Nonspreading wave packets in quantum mechanics

Foundations of Physics, Vol. 8, Nos. 7/'8, 1978 Nonspreading Wave Packets in Quantum Mechanics V. K. Ignatovieh 1 Received June 20, 1977 In this paper a nonspreading, unnormalizable wave packet satisfying the Schr6dinger equation is constructed. A modification of the Sehrb'dinger equation is considered which allows the normalization of the wave packet. The case is generalized for relativistic mechanics. It is a c o m m o n opinion that the Schr6dinger equation has no solution in the f o r m o f a nonspreading wave packet. However, we wilt show that such a solution exists if one takes as a packet the following function: W(r, t) = f ( r - - vt) exp(ik • r - - ioJt) which is not necessarily normalizable and where f ( r - vt) has a m a x i m u m at r = vt. In what follows it is convenient to take instead o f the time variable t the c o m b i n a t i o n ht/mo, which has dimension cm ~, and denote it by t. The function W then takes the f o r m ~(r, t) = f ( r - - kt) exp(ik • r - - ieot) (1) where k is the wave vector and velocity o f the particle described by the function W at one and the same time, and co is the frequency, which equals k~/2 when f ( r -- kt) = const. F o r f v~ const, the case we are interested in, ~ k~/2. Let us denote by s2/2 the difference between w and k2/2; we take it first as positive and write the amplitude f ( x ) as a Fourier integral: f(x) = f f(p) exp(ip • x) d~p x Joint Institute for Nuclear Research, Laboratory of Neutron Physics, Dubna, USSR. 565 0015-9018[78/0800-0565505.00/0 © 1978 Plenum Publishing Corporation 566 Ignatovich T h e n Eq. (1) is t r a n s f o r m e d as follows: 1P(r, t) f f(p - k) exp{ip • r - ½ip2t + ½it[(p - k) 2 - - s~]) d3p (2) I f we require t h a t ~P"satisfy the SchrSdinger e q u a t i o n Ls T : 0 (3) where Ls : i~ + A/2 (4) t h e n the last t e r m in the e x p o n e n t m u s t be zero. F o r a given s a n d k this h a p p e n s only i f f ( p ) contains a factor 3(p 2 - - s~). L e t f b e f(p) = c3(p 2 - - s ~) (5) where c is a c o n s t a n t ; then T ( r , t) = 27rc sin s l r - - k t I e x p ( i k • r - - loot) (6) Ir--ktl and ~o = ½(k 2 + s 2) (7) N o t e the d e p e n d e n c e o f T on the p a r a m e t e r s. W i t h the a s s u m p t i o n t h a t c is i n d e p e n d e n t o f s, for s --+ ~ , one has T ( r , t) ~ 3(t r - - k t I) exp(ik • r - - ioJt) (8) b u t if it is s u p p o s e d t h a t c N 1/v/~, then I T ( r , t)] ~ ~ ~(l r -- k t ]) (9) The p a r a m e t e r s characterizes the w i d t h o f the packet. W h e n s ~ 0 t h e p a c k e t b e c o m e s b r o a d e r , a n d when s reaches zero, ~ t r a n s f o r m s into a p l a n e wave. This t r a n s f o r m a t i o n occurs if c is p r o p o r t i o n a l to 1/s. T h e n trf(r, t) -~-_;6~exp(ik • r - - ioJt) (10) I t seems very attractive to assume t h a t s = c~k (11) where c~ is a p h e n o m e n o l o g i c a l p a r a m e t e r . Then, if c is t a k e n to be a function o f s such t h a t c(~) = f 1/v/~' ~ ~ ~ I1/s, s --. 0 (12} Nonspreading Wave Packets in Quantum Mechanics 567 an increase in particle velocity will be accompanied by a continuous transition f r o m wave optics to geometrical optics. The relationship (1 !) allows us to interpret oJ in (7) as the kinetic energy of a particle: co = ½k2(1 q c~2) (13) with renormalized mass m = m0/(1 q- c~2) (14) and not with m 0 , the mass of the particle. Let us note that the wave packet (6) is unrenormalizable. This does not affect the investigation of scattering processes, for which the n o r m is not essential, but causes problems in the determination of the probability p = [ ~ ]2. Can this be avoided? The answer to this question is in the nega- tive. O f course, the construction of the packet (6) implied some arbitrariness First, the Fourier coefficients in (5) might have been taken in a different form, and second, the sign of s 2 in (7) might have been taken negative rather than positive. Nevertheless, as is seen below, the arbitrariness is not too great and the expression (6) in fact is unique, since only this function has a spherically symmetric amplitude with a m a x i m u m at r -- kt. So, in order to m a k e the packet normalizable, it is necessary to choose coefficients f(p) in expansion (2) essentially different f r o m (5). But in this case the action of the SchrSdinger o p e r a t o r (4) on function W will give the following expression: /. LslP " = - - ½) [(p--k) 2-s z]fip-k) × exp{ip • r -- ip2t/2 q- it[(p -- k) 2 -- s2]} d3p (15) different f r o m zero. Let us take c 2 tiP) -- p= - - s 2 zL iE (27r) a (16) Then the expression differs f r o m zero only in one point Ls 7t = - - c 3(r - - kt) (17) The Fourier coefficients (16) do not provide a normalizable g*. F o r example, in the case of coefficients (16) with positive s ~, W(r, t) = c e x p ( ~ i s j r - - kt j) (18) } r -- kt I exp(ik • r -- ioJt) 568 Ignatovich In particular, for s = 0 we have one of the expressions proposed by de Broglie. m To ensure normatizability it is necessary to change the sign o f s z, i.e., to suppose that s~ = _q2 < 0 (19) In this case the normalized function T is equal to ( exp( 7 q ! _ r -- kt 2) exp(ik • r -- ioJt), oJ = 1 (k 2 _ q2) W(r, t) = \~-~-] q ~k-il (20) This expression was also proposed by de Broglie ~1) and investigated in- dependently in Ref. 2. Now let us find all the functions presentable in the form (1). For this purpose function (1) is substituted into Eq. (3) and, as a result, the equation for f ( r -- kt) is obtained: [i~ + ikV~ + ½(A~ -}- s~)]f(r -- kt) = 0 (21) where v~ and A~ mean differentiation over coordinates r. Let us introduce new variables ~ = r -- kt; then Eq. (21) transforms into (,de z + s~)f(g) = 0 (22) The general solution is given by the expansion fig) =- ~ aiM Y1M(~).A(S~) (23) where the YZM are spherical harmonics, jz(s~) are spherical Bessel functions, and arm are the expansion coefficients. It is seen from (23) that the spherically symmetric solution is a unique one, and the expression for W corresponding to it coincides with (6). It is evident that the Neumann or Hankel functions should not be included in the expansion (23), since these functions are not solutions of Eq. (22), but instead of the equation (Ae + s 2 ) f = 8(~) (24) Ifs 2 in (22) is assumed negative (s 2 = _q2 < 0), then the general solution is expressed through Bessel functions of imaginary argument that diverge at infinity, and for this reason are not attractive from the point of view of physical interpretation. If one puts s 2 = _ q 2 in (24), then the general solution is obtained in the form f(~) = K(ol)(q~) + ~ azMY,~t(~) j~(is~) (25) Nonspreading Wave Packets in Quantum Mechanics 569 where K(oll(x) is the MacDonald function and approaches zero at infinity proportionally to e -~. Here it is seen that the normalizable solution is a unique one and the expression for W corresponding to it coincides with (23). Nevertheless, it is necessary to emphasize that this function is not the solution of the Schr6dinger equation, but of another one, viz., (iSt -]- ½-A) ty = e 8(r -- kt) (26) where c is a constant determined by the normalization of 7 t. The possibility of obtaining a nonspreading packet, even if unnor- realizable, provides an opportunity to look in a new way at scattering processes. In particular, this applies to phenomena where a characteristic such as packet width manifests itself. Several concrete processes will be considered in a forthcoming paper. Here we consider on/y the principal ways of solving scattering problems. Since a packet is a nonstationary construct, as its position changes with time, the scattering problem even with a static potential is nonstationary. Nevertheless, since a free packet is rep- resented by a plane wave expansion and the Schr/Sdinger equation does not change the expansion coefficients, the scattering problem can be solved by using stationary methods for each partial plane wave. As a result we obtain the wave function in the external field as follows: 7t(r, t) = f f(p -- k)[exp(--ip2t/2)] X(P, r) dZp (27) where X(P, r) is the solution of the stationary scattering problem with incident wave equal to exp(ipr). For t - - * - - o o we have an asymptotic W(r, t ) - + T0(r, t, k0), where 7t0(r, t, k0) is given by expression (6), k - k0, and k0 is the velocity of the incident particle. For t --~ q- ~ , U(r, t) : : f g ( r l , kl) tY0(r -- r~, t, kl) darl d~#~ (28) i.e., a set of packets scattered at time t = 0 at points r~ and propagating with velocity ka. An analogous procedure is also possible for the packet (20) if quantum mechanical equations for partial plane waves are modified accordingly. Let us see now whether it is possible to find a packet-type solution for the relativistic Klein-Gordon equation. We write this equation as follows: [at2 - A q- 1 ] ~ = 0 (29) where time t and coordinate r are dimensionless, due to the introduction of units to = h/rno cz and r0 = h/moc, and rn 0 is the mass of the particle. Let us look for a solution of the type 7 t = f ( r -- ~t) exp(ik • r -- iwt) (30) 570 Ignatovich where 13 = v/c, v being the velocity of the particle. We substitute (30) into (29) and find the following equation for f ( g ) , where ~ = r -- 13t: [k 2 -- co2 -}- 1 -1- 2ico(13 • V) - - 2i(k" V) + (13' V) 2 -- A ] f ( ~ ) = 0 (31) I f the linear derivatives meet the requirement of being mutually compensated, then k = p13 and co = p, where p is some p a r a m e t e r of proportionality between k and 13. Writing k 2 -- o~2 + 1 = --s 2, one obtains an equation in the following form: [A -- (13. v) 2 + s~]f(~) = 0 (32) Let us choose the coordinate system in such a way that 13 is parallel to the z axis. Then, after t r a n s f o r m a t i o n to a new variable ~ ' = ~(1 --/32)-1/2, the equation is reduced to (22). A n d the general solution of this equation is identical to (23). A particular solution that is spherically symmetric in coordinate g' is equal to jo(~'s). Before writing down the final expression for ~ , let us find the relation between p and s. It is p = [(1 -}- s2)/(1 --/32)11/2 (33) After switching over to natural units, one obtains the final expression for W: 7 " = c ~sin s'R e x p ( i k , r -- loot) (34) where R= [x 2 + y2 + (Zl--_ vt)213 e ]1/2, k--hl (1--mv/32)1/2 (35) WtC~ 1 S' m°c 60 - - ~ s - - (1 -/3~)1/2 h' h and the mass mo of the particle is related to the mass m entering the expression for k and ~o as follows: mo -- m/(1 + sZ)1/2 (36) The function (34) is of the packet type, but again it is unrenormalizable. I f the relationship between masses is taken as m o = m/(1 -- q2)1/~ (37) and the equation for the free p a r a m e t e r is supposed of the type (24), then the wave function can be written in the form a) W = C e x p ( - - q ' R ) exp(ik" r -- icot), q' -- qmoc/h (38) R Nonspreading Wave Packets in Quantum Mechanics 571 If the parameters s' and q' in expressions (34) and (38), respectively, are taken to be proportional to k, s" = sok, q' = qok (39) then, for v --, 0, we have again the continuous transition from geometrical to wave optics. ACKNOWLEDGMENTS The author is grateful to Drs. V. I, Luschikov, Yu, N. Pokotilovsky, and A. V. Strelkov for their interest in this work and for many fruitful discussions. He also expresses his gratitude to Drs. G. G. Bunatian, E. Kapustsik, and M. I. Shirokov for useful remarks. REFERENCES 1. L. de Broglie, Nonlinear Wave Mechanics (Elsevier, Amsterdam, 1960). 2. V. K. Ignatovich, JINR, E4-8039, Dubna (1975), 82518t718-s