on the quantization op the damped harmoni: oscillator

ON THE QUANTIZATION OP THE DAMPED HARMONI: OSCILLATOR

I. A. PEDROSA and B. BALEIA

Universidade Federal da Paraíba

Departamento de Física - CCEN

58.000 - João Pessoa (PB) - Brazil

Abstract

transformation and

to quantize the damped

can reproduce the proper

classical limit by this treatment. The question about the zero-

We present, by using a canonical

coherent states, an alternative treatment

harmonic oscillatori. It is shown that one

point energy is also discussed, (H^-IM Mf-

PACS numbers: 03.20. +i , 03.65. -w.

UFPb-DF 06/84

6e8St) f 3 S

I. INTRODUCTION

The quantization of the damped harmonic oscillator received

in the last few decades widespread attention, as or.2 can verify in some

1 2 excellent review papers published * . At present, there exist

essentially, four methods to achieve this end, namely, the explicitly

3—8 9 10 time-dependent Haràltonian t the method of dual coordinates ' ,

the nonlinear Schrodinger equation ' ' , and the so-called method

of the lóss-reservoir ."" . However, it is about the first method

that we want to dwell upon here.

In the physical literature, different.treatments have been

employed to quantize the damped harmonic oscillator by the first

method. The main purpose of this paper is to exhibit an alternative

treatment to quantize this system. To this end, we use a canonical

transformation and the well known coherent states for the harmonic

oscillator. These rtates became popular during the I960*s for their 18 usefulness in describing the radiation field and now they serve

as a very convenient tool for solving various problems in almost all

fields of physics. Furthermore, with the aid of these states we show

that one can reproduce the proper classical limit by means of such a

treatment. We also discuss the question about the zero-point energy

in the damped harmonic oscillator.

This paper is organized in the following.manner. In Sec. II,

we specify the Hamiltonian and the canonical transformation to be

employed. In Sec. Ill, wc briefly discuss the coherent states for the

harmonic oscillator and study the problem of the damped oscillator.

Sec. IV contains a brief discussion about the quantization of this

dissipative system based on the fisrt method above mentioned.

3

11. EXPLICITLY TlMg-DfcPENUENft* HAMILTONIAN

AND CANONICAL TRANS FORMATION

The nechanical system we are in

by the equation

q • Xq" • e*q » 0

where q is the linear coordinate of the sbsten

positive constants. This equation of aotipn

Lagrangian

L - e (—q - —^— q f,

:erested in is that described

as one can easily verify. The canonical

P Ü V '

where p. * «nq is the usual kinetic nonentjw. The Lagrangian (2.2)

leads to the Kamiltonlan

2» 2

(2.1)

and \ and at are

nay be obtained frost the

(2.2)

nttua associated with q is

(2.3)

J t

The Hamilton equations associated with (2.4) is given by

9 * • = • • r p - - i

(2.4)

T ' (2.5)

4 *

which aro oqutvnlont to (2.1). On tho other hand, in *i»rms of II,

the mechanical energy of the system, i.e., the sum of kinetic and

potential energies

a mw* B - y q * + — q1 (2.6)

is given by

E - e~XtH. (2.7)

8 19 Now, performing the canonical transformation ' given by

the generating function

P(q,P,t) «qPe X t / l - aiq*e X t (2.8)

which yields the change of variables

0 - eXt/*q , (2.9)

p . e - ^ / aP • SLe

X t / lq , C2.10)

we obtain a new Hamiltonian given by

wnere ! » I

I i

P.2 - u2 - X7/4 ! (2.12)

is a positive constant, since we ere only linterested in the case of

2 ur-dercriticaily damped oscillator, i.e., X < 2-x . Here, we mention i

thaz r.he procedure adopted in this section ;is similar to that recently

developed by G^yl". In particular, this autjhor also used the canonical

transformation determined by Í2.S) and shoved that it can be implemented i

as 2 unitary change of representation. I i

To finish tnis e<:tion we should prenark that, according to i

Iter. 5 , so lv ing Í2.5) i s e q u i v a l e n t t o s o l v i n g t h e Harailtonian p a i r

s s s c c i a t e c with (2..11} , and with t h e use oft (2 .9) and (2.10} o b t a i n i n g

the s o l u t i o n to ( 2 . 5 ) . ; '

I I I . QUANTUH-MECfiAIÍICAL SOLUTÜOK

A. Coherent s t a t e s of a s i n g l e o s c i l l a t o r

T.i tka quantum c a s e , q and p a r e t a k e n a s o p e r a t o r s i

s a t i s f y i n g the commutation r e l a t i o n I

[q,p] » i*n . | (3.1)

|

Í Let us observe that \o.,v\ " fa'p] whi<-n impl ies that the commutation

r e l a t i o n s remain the same in bcth coord inates .

We now cons ide r t he Hantiltonian

H « ^ - P 2 + ^ 1 Q 2 (3.2) * 2JB 2

which i s formal ly equal t c the Hair.iltonian of t he harmonic o s c i l i s t o :

Then, by us ing t h e a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s

i - J 2TÍmí2 j (mílQ + i p ) , ( 3 . 3 )

i+* fa ina] 2

(mfi Q - i p ) , (3 .4)

we can r e w r i t e (3.2) as

H, » £ f t ( 3 + a + - ) . (3.5)

The operators a and a have the properties

[a,a4] - 1 , (3.6)

a l n > j = n ' | n - l > , (J.7)

a+|n>» fn + l)'/2|n + 1> , (3.8)

(a+)n

n> m - J — 1 — I o > , • (3.S)

< n i ) , / J

where in> are the number states and i 0 > is the oscillator ground

state.

13 20 The so-called coherent states can be defined ' as

(a) eigenstates of the annihilation operator, (b) states created by

a particular unitary displacement operator, or Cc) minimum-uncertainty

states. Notwithstanding, for the haimonic oscillator these three

2D different definitions are all equivalent , so that we can adopt- one

of ther as the definition of the harmonic-oscillator coherent states.

In this paper, v/e adopt the definition (a) and present (b) and (c)

as properties of these states.

Define the coherent states of the harmonic oscillator at

t= O as thi eigenstates of the annihilation operator, with

eigenvalue a:

a|a> = ala> , (3.10)

where a = u + i v is a complex number. These states can be written

20 in terms of the number states as

a> » exp(-i-|a!z) 1 ——~- |n> . (3.11) 2 "=0 („!)'/'

Application of <3.9) leads (3.11) to

a> - e x p { - - | a | t > e x p ( a a * ) | 0 > . (3.12)

In what follows, we give 3ome properties of the coherent s ta tes . They

are generated by the unitary displacement operator

8

D(a) = expia a - a*a) (3.13)

acting on the ground state

|a> » D(a) |0 > . (3.14)

That this is corret follows from the Baker-Carapbell-Hausdorff

identity21

exp(A)exp(B) = exp(A+B+-[A,Bj) , (3.15}

valid when A and B commute with their coianiutator f A,3j .

To obtain the time dependence of the coherent states we

note that the eigenenergies for the harmonic oscillator are

> 1 Hl |n > = fin(n + -) | n > .

So, using (3.11) and (3.16), one has

| a , t > - exp( - iHj t / f t ) i a >

(a ) n exp] - i f i t (n + - ) j

(3.16)

1 , .2 * e x p ( « - a Z

2 1 n=0 to!)«/2

exp( - i f i t / 2 ) ja ( t ) > , (3.17)

where

a ft) « a e x p ( - i f i t ) . (3.13)

• Then, t r i v i a l l y , from (3.5) and (3.17) we obtain the expectation t

value of Hj ,

<cs,t | H. | o , t > = ' f i n d a i 1 * T) = < a | H. [ a * . (3.19)

Now, by calculating the uncertainty in 0 and P in the

state joi,t>, one finds

(àQ)s * <Q*> - <Q>2 = -2-2mfi

(3.20)

and

(AP)2 » <P2> - <P> 2 _ W Í > 'h* (3.21)

Thus, from (3.20) and (3.21) we see that the coherent states are

states cf minimum uncertainty, i.e.,

&Q&P » V 2 . (3.22)

Using (3 .17) , we obtain the expectat ion value of Q for

the coherent s t a t s | a , t > ,

< t t , t | Q l a , t > = 2^i«!2

mfí s i n ( Ò t + 4>) , (3.23)

where $ i s a r g ( a ) , a * u + i v . Therefore, with the usual prescription'

•$. - O, |er | •* • in such a way t h a t C 2 ^!« l A R) * A0 ( f i n i t e ) ,

we recover the c l a s s i c a l so lu t ion for the motion of an o s c i l l a t o r

18

I n

with initial amplitude A . Henceforth, for the sake of simplicity,

the prcT.i-ript. ion .lbuvi- iwuliunuil will be refined to a:: clar::.if limit.

8. Damped harmonic oscillator

With the aid of the coherent states presented in part A

of this section and having in mind the comment of the last paragraph

of Sec. II, we now investigate the problem of the damped harnior.ic

oscillator.

Let us begin by calculating the expectation value of the

coordinate q in the state ja,t> . Using (2.9) and (3.17) one obtains

, T / 2

<a,t[eja,t> - e -Xt/a 2Ú\:

m£i sinifit + $) (3.24)

which, in the classic limit, becomes

<o , t | q | o , t > » A e ~ X t / 2 s i n ( G t + <*>) . (3.25)

We recognize (3.25) as the classical solution for the motion of the

damped harmonic oscillator with initial í,inolitucHí A and reduced • o

frequency fl , as given in (2.12).

On the other hand, from (2.10J and (2.17) we obtain the

expectation value of the kinetic momentum p. for the coherent state |a,t >,

< a / t | p k | a , t > » me -xt/2 2 £ U 2 | Vs

{n cos (at + $)

- i- s in ( a t • $ ) } . (3.26)

11

In the classic limit, (3.26) is expressed as i

<a,t;pk!a,t> = m AQ e"X fc/2 f Q cos( Qt + <fr)

i

-ysin(Ot + • ) ! , (3.27)

which i s the c l a s s i c a l value of the k i n e t i c momentum for the damped

harmonic o s c i l l a t o r .

Next:, we c a l c u l a t e the uncertainty in q and p in the s t a t e

| c t , t > . After some c a l c u l a t i o n , one f inds

U q ) 2 = < q 2 > . - < q > 2 = - A _ e " A t (3.28) 2 row

and

(Ap)2 - < p 2 > - < p > 2 » roa>2* e X t . (3.29) 2fí

Thus, from (3.23) and (3.29) we obtain the uncertainty r e l a t i o n

A q A p « JÍ2L , (3.30) 2fl

which modifies the minimum incertainty. This means that the presence

of dissipation causes an increase in the minimum uncertainty relation,

as one would expect from physical grounds. Let us remark that as

X * 0 (3.30)' coincides with (3.22). ;

We can also compute the expectation value of the mechanical

energy in the coherent state |a,t> . From (2.7). and (3.17) one obtains

12

< o , t | E | a . t > = e " X t ( | a | 2 + | > - ^ p • (3.31) f

I t i s i n t e r e s t i ng to not ice t h a t a r e s u l t s imi lar to (3.31) was 4 5

a l t e r n a t i v e l y obtained in the past by Kerner and Stevens axià, ntore Q

recently, by Gzyl . Now, observe that as X •*• 0

< a , t | E | a , t > —• - h u d a ' 2 + - ) (3.32)

as it should. However, <a,t[B|c;,t> • 0 as t + » . This result

causes the system to go ul* iinately to a state of zero energy instead

of going to the quantum ground state.

-IV. DISCUSSIONS

In this paper, we have presented an alternative treatment

to quantize the damped harmonic oscillator based on the Hamiltonian

(2.4). This Hamiltonian, which is generally called the Baterr.an-

3 22 23 Caldirola-Kanai Hamiltonian ' ' , has given rise to controversial question of whether it is able3"8'23'24, or not25"29, to describe

25-29 the damped harmonic oscillator. According to several authors ,

the quantization of this dissipative system through this Hamiltonian

is unsatisfactory because is violates the Heisenberg uncertainty

relations. This criticism is due to the fact that the commutation

relation (3.1) implies for the non-conjugate variable p. (kinetic

momentum) and coordinate q a commutation relation of the type

[q,pjj - iiexp(-Xt) (4.1)

13

from which i t follows

AqAp. £ - ~ e x p ( - A t ) ( 4 . 2 )

Hence, the uncertainty product for the kinetic momentum p. and the

23

coordinate q tends to zero as t + • . As Caldiro'Va observed,

although this result may appear at first sight rather strange, it

cannot be taken as an argument for the incompatibility of the

Kami1tonian (3.1) with quantum mechanics, since the axiomatic

apparatus of quantum theory requires that the uncertainty relations

must hold only for the conjugate canonical variable. Let us recall

that in the Lagrangian and Hamiltonian formalisms the conjugate

^omentum p is, in general, different from the kinetic momentum p. .

The problem to be explored now is that of the zero-point

energy. As we have already pointed out, the expectation value of the

mechanical energy [see eq. (3.31)1 tends to zero as t +,» . In other

words, this result do not predict the correct zero-point energy. 29-31 According to some authors , this disturbing point is due to the

tact that there are a number of Hamiltonians which describe the

damped oscillator and this nonuniqueness of the Hamiltcnian makes

the canonical quantization of the damped oscillator necessarily

questionable. On the other hand, note that the canonical transformation

given by the generating function (2.8) is a'non-inertial transformation. 32

In fact, we have shown that the change of variable (2.9) leads one

to a non-ineitial frame of reference. This feature has not been

taken into account in the literature to quantize the damped harmonic • 33

oscillator through the Hamiltonian (3.1). But it is known that in *

quantum mechanics the non-inertiality of a frame of reference affects

14

somehow the energy of the system. We are investigating this question

and suggost that the problem of the zero-point energy has to do with

the non-inertiality of the canonical transformations used to quantize

the damped narmonic oscillator through the Bateroan-Caldirola-Kanai

Hamiltonian.

ACKNOWLEDGMENTS

Vte thank M. Kyotoku for some discussions on this work.

Also, one of us (B.13.) wishes to thank partial financial support

from Conselho Nacional de Desenvolvimento Científico e Tecnológico,

CNPq, Brazil.

i:>

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