account for in aphenomenological manner - the Stokes linearfrictional force proportional to the velocity v,Diracs radiation damping v, and the viscousforce 2v [5] to name a few.

In the microscopic realm, dissipation can bepointed out by considering the behavior of elec-trons in an insulating media under the influenceof a large external field. These electrons havethe tendency to transfer appreciable amountsof energy locally to the lattices through thecoupling with optical phonons. Moreover, itwas found out experimentally that the opticalphonons were capable of dissipating the elec-

trons energy, as cited[4]

. Furthermore, the en-ergy dissipated by the actual oscillator of inter-est is absorbed at the same pace by an artificialdevice that takes place in the physical reservoirto which this system is coupled [5]. Unlike inclassical mechanics, accounting for the dissipa-tive forces in quantum systems are not that easyto do.

The dissipative and irreversible behavior ofthe majority of physical phenomena have deeplytroubled science with an intriguing yet enduringproblem. It is because of this that a growing in-terest arose about the problems of dissipation in

quantum mechanics. Consequently, more andmore have shown interest in providing a consis-tent description of these kinds of systems whichhas found many applications especially in quan-tum optics and nuclear physics.

Being the simplest model for the dissipativesystem, it is equally important to provide a

Email address: cubero.ryan@yahoo.comEmail address: jbornales@yahoo.com

quantum description of the damped harmonicoscillator. One of the ways of providing a de-scription of the entire dynamics of this systemis by obtaining the quantum mechanical propa-gator. This nonrelativistic mathematical objectis often referred to as the transition probabilityamplitude which propagates the wave functionfrom a starting point x0 at time t0 to the finalpoint x1 at the final time t1

[8]. Carried by thepropagator are important informations of a par-ticle such as the wave function and the energyspectrum which, in turn, describes the particleconsidered.

In this paper, we shall evaluate the quantumpropagator of a damped harmonic oscillator us-ing the white noise path integral approach. It

should be noted that the interpretation diffi-culties of the Caldirola-Kanai Lagrangian aredisregarded, that is, the Caldirola-Kanai La-grangian is assumed as a passable descriptionto the system mentioned. Also, from this pointthroughout, the m and h are set to unity forconvenience in the calculation.

1. The Caldirola-Kanai Model:

Damped Harmonic Oscillator

The study of dissipative quantum systems,especially the damped harmonic oscillator, has

a rather long history. About half a century ago,Kanai adopted a Hamiltonian,

H=p2

2exp(t) + 1

2exp(t)2x2, (1)

where p is the momentum, x is the coordinate, is the constant damping coefficient, and is theangular frequency. Through the Lagrange equa-tion, a canonical transformation of the Hamil-

8/8/2019 On the White Noise Path Integration Approach to the Damped Harmonic Oscillator

2/3

On the White Noise Path Integration Approach to the Damped Harmonic Oscillator 14

tonian gives us the Lagrangian

L =1

2(exp(t)x2 exp(t)2x2). (2)

Eq. (2) is the Lagrangian of a damped har-monic oscillator and this is usually referred to as

the Caldirola-Kanai Model. This was first pro-posed by Caldirola in 1941 and subsequently,Kanai in 1948 in order to study the quantummechanics of damped harmonic oscillators.

2. Evaluation of the Propagator using

White Noise Path Integration

With Eq. (2), we are now ready to set-upthe Feynman integrand and translate it to thelanguage of white noise calculus for evaluation.Using Eq.(2), the corresponding action for adamped harmonic oscillator is given by

S=

t1

t0

1

2(exp(t)x2 2 exp(t)x2)dt (3)

and the Feynman propagator

K(x, x, t) =N

exp

t1t0

i

2exp(t)x2dt

exp iSV[x]

D[x] (4)

where SV[x] is a function that depends onthe potential V = 12 exp(t)

2x2 and D[x] isthe infinite-dimensional Lebesgue measure. It

should be noted that the integration in theFeynman propagator has long been criticizedfor its lack of mathematical meaning with itsinfinite-dimensional flat measure D[x] whichdoes not exist in infinite dimensional spaces.[8]

To evaluate Eq. (4) in the framework of whitenoise analysis, the paths x(t) of the action mustbe parametrized by introducing trajectories ofx consisting of a sure path y plus a Brownianfluctuation [6]

x() = y() +hm

1/2B() (5)

where B = ()d is the Brownian motionand is the white noise variable.

Thus, the power of the first exponential in (4)can be expressed ast1

t0

i

2exp(t)x2dt =

t1t0

i

2exp(t)2d. (6)

The potential V(x) = 12 exp(t)2x2 thus,

SV[x], depends at most quadratically on Brow-nian motion. By Taylor Series expansion and

setting y in Eq. (5) to zero, SV[x] can be ex-pressed as [6]

SV[x] 12, SV. (7)

where the inner product , d and

SV(1, 2) =hm

t1

12

V(x())d, (8)

where V = exp(t)2 and 1 2 =min(1, 2) = 2 are some arbitrary time be-tween the initial time t0 and the final time t1 ofthe propagator.

With Eq. (6), together with Eq. (7), theexponential of the action in terms of the whitenoise variable, , can now be written as

exp

t1t0

i

2exp(t)x2dt iSV[x]

= expt

1

t0

i2

exp(t)2d i2, SV

.

(9)

Bearing in mind the correspondence betweend() and dx [6]

d() =N exp1

2

()2d

d (10)

whereN is the normalization factor and expo-nential in d() is responsible for the Gaussianfall-off. The Gaussian measure d() is notedas a natural measure to use for infinite dimen-

sional analysis[8]

, unlike the Lebesgue measurewhich only exists in finite-dimensional spaces.

The parametrization of the paths x(t) in Eq.(5) shows that only the initial point x0 is fixedfrom where the random Brownian motion be-gins. Thus, the endpoint needs to be fixed witha Donsker delta function (x(t) x1) such thatat time t1, the particle is at x1.

With Eqs. (9) and (10) and expressing theDonsker delta function in its Fourier represen-tation, the Feynman propagator in the frame-work of white noise analysis is now given as

K(x0, x1, t) =

IDHOd() (11)

where x0 and x1 are the initial and final pointsand IDHO is the white noise functional given by

IDHO =Nexp12 ,(i exp(t) + 1)

exp 12 ,iSV 12

expi

, 1[0,t] x1 d. (12)R.J.A. Cubero, J.B. Bornales October 2009

8/8/2019 On the White Noise Path Integration Approach to the Damped Harmonic Oscillator

3/3

15 On the White Noise Path Integration Approach to the Damped Harmonic Oscillator

with 1[0,t] as a unitary vector.We note that in Eq. (11), the integration over

d() is just the T transform of IDHO .

3. The T transform of the White NoiseFunctional IDHO

We shall now evaluate the T transform ofthe functional IDHO . To do this, we shall firsttake the S transform of IDHO , that is

SIDHO() = C()

IDHO exp(, )d()

=1

2C()N

eiy

e,+i,[0,t]

e 12 ,(i exp(t)1+iSV)dd()= C() 2te, (i exp(t) + iSV)1e

det i exp(t) + iSVi exp(t) 1/2

exp

1

2, (i exp(t) + iSV)1

+1

2e, (i exp(t) + iSV)1e

xt i, (i exp(t) + iSV)1e

2

(13)

where C() = exp(122d) is the charac-

teristic functional which is the Fourier trans-

form of the probability measure on the dis-tribution space of the generalized functions,e = t1/2[0,t] is a unit vector and is an op-erator.

With Eq. (13), the corresponding T trans-form ofIDHO , that is

[9]

TIDHO () = C()SIDHO (i) (14)

can be calculated and the result yields

TIDHO () =

2te, (i exp(t) + iSV)1e

det

i exp(t) + iSV

i exp(t) 1/2

exp

1

2, (i exp(t) + iSV)1

+1

2e, (i exp(t) + iSV)1e

xt i, (i exp(t) + iSV)1e

2.

(15)

We are now left with the evaluation of the in-ner products e, (i exp(t) + iSV)1e and thedeterminant

det

i exp(t)+iSV

i exp(t)

1/2in Eq.

(15). These are the next calculations that wehave to make.

4. Concluding Remarks

In this paper, we have studied the dampedharmonic oscillator in the framework of whitenoise analysis. Using the Lagrangian in Eq. (2),we have set-up the white noise functional IDHOin Eq. (12). By using the S and T trans-forms in white noise analysis, we have shownthat the propagator for the damped harmonicoscillator can be obtained using the white noisepath integral approach.

5. References

1. Cheng, B. K. Exact evaluation of the propagator forthe damped harmonic oscillator. J. Phys. A : MathGen. Vol 17, p2475-2484. (1984).

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

3. Pepore, S., et al. Path Integral for a Harmonic Os-cillator with Time-Dependent Mass and Frequency.ScienceAsia. Vol 32, p173-179. (2006).

4. Thornber, K. Applications of Path Integrals in Dis-sipation. Path Integrals and their Applications inQuantum, Statistical, and Solid State Physics. Eds.Papadopoulos and Devreese. Belgium. p359-382.(1977).

5. Dekker, H. Classical and Quantum Mechanics of theDamped Harmonic Oscillator. Phys. Reports. Vol.

80, p1-110. (1981).6. Hida, T and Streit, L. Generalized Brownian Func-

tionals and the Feynman Integral. Stoc. Process.and their Appl., Vol. 16, p55-69. (1983).

7. Grothaus, M., et. al. The Feynman integral for time-dependent anharmonic oscillator. J. Math. Phys.,Vol. 38, p3278-3299. (1997).

8. Kubo, H.K. White Noise Theory. Handbook of Sto-chastic Analysis and Applications. Eds Kannan andLakshmikantham. p107-158. (2002).

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

R.J.A. Cubero, J.B. Bornales October 2009