Physics Letters A 184 (1993) 23-28 PHYSICS LETTERS A North-Holland

Schwinger's method for a harmonic oscillator with a time-dependent frequency

Carlos Far ina Instituto de Ftsica, Universidade Federal do Rio de Janeiro, CEP 21.945-970, Rio de Janeiro, R.J., Brazil

and

Antonio J. Segui-Santonja Departamento de Flsica Te6rica, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received 14 September 1993; accepted for publication 4 November 1993 Communicated by J.P. Vigier

We use Sehwinger's method, which was applied with success in the computation of relativistic Green functions with external electromagnetic fields, to obtain the Feynman propagator for a harmonic oscillator with a time-dependent frequency.

Although Schwinger's method was introduced in the context of relativistic quantum mechanics [ 1 ] (RQM), and since then its use has been restricted practically to relativistic problems, it is also well fitted for non- relativistic quantum mechanics (NRQM). Despite the fact that it appeared four decades ago, this method has shown to be a powerful field theoretical method in recent developments of anionic superconductivity, partic- ularly in the derivation of the renormalized Chern-Simons term [ 2 ]. This method enters in the computation of the tadpole corrected fermion propagator, which happens to be the Green function of a free fermion in a fictitious magnetic field with a chemical potential/z. In fact, Lykken, Sonnenschein and Weiss [2 ] showed that the vanishing of OR (the renormalized Chern-Simons coefficient) leads, under certain assumptions, to su- perfluidity for neutral anions, and to superconductivity for charged anions.

The purpose of this paper is to apply this method in an important non-relativistic problem with a time- dependent Hamiltonian operator, that of a harmonic oscillator with a time-dependent frequency. This problem was solved by Khandekar and Lawande [ 3 ] and since then it has been considered by other authors in different contexts [4-7 ]. In fact, it is related to many interesting problems. For instance, it may appear in cosmology if one considers a three-dimensional isotropic harmonic oscillator with a constant frequency COo in a spatially flat universe such that gij=R(t)J~j, with R(t) being the scale factor at time t. It was shown by Lemos and Natividade [ 5 ] that for this metric the last problem turns out to be that of a harmonic oscillator with a time- dependent frequency CO(t) such that c o 2 ( t ) = c o 2 _ ~ ( t ) / R ( t ) .

Since Schwinger's method is rarely used in NRQM (as far as we know, this method was applied only once for non-relativistic problems, see ref. [ 8 ] ), we shall make a brief introduction of the method directly in the context of NRQM.

Consider the non-relativistic Feynman propagator defined as

K(x, x'; T) =0(T) (x lexp [ - ( i / f i )~T] Ix ' ) , ( 1 )

where 0(~) is the step function and for the moment we are considering a time-independent Hamiltonian ~ = ogC(X, P). In our notation the kets Ix) and Ix ' ) are eigenkets of the position operator X (in the Schr6-

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Volume 184, number I PHYSICS LETTERS A 27 December 1993

dinger picture) with eigenvalues x and x' respectively. For r> 0 we have from (1) that

ihO~K(x, x'; z)= ( xl ~ e x p [ - ( i / h ) ~ z ] ix ' ) .

Now, inserting in the r.h.s, of this expression the unity I = exp [ - ( i /h) ~ z ] exp [ ( i /h) ~¢t~r ] and using the well known relation between operators in the Heisenberg and Schr6dinger pictures, we get the equation for the Feynman propagator in the Heisenberg picture,

ihO~K(x, x'; r) = (x(z) [~(X(z), P(r) ) Ix'(0) ) , (2)

where we also used the relation between Heisenberg and Schr~Sdinger bras and kets. Observe that Ix(r) ) and Ix' (0 ) ) are respectively eigenvectors of the Heisenberg operators X(z) and X(0) with the corresponding ei- genvalues x and x'. Besides, X(v) and P(T) satisfy the following Heisenberg equations,

ih-~--~- ( z )= [X(~), o~], ih (z) = [P(z), o~] . (3)

Now we are ready to state the essential idea of Schwinger's method. The starting point is eq. (2). The method, then, consists of the following steps:

(i) Firstly, we solve the Heisenberg equations for X(r) and P(~), and write the solution for P(z) only in terms of the operators X(~) and X(0).

(ii) As a second step, we substitute the previous solutions into the expression for o~f(X(~), P(~) ) and, using the commutator [X(0), X(~)], we write o~f with the operators ordered such that all operators X(r) stay on the left, while operators X(0) stay on the right of each term of ~ .

(iii) With such an operator ordering, and observing that in the Heisenberg picture the propagator is written as K(x, x'; r) = (x ( z ) Ix ' ( 0 ) ), eq. (2) can be readily cast into the form

ihO~K(x, x'; z) =F(x, x'; z)K(x, x'; z), (4)

with F(x, x'; 3) being an ordinary function (and not a quantum operator) defined as

F(x, x'; ~) = (x(z) I ~e(X(,) , P(z) ) Ix'(0) > (5) <x(r) Ix'(0)>

Hence, integrating in ~ the Feynman propagator takes the form

K(x,x';'c)=C(x,x') exp - ~ F(x,x'; r ') dr' , (6)

where C(x, x') is an integration constant independent of z. (iv) The last step, of course, is concerned with the evaluation of C(x, x'). This is possible after we impose

the correct relations between momentum and position operators on the extremes, that is,

-ihOx (x(z) Ix'(0) ) = ( x ( z ) I e ( r ) Ix'(0) ) , (7a)

ihOx, (x(~) Ix'(0) ) = (x(z) IP(0) Ix'(0) ) , (7b)

as well as the initial condition {K(x, x'; T)}~o+ ~6(x -x ' ) . We should remark here that to impose conditions (7a) and (7b) means to substitute in the l.h.s, of these

equations the expression for ( x ( r ) I x ' ( 0 ) ) given by (6), while for the r.h.s, we must write P (Q and P(0) respectively in terms of the operators X(T) and X(0) with the correct ordering (which is possible once we have already solved the Heisenberg equations).

For the problem at hand, notice that it is described by a time-dependent Hamiltonian operator (~,~e(z) = ½P2+ ½co2(T)X2), and therefore, the corresponding Feynman propagator (for r> 0) is given by

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Volume 184, number 1 PHYSICS LETTERS A 27 December 1993

i i~rg(t) dt)][x') , (8) K(x,x'; z')= (x[Jr[exp(- ~ o

where ~r is the time ordering operator and (x[ and Ix ' ) are in the Schr6dinger picture. However, despite the time-dependent character of g , the equation obtained for K(x, x'; z), after the application of O, to both sides of (8), is precisely eq. (4) with F(x, x'; ~) given by (5).

Thus, in order to use Schwinger's method to this case, we must follow the same four steps stated above. The first thing to do is to solve the Heisenberg equations for the operators X(z) and P(z) . With this purpose, let us recall that the classical solution for a harmonic oscillator with a time-dependent frequency to (t) can be writ- ten in the form [ 5 ]

xd(t) =S(t){A cos[7(t) ] + B sin [7(t) ]}, (9)

with S(t) and ?(t) satisfying

~ - ~2S +to2S=O , (10a)

pS 2 = C 2 , (10b)

where the dot means differentiation with respect to time and C is a constant independent of the initial conditions. An expression for Pd (t) follows immediately from (9). We can now impose the initial conditions xc~ (0) = Xo

and pc~(0) =Po to determine A and B and then write the classical solution in terms of the initial conditions. After some algebraic manipulations it can be shown that

X¢l(Z) = xof(~) + P o g ( r ) , ( I 1 )

where

f(r)=S(z){acos[?(T)]+csin[7(r)]} , g(v)=S(f){bcos[?(r)]+dsin[7(r)]} , (12)

with a, b, c and d given by

1 so a = So [coS(7o) + (Sogo/C 2) sin(yo) ] , b= - ~-i sin(?o),

1 So c= So [sin(7o) - (Sogo/C 2) cos(7o) ] , d= ~ cos(yo). (13)

Once the Heisenberg equations are formally the same as the classical ones and since for this case there are no ordering problems, the solution for the quantum operators X(z) and P(z) are given respectively by

x(T) = x(0)f(z) +e (0 )g ( r ) , (14a)

P(v) =X(O)f(z) + P ( O ) g ( v ) . (14b)

Using (14a) to eliminate the operator P(O) from (14b), the solution for P(z) takes the form

g(~) P(z) = X ( 0 ) g ( z ) I f ( z ) /g (z ) ] "+X(z) g(v'--'~" (15)

Inserting ( 15 ) and (14a) into the Hamiltonian operator for a harmonic oscillator with a time-dependent fre- quency to(r) , we get

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1 Je(r) = ~ (x~(T){ [g(r)/g(~) ]~ + co2(T)} +x2(0)g2(r){ [f(r) /g(r) 1"} 2

+ 2X(z)X(0)g(z) [f(z)lg(z) 1"+ IX(0), X(T) ]g(z) [f(r)/g(z) 1"), (16)

where we have added and subtracted the operator term X(z)X(O)g(z) [f(z)/g(z) ]" in order to write the Ham- iltonian with the appropriate operator ordering.

Substituting the commutator IX(0), X(v)] =ihg(z) into the last equation, observing that X(z) lx(z) ) = x l x (z)) and X(0 ) I x' (0) ) = x' I x' (0) ), and using eqs. ( 5 ) and ( 6 ), the corresponding Feynman propa- gator takes the form

( i i (x(z')l~f~(X(z'),P(z'),z')'x'(O))dz,) K(x, x'; z) = C(x, x') exp - ~ (x(z ' ) Ix'(0) ) ~r

i i {½x2 [gZ(r')/g2(z) + °92 (z') ] + ½x'Zg2(z'){ [f(z')/g(z') 1"} 2 =exp( -

+xx'g(z') [f(z')/g(z') ] '+ lihg(z')g(z') [f(z ') /g(z ') ]'} d r ' ) . (17)

All the integrals on the r.h.s, of (17) can be done in a straightforward way. Using the results quoted in the appendix, the Feynman propagator becomes

K(x,x';z)=C(x,x') si 7') exPk2-hsX )

i XexP(2fi s in(?-7 ') [ (Tx2+~'xa) cos(?-~,') -2xx'x/~ l ) - (18)

As a final step, we still need to evaluate C(x, x') by imposing conditions (7a) and (7b), as well as the initial condition. From (7b) and (18) it follows that

( ' ) iOx,- ~;x' C(x,x')=O, (19)

whose integration readily yields

C(x, x') = C1 (x) e x p ( - iS' t2 ~ ~ x ) . (20)

Inserting (20) into (18) and using condition (7a) we simply get 0~C1 (x)=0, which means that C~ (x) is a constant independent of x. Imposing then the initial condition we obtain C~ (x)=x/~/2xih. With this value for C~ (x) and eqs. (20) and (18), we finally obtain, after bringing back the mass, that

K(x, x'; \2~ih sin(~-~,')] exp ~-~ S' ]d

im XexP(2~ sin(y_~,,) [ (~xZ+ ?'x a) cos(~,-y')-2xx'x/~']) , (21)

in agreement with previous results [ 3-5 ]. Of course, the propagator for a harmonic oscillator with a constant frequency, say to ( t )= O~o, corresponds

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to a particular case of (21). In fact, for oJ(t)=tOo eq. (10a) admits as a solution S(t)=const, which implies that ~ (t) = COo or, after integration, that ? - y' = too~. Substituting these results into (21 ) we immediately get the well known propagator for a harmonic oscillator with a constant frequency, as expected.

Wc would like to finish this Letter emphasizing that Schwinger's method is actually a powerful technique not only for computing relativistic Green functions, but also for computing non-relativistic propagators as well, as we tried to show here by solving the harmonic oscillator with a time-dependent frequency.

This work was partially supported by Conselho Nacional de Desenvolvimento Cientifico e Tccnol6gico of Brazil (CNPq). One of us (C.F.) would like to thank the Theoretical Physics Department of Zaragoza for hospitality, where part of this work was done.

Appendix

This appendix is included just to make this Letter more self contained. Here, we shall briefly quote the main steps that will lead to eq. ( 18 ) if we start from eq. (17). Looking at (17), we see that in order to compute the integration over z', we need the expressions for g, (f/g) ", g2/g2, etc.

Firstly, observe that from (12) and (13 ) f ( t ) and g( t ) can be written as

S(t) (cos[7'-y(t) f( t) = - - ~ ~ ] - (S'~'/C 2) sin [7(t) - 7 ' ] ) (A.I)

S(t)S' g ( t ) - C--- F- sin[y(t)-7'] , (A.2)

where the prime means evaluation at t = 0 (for instance, 7 '=? (0 ) , etc. ). From the above results we readily get

~(t)S(t)S' ~(t)S' g ( t ) - C2 cos[y(t)-?']+ ~ s i n [ y ( t ) - ? ' ] , (A.3)

S' g(t)g(t) = -~ {S(t)c~(t) sin2 [ ? ( t ) - ? ' ] +S2(t)~(t) sin [?(t) - 7 ' ] cos[?(t) - 7 ' ] } , (A.4)

C2~,(t) (f/g) "= $2 ~ c o s [ ? ( t ) - 7 ' ] . (A.5)

Let us then analyse each term of (17) separately. It is worth mentioning that in the following calculations eq. (10b) will be used repeated times to conveniently eliminate the constant C 2 in favor of ~)(t) and S(t).

Using (A.2) and (A.5), we have

- 2--h [g2(t)/g2(t)+to2(t)] d t = - ~-~ \ S2(t ) +y( t ) c s c 2 [ ? ( t ) - ? ' ] - y c o t [ y ( t ) - ? ' ] dt

T

- -~ {~°(t)/S(t)+~'(t) cot[?(t)-?'l} d t= ~ - ~ x + ~-~ ~x 2 c o t ( ? - ? ' ) , (A.6)

where ? means ?(T), S means S(T), etc. Using (A.2) and (A.5), we get

i a f i ,2., f i ,2., , -~ -~x g2(r){[ f ( t ) /g( t ) l ' }2dt=-~-~x ? jS,(t) c s c 2 D , ( t ) - ? ' l d t = - ~ x ? c o t ( ~ - ? ) . (A.7)

Next, using (A.3) and (A.5), we have

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f ixx'f -ixx' g(t)[f(t)/g(t)]'dt= ~ S ' J {~;(t)~,(t) csc[~,(t)-y']+~,2(t)S(t) c o t [ 7 ( t ) - 7 ' ] c s c [ 7 ( t ) - y ' ] } dt

= - c s c [ y ( t ) - y ' ] - C 2 ~ ' ( t ) ~ c o t [ y ( t ) - 7 ' 1 c sc [y ( t ) - 7 ' 1 dt

T

= - ~ x x ~ / 7 {,,/~'~-~ csc [7 ( t ) - 7'] } de= - ~

where we used that ~(t)R(t)=- ½~(t)S(t)= -~(t)C2/2V/~. Finally, using (A.4) and (A.5) , we get

½ i g(t)g(t) [f(t)/g(t) l" d t = - ½ i {S(t)/S(t) +)(t) c o t [ y ( t ) - 7 ' 1 } dt

= - 5 ~ (ln[S'S(t) l + l n { s i n [ y ( t ) - Y ' ] } ) at - - - ½ l n ( C 2) + ½ l n ( ~ / ~ ) - ½ ln[sin(7-7') ]

so that

T exP(½ f g(t)g(t)tf(t)lg(t)]'dt)=(const)t,f~lsin(7-7')] '/2 . CA.9)

Subst i tut ing ( A . 6 ) - ( A . 9 ) into eq. (17) , one readi ly gets eq. (18) as assumed in the text. I t is worth remem- ber ing that all the above integrals are indefini te , and that the corresponding integrat ion constants were already inc luded in the constant C(x, x'), de te rmined by condi t ions (7) as well as the ini t ial condit ion.

References

[ 1 ] J. Schwinger, Phys. Rev. 82 ( 1951 ) 664. [ 2 ] J.D. Lykken, J. Sonncnschein and N. Weiss, Int. J. Mod. Phys. A 6 ( 1991 ) 5155. [ 3 ] D.C. Khandekar and S.V. Lawande, J. Math. Phys. 16 (1975) 384. [4] B. Berger, Phys. Rev. D 12 (1975) 369. [ 5 ] N.A. Lemos and C.P. Natividade, Nuovo Cimento 99B ( 1987 ), 211B;

C.P. Natividade, Am. J. Phys. 56 (1988) 929. [6] L.S. Brown and L.J. Carson, Phys. Rev. A 20 (1979) 2486;

L,S. Brown, Phys. Rev. A 36 (1987) 2463. [ 7 ] B.R. Holstein, Am. J. Phys. 57 (1989) 714. [8 ] N.J. Morgenstern Horing, H.L. Cui and G. Fiorenza, Phys. Rev. A 34 (1986) 612.

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