Volume 123, number 6 PHYSICS LETTERS A 17 August 1987

THE PROPAGATOR FOR A TIME-DEPENDENT MASS SUBJECT TO A HARMONIC POTENTIAL WITH A TIME-DEPENDENT FREQUENCY

Carlos FARINA DE SOUZA Instituto de Fisica, Universidade Federal do Rio de Janeiro, Cidade Universitdria, 21941 Rio de Janeiro, Brazil

and

Alvaro DE SOUZA DUTRA Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud 150, 22.290 Urca, R J, Brazil

Received 2 March 1987; revised manuscript received 8 June 1987; accepted for publication 9 June 1987 Communicated by A.A. Maradudin

We find the quantum propagator of the harmonic oscillator with a time-dependent mass solving directly the Schr'6dinger equa- tion through a change of variable and a time reparametrization.

The interest in solving problems involving harmonic oscillators with time-dependent frequencies or with time-dependent masses (or both simultaneously) has increased in the last years [1-8] (see also ref. [9] and references therein). One of the reasons for that is the connection among these problems and many others belonging to different areas of physics like quantum chemistry, plasma physics, quantum optics etc. For a spe- cific example, Colegrave and Abdalla [ 10] studied the harmonic oscillator with a constant frequency and a time-dependen~ mass in order to describe the electromagnetic field intensities in a Fabry-P6rot cavity.

In this letter we will be concerned with the harmonic oscillator with both frequency and mass being arbitrary given functions of time. Many approaches have been used to attack this problem. For instance, one could use the ideas developed in recent works of Lawande and Dhara [ 11 ], where they demonstrated the usefulness of the time-dependent invariants (TDIs) of a system in performing the path integral. Specifically, they showed how the Feynman propagator for an explicit time-dependent lagrangian can be obtained from the propagator for an associated time-independent one if one uses a TDI known a priori. Another approach is done in a more recent paper by Nassar et al. [ 8 ]. They made use of the nonlinear superposition law of Ray and Reid [ 12 ] to obtain the space and time transformation which changes the classical actions for the time-dependent quadratic lagrangian into eitber an associated time-independent quadratic lagrangian or that for a free particle. In fact, the transformation of a general quadratic path integral into a free particle one was done first by Junker and Inomata [ 13 ]. Similar ideas were used more recently by Cbeng [ 6 ]. For the transformation from the harmonic oscillator to a free particle the reader is referred to rcfs. [ 16-18 ].

However, we will attack this problem in a different way. We shall solve directly the Schr'6dinger equation through an adequate change of variable and time reparametrization. The essential idea of our method is to introduce two arbitrary functions, say, s(z) and g(z) , where T is the new time parameter, which will permit us to reduce the original Schr'6dinger equation to the equation for the standard harmonic oscillator with con- stant frequency and mass. This will be done through a convenient choice of s(z) and/J(z) . Of course, after the calculations~ we have to come back to the original variables x and t.

The Schr'6dinger equation for our problem is

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Volume 123, number 6 PHYSICS LETTERS A 17 August 1987

1 02 0 2re(t) Ox 2 ~t(x, t) - ½m( t)to2( t)xZ~t( x, t) =i ~ ~t(x, t), (1)

where re(t) and to(t) are given functions of time and we have set h = 1. Let us make the following transformation,

x=s(z).~, (2)

where z is a single-valued function related to t by

t

T(t)= J#(~) d~, (dT( t ) /d t=la( t ) ) . (3)

In order to write the Schr6dinger equation in terms of the new variables ~ and T, we also have to use the changes in the partial derivatives, that is,

0 Or 0 O~ 0 =l~ s 0~] Ot - Ot Or + Ot O~ -~z - ' (4a)

0 Oz 0 Og 0 1 0 + (4b)

Ox - Ox Oz Ox O~ - s 0 ~ '

where the prime denotes differentiation with respect to the parameter z. Using (2) and (4) we write eq. (1) in the form

(.O.S'_O 1 0 2 ) 1/~-~z - tUsX~--~+ 2ms~ OY ~ ½ms2tn2g 2 0(g , z )=0 , (5)

where we defined ¥(x(& T), t(~, 3)) - ~ ( 2 , z). The function 0(x, z) can be regarded as the wavefunction of the original problem written in terms of the new variables. If we find 0(Y, z), all we will have to do is to substitute g = [s ( r ( t ) ) ] - ~x and T(t) to get the solution.

Let us make the ansatz

q~(.g, x) = exp [if(.~, T)] xC~, z). (6)

Substituting (6) into (5) we get

( ) ( s) O 1__ L 02 O z 1 0 f _ u . ~ _ 2 i#~-~z + 2 m s 2 0 ~ ~ --½msZto2X 2 X(X,O+i~-~(X,O m~2 0X

+ l _ O f _ u ~ } Z ( X , z ) = O . (7) r 02s_ (0s) 2 , Lax kox/]+'UsX We will choose f(~, t) in order to make the term proportional to OX(~, z)/O2 vanish. With this choice we have

O f / O ~ = rnltss' X, (8)

which leads to the solution

f ( x , ~) = ½mltss'X z +f~ (~), (9)

where f~ (~) is an arbitrary function of T still to be determined. Inserting (9) into (7) and rearranging terms we get

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0 1 02 ) ilt ~ + 2ms--- 2 OX ~ ½ms20)2pC 2 ,~(2, Z)

- - - t iUs -u-a-~-T + ½ ~ 2 m s ' x 2 - ½ ~ (muss') x 2 Z(x, 0 . (10)

Now, we will try to find f (z) in such a way that the right-hand side of (10) can be written in the following convenient form,

-U--~z +½u2ms'2X2-½U d (mltsS') X2)=-lms272(z)X2, (11)

where 7(z) is a function only of • to be determined later on. Observing that the first two terms of the left-hand side of (1 1 ) depend only on z, our aim can be achieved by setting

df~/dz=½is'/s (12a)

and

72(z) = - + ~s2~zz (mpss'). (12b)

Integration of (12a) leads to

f, (z) = i ln(s'/2), (13)

where we have, appropriately chosen the constant of integration. Having in mind results (12) and (13) we can substitute (11 ) into (10) and write the Schr6dinger equation in the compact form

( . 0 , 0 ~ ) l l t ~ + 2ms~ 0:¢~ - ½ms2(o~ 2 +y2)~2 Z(x, z) =0. (14)

Now, we are ready to choose the arbitrary functions s(z) and p(z) in order to reduce this complicated dif- ferential equation into a much simpler one, with no time-dependent terms. Then let us make

ms21z=Mo;, Mo=const; toz+y2=.(Moto~/ms2)lt, tOo2=const. (15)

Substituting (1.5) into (14) we obtain

(0 ) /t l~-~z + 2Mo Og 2 ½Moo~2g 2 X(g,z)=0, (16)

that is, we factorized the time dependence of all terms. This reduces the original problem to the well-known harmonic oscillator with both mass and frequency constants, given respectively by Mo and O~o 2. Therefore, the desired solution is given by

~(x, t) = {exp[if(x, r)] y(X, z) }x=x/st,) . . . . t,). (17)

However, let us obtain explicitly the propagator K(x, t; xo, to) of our problem, instead of writing the wave- function ~t(x, t). The propagator is simply a special solution of the Schr6dinger equation for t> to, subject to the condition

lira K(x, t; Xo, to) = 8(X-Xo). (18) t ~ l O

It gives us the solution for any arbitrary initial state g(Xo, to), that is,

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7 V ( x , t ) = J K(x,t;Xo, to) V(Xo, to) dxo. (19)

- - o o

Analogously, for the simple harmonic oscillator also holds

X(g, z) = i K,.o.(g, Z;Xo, 30) Z(Xo, Zo) dXo. (20) - - o o

where Kh.o.(g, z; Xo, 30) is the respective propagator. With these results in mind, we may write

K(x, t; Xo, to) = {exp [ if(g, z)l K,.o.(g, 3; Xo, 30) exp[ - i f * (2o, 3o)1 )x=x/s(~) . . . . (,), (21)

wheref*(~o, 3o) is the complex conjugate. Substituting (9) and (13) into (21) we finally obtain

( Motoo ~1/2 [ i (.m~.£ "2 mJo g ~ ) ] K(x,t;xo, to)= \2nihssosin[tOo(r-Zo)]J exp ~ So

×ex (2 sin[too(Z-30)] x=x/~(o . . . . (,)

(22)

where we have brought back h, used the well-known result for Kh.o.(X, z;2o, %) [14] and made the identifi- cation gs' =g(dsldt) dt/dr =J (the dot here means differentiation with respect to t). It is understood in (22) that s and z(t) =f 'g(~)d~ are given by (15) and (12),

It is interesting to note that f~ (T) is imaginary and thus, exp [ if~ (3)] is not a phase, but assumes a real value, contributing to the pre-exponential factor. In other words, this term is exactly the jacobian arising from the change of variable in the path integral measure when one works with Feynman's formalism. In our approach, it appears as a solution of a simple differential equation (see formulae (12a) and (13)). A final comment is that if in (15b) we choose tOo=0, the problem is reduced to the free particle one, and the solution is given simply by putting tOo = 0 in (22). This result is also contained in the beautiful work of Junker and Inomata [13].

The application of variable transformations in the Schr6dinger equation can also be applied to more complex problems, Some cases including time-dependent electromagnetic fields are being studied and will be published elsewhere.

We would like to express our appreciation to the referee for valuable suggestions.

References

[ 1 ] P. Caidirola, Nuovo Cimento B 77 (1983) 241. [2] V.V. Dodonov and V.I. Man'ko, Phys. Rev. A 20 (1979) 550. [3] M.S. AbdaUa, Phys. Rev. A 33 (1983) 2870. [4] R.K. Colgrave and M.S. AbdaUa, J. Phys. A 14 (1981) 2269. [5] R.K. Colgrav¢ and M.S. AbdaUa, J. Phys. A 15 (1982) 1549. [6] B.K. Chcng, Phys. Lett. A 113 (1985) 293. [7] P.G.L. Leach, J. Phys. A 16 (1983) 3261. [8] A.B. Nassar, J.M.F. Bassalo and P.T.S. Alencar, Phys. Lett. A 113 (1986) 365. [9] M.S. Abdalla, Phys. Rev. A 34 (1986) 4598.

[ 10] R.K. Colgrave and M.S. Abdalla, Opt. Acta 28 (1981) 495.

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[ l 1 ] S.V. Lawande and A.K. Dhara, Phys. Lett. A 99 (1983) 353; Phys. Rev. A 30 (1984) 560; J. Phys. A 17 0984) 2423.

[ 12 ] J.R. Ray and T.L. Reid, J. Math. Phys. 22 (1981 ) 91. [ 13 ] G. Junker and A. Inomata, Phys. Lett. A l 10 (1985) 195. [ 14] R.P. Feynman and A.R. Hihbs, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). [ 15 ] V. de Alfaro, S. Fubini and G. Furlan, Nuovo Cimento 34 (1976) 569. [ 16] R. Jackiw, Ann. Phys. (NY) 129 0980) 183. [ 17] P.Y. Cai, A, lnomata and P. Wang, Phys. Len. A 91 (1982) 331.

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