ANOTHER DEFINITION FOR TIME DELAY* H. Narnhofer. Institut für Theoretische Physik Universität Wien

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1 RTS1OO6O0 UWThPh-80-6 ANOTHER DEFINITION FOR TIME DELAY* H. Narnhofer Institut für Theoretische Physik Universität Wien Abstract Time delay is defined by geometrical considerations which work in classical as well as in quantum mechanics, and its connection with the S-matrix and the virial is proven for potentials with V(x) and xv(x) vanishing as r for r ) Work supported in part by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich, Project Nr

2 1 The idea of time delay as a characteristic!» of a scattering process was given a precise definition by Jauch et al. [l]. It is closely related to the existence of resonances and to the tin«of sojourn, for which upper bounds are found e.g. in [2] (though here generalized to long range potentials). The notion of time delay appears already in classical scattering. Here it can be defined easily by essentially geometrical considerations. Its connection with the virial and with the scattering matrix can be shown in the same way in classical and in quantum mechanics. He will not worry about domain problems though we are dealing with unbounded operators. The existence of the relevant quantities follows from estimates in [3] and references therein, if the potential V(x) and jxvv(x)j decreases faster than 1/r and therefore its square root H - and H smooth, so that o the relevant quantities in the virial are integrabel in time. Geometrical Considerations The existence of scattering theory implies the existence of lim t-*±» p(t) p lim [x(t) - p(t)t] = lim x(t) - x ± t-+± t-^i"* Then x x - x_ is space delay in comparison to free time evolution. This operator has the disadvantage that it depends not only on the path but changes under time translation {x(0),p(o)} + {x(z),p(z)> X ± * X ± + P i Z ' X R * X R + ( P + * P -, Z * This ambiguity can be removed if we take only the part of x parallel to p +. Then we lose the vector property and it is more natural to speak of time delay (we put m * 1)

3 »»._ $. T *+ P + *- P - This time delay still does not only depend on the path of the particle but also on the choice of the origin in space and changes accordingly to P + " *-> a ( x-*x + a, T -» T + But since neither the interaction hamiltonian nor the S-matrix is invariant under space translation this ambiguity is to be expected. He can compare our definition of time delay with the one given in [1]. Consider a sequence of balls S with radius R and center at x «0. Take T. to be the difference between the time a particle spends in the ball S, if it starts at t» - with the same initial condition and moves either freely or accordingly to the interaction hamiltonian. Then the limit T_ exists and coincides with the one of our definition. Suppose V 0 for x > R. If the orbit enters the ball at t - T. and leaves it at T. we have x_ - x(-tj) + Tjp_, x + - x(t 2 ) - T ^, from which we obtain, as can be seen in fig. 1, the time the particle really spends in the ball equals / R 2_ b 2 _ x + p + /R2-S 2 + x_p_ [> + T- I 1 + -I 1 " fpj pj A free particle would spend 2 /R 2 -a 2 /p and the difference becomes in the limit R -v v* - x - p - p * We can already see the connection with the S-matrix which transforms x_ into x + [4]: For a central force S(,L) becomes for r + the matrix S(p 2 /2,L) exp(2i«(p r/ L) that <r,6> * (r - 2 f!", 6-2 f)

4 is a canonical transformation with 86/3p the change in r. Connection of T With the Scattering Matrix It was shown in [5] that for potentials vanishing sufficiently at -4-e infinity (as r ) x_ converges and becomes K w lim T - - is _ 1 /de 6(H -E). K ' 0 «E The proof is rather involved and the restriction for the potential is too strong. We will define the time delay by 1... "ih t ih t, _..,.. ^.i iht 0-0 -ihti.^ T * -T [lim <$ e e De e ^> - t-*» ' t^ -i f l t -ih t._. ± I -iht 0 - O iht I ±, - lim <$ e e De e $>] with D» j {aqp + px}, D-~{j-D + D^->andB- (x(t)p(t) + p(t)x(t)}. u 0 O Then T = \ [lim <(j) - S + D* - 1*> - lim <$ - D + D* - 1*>] - t-*» p 2 (t) p2 ( t ) t^_ p2 ( t ) p 2 ( t ) - ~ <* ß + 5 es - n_ 5 &\+> - { ^Js' 1 5 s - 5 <> in > - " 4 < *ij S ~ 1 H" td ' S] + S " 1 [D ' Sl i^v 0 0 taking into account that [H,S] = 0. Now we write o S - / de 5(H -E) S(E), ' 0 with S(E) being a function of the angles only such that [S(E),D] 0. Therefore [D,S] «-i jj /db 3(o" 2 H o -E)S(E)[ < i /db H fl «(H^E) -2 pl

5 4 and T - i f 1 / « * ^ 3E It should be noted that we have to choose 4 c p»»h» a > O, so that,. E P.H. Then we replace 1/p in (a,«) 2 by l/(p 2 +e) and take finally the m (a,») limit E -* 0. Connection With the Virlal The connection between time delay and the virial was already observed in [6] though here restricted to a central field and expressing the relevant terms by the phase shift and not by the S-matrix. He'want to prove the equivalence in our context. With D as before +T. r _ 1 iht ft3 _, -iht 1, iht _ -iht -iht _ iht. L / dt e [H,D] e» (e De - e De ) = H H _1> = ~(p 2 (T)T + p 2 (-T)T + D(T) - D(-T)). n On the other hand it equals +T +T - ~ d J \ dt. - 1 e iht e iol He -iod e -iht»/dt \ ** *# (p* 2 - ^o.«xvv) e " i H t an ' H ' n -T -T = +T = 4T - / dt i e iht (2V + xvv) e~ i H t. H -T If we take into account that 1/p 2-1/p 2 1/2H and further, that our V is tntegrabel in time aid its derivative with respect to time is bounded, so that p 2 (+T) and p 2 (-T) converge to 2H faster than 1/T, we can conclude +T T = - lim \ J e i H t [~(2V + x V) + (2V + x V) ] e" i H t dt. T-w T Through this expression we see that T exists for potentials such that V ar.d xvv fall off like r" 1 " 6.

6 5 Furthermore one can make conclusions about the sign of the time delay. Consider e.g. a purely repulsive potential: Due to repulsion the path of the particle will become shorter, due to energy conservation the velocity of the particle will become smaller and the virial tells us which effect -v dominates. Take V(r) = c r. Then '-T/*'l«"5l For v = 2, t = 0, the phase shift is independent of the energy. For v < 2, i < 0, so the change in the velocity is the dominant effect, whereas for v > 2 (where scattering theory exists for repulsive potentials) the length of the path is the relevant one. We also see that for these potentials the phase shift is araonotonicfunction of the energy. Time Delay for n Particles Based on the definition of [1] the idea cf time delay was generalized to n particle scattering theory in [10], We will sketch how this generalization can be done for our definition and leads to the same result. We ignore our insufficient knowledge about the existence and completeness of wave operators. As usual let H be the channel Hamiltonians, x, p the coordinates and momenta between the clusters and K the corresponding kinetic energy. Assure we start with a <J>. state. With Q ^ = projection ß i n a iht -ih t on the range of st lim e e a P we write t-*±.. -ih t - a" -iht n 1 r,. ^.i_ iht a D e e Q - T = > lim <* Q, e e a ct+ ~ i H t *V = " i H 3 t iht n u - Q & _ e e w D g e e Qg_l* > where D = -r {x p +px} and D = { D + D }. Then, using a 2 a a a a a K a a K a a i s" s p = p aß ctg ß ß T = V <$ s :! [5 S - S D lit > B 2 L ß in 1 aß 1 a aß aß ß J > ß in a

7 6 Writing S = /db 6(K -E+E ) S Q(E) =/des JE) 6(K - E + E.), aß ' a ot aß ' a ß ß ß E, E the energies of the clusters, we obtain again by partial integration a 6 5 a S aß " S aß 5 ß " " 4 i ' de 6 { \ ' E ) ät" and the whole time delay becomes 3S (E) x --2il8«JdB6(H a -B)-^-. a»s (E) ' The connection with the virial can be found by writing T o r i.n fj^ ih t,_ -, -iht. (.. iht,_ -, -iht, T = I lim <*[i /dt e l 1^ 0 e ^ + i Jdt e l 1 «' 0 «' e ' " a T-x» o -T - D (0) + D.(O)!*>. a ß Unfortunately results on H - and H-smoothness are missing. Also nothing can be said about ".he sign of time delay essentially due to the fact that also in the one particle case results are only available for central forces and for many body theory this has no meaning.

8 » _ "»" Ack now1edgeme n t I want to thank Dr. M. Breitenecker and Dr. P. Gesztesy for making me aware of ref. [6] and [7] and Prof. W. Thirring and Dr. M. Combescure for encouragement and stirnulttting discussions. References [1] J.M. Jauch, B. Misra, K.B. Sinha: Helv. Phys. Acta 45_, 393 (1972) [2] R.B. Lavine: Constructive Estimates in Quantum Scattering, Preprint (1076) [3] M. Reed, B. Simon: Methods of Mathematical Phyrics III, Academic Press, New York (1978) [4] w. Thirring: Lehrbuch der Mathematischen Physik III, p.156, Springer, Wien (1979) [5] Ph.A. Martin, CMP 4J_, 221 (1976) [6] Y.N. Demkov: Soviet Phys.-Dok, 393 (1961) [7 J.O. Hirschfelder: Phys. Rev. A, J2' H979) [8J T.A. Osborne, D. Bolle: J. M. P. JU3, 437 (1977) [9] E. Wigner: Phys. Rev. 98, 145 (1955) [lo] T.A. Osborne, D. Bolle: J.M.P. 20, 1121 (1979) i

9 Acknowledgement I want to thank Dr. M. Breitenecker and Dr. P. Gesztesy for making me aware of ref. [6] and [7] and Prof. W. Thirring for encouragement and stimulating discussions. References [1] J.M. Jauch, B. Misra, K.B. Sinha: Helv. Phys. Acta 45_, 398 (1972) [2] R.B. Lavine: Constructive Estimates in Quantum Scattering, Preprint (1976) [3] M. Reed, B. Simon: Methods of Mathematical Physics III, Academic Press, New York (1978) [4] W. Thirring: Lehrbuch der Mathematischen Physik III, p.155, Springer, Wien (1979) [5] Ph.A. Martin, CMP 41_, 221 (1976) [6] Y.N. Demkov: Soviet Phys.-Dok 6_, 393 (1961) [7] J.O. Hirschfelder: Phys. Rev. A, \$_, 2463 (1979) [8] T.A. Osborne, D. 3olle: J. M. P. ^i» 4 3? (1977) [9] E. Wigner: Phys. Rev. 98, 145 (1955)

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