Propagator relative to the step potential

IL NUOVO CIMENTO VOL. 108 B, N. 8 Agosto 1993

Propagator Relative to the Step Potential.

L. CHETOUANI(1), L. GUECHI (i), T. F. HAMMANN(2) (*) and Y. MERABET(1) (1) Dgpartement de Physique Thgorique, Institut de Physique, Universitg de Constantine

Constantine, Algeria (2) Laboratoire de Mathdmatiques, Physique Mathgmatique et Informatique

Facultg des Sciences et Techniques, Universitg de Haute Alsace 4, rue des Fr~res Lumi~re, F-68093 Mulhouse, France

(ricevuto il 17 Settembre 1992; approvato il 30 Marzo 1993)

Summary. -- The propagator associated to the Heaviside potential VoO(X) is obtained under its compact form by summation on all the states of the particle. It is proved that the base of the states is complete indeed and that for Vo--~ ~, the propagator tends towards the well-known propagator of a particle subjected to move along a semi-axis.

PACS 03.65.Bz - Foundations, theory of measurement, miscellaneous theories. PACS 03.65.Ca - Formalism.

1. - Introduction.

The purpose of this paper is to build very explicitly the propagator relative to the step potential or Heaviside potential

(1) V(x) = VoO(X) (Vo > 0),

by summing up all the ~_ (x) and ~_~ (x) states of the particle; (we use here the standard convention according to which the arrow in the wave function indicates the way in which the particle moves).

The resolution of Schr5dinger's equation and the calculation of the transmission and reflection coefficients are trivial, whereas the calculation of the propagator leads to highly interesting mathematical and physical problems. Summing up, the wave functions often proves to be a method more manageable than the path integral approach: as an example, we can quote the Kepler-Coulomb problem for which the Green's function has been obtained through the traditional method [1] long before it could be solved in the framework of the path integral approach [2].

Finding a solution to the Heaviside potential within the path integral for- realism[3] has been tried, but this solution has not proved satisfactory since, for

(*) To whom request for reprints should be addressed.

5 9 - I l Nuovo Cimento B 879

880 L. CHETOUANI, L. GUECHI, T. F. HAMMANN and Y. MERABET

Vo-~ ~ the well-known propagator associated with a particle subjected to move along a semi-axis, could not be found.

In this paper we aim at obtaining an accurate expression for the propagator.

2. - P r o p a g a t o r for the f in i te - s tep potent ia l .

The solutions to SchrSdinger's equation

he d 2 ] + VoO(X) ~ _ ( x ) = E~_+(x)

(2) 2m dx 2 +-

are well known. When the particle moves from right to left, for E > V0 energies, the wave function can be written as

m ]1/2 (3a) ~+ (x) = 2r~h2k 2 [ e x p [ - i k 2 x ] - c~ exp[ik2x]], x > 0,

m 11/2 (3b) ' ~ ( x ) = 2rch2k 1 5 r e x p [ - i k l X ] , x < 0,

with

ki=[2 llJ2 k2 0 ]1J2 h 2 J ' h2 '

and where

kl - k2 2(kl k2) 1/2 - and ~ - ,

kl + k2 kl + k2

respectively, represent the reflection and transmission coefficients. When the particle moves from left to right with a positive E energy, the wave function can be written as

m 11/2 5r exp [ - ik2 x ] , x > O, (4a) ~_~ (x) = 27~h2 k 2

m ]1/2 (4b) ~+(x) = 2=h2k1 [exp[iklX] + g2 e x p [ - i k l X ] ] , x < 0,

the [m/2rch2kl] 1/2 and [m/27~h2k2] 1/2 constants being obtained by normalization of the wave functions,

- E )~ ,__ (5) : : -+-+,

PROPAGATOR RELATIVE TO THE STEP POTENTIAL 881

through the formula

r

I i (6) d x e x p [ i ( k + i0)x] - k + i0 0

The propagator is linked to these wave functions by the following summation on the states

I i E T i T (7) K(b, a; T ) = d E ~ ( b ) r - - - ~ + dE~___)(b)r - - - .

vo o

We shall keep this notation for E < V0, where the k2 constant then becomes imaginary.

In order to put eq. (7) into a compact form, we shall consider all the possible cases according to the initial and final positions of the particle:

Firs t case: b < 0 and a < O.

In the case, the r and ~_~ wave functions of eq. (7) are given respectively by eqs. (3b) and (4b), and the propagator

(8) K<<(b, a; T) = Ko(b, a; T) + K~< (b, a; T) ,

features a part which is the propagator of the free particle

(9) Ko(b, a; T) =

1 I dkl exp [ - i [ak~ + kl (b - a)]] = m 1/2 i m 27:_~ ~ exp ~ ( b - a ) 2 ,

where

hT

2 m

and also a part which is the propagator resulting from the reflection by the potential

{( 1 dkl exp [ - i ak~] [,c/~ e x p [ - i k l ( b + a)] +c.c.] + (10) K~<(b, a; T) = 2--~ d

+ Idkl e x p [ - i a k l ~] .c2 [ e x p [ - i k l ( b + a)] +c.c.]

where c.c. stands for the complex conjugate expression, and y = 2 m V o / h 2.


Thanks to the formula

exp[-iak~] = [-~ ) _J dpexp[i(ape ~ [3kl)], with fl= 2~p,

allowing us to linearize k~, eq. (10) can be written as

- - - - dp exp [i~p2] �9 (11) K~<(b 'a;T)= 2~[i7~] _~

I; Y } - dk~[ , (2exp[ - i k~ (b+a- /? ) ]+e . c . ]+ d k ~ / C [ e x p [ - i k ~ ( b + a - f l ) ] + c . c . ] . o ,~

The change of variable ka = ~ cos 0 and the use of Fourier series [4]

(12) exp[iucosO] = ~ i~J~(u)exp[inO], n = - ~ c

allow us to calculate the J1 integral:

J i = J d k l { ~ e x p [ - i k l ( b + a - / ~ ) ] + c . c . } . 0

Similarly, the change of variable kl = V~cosh 0 and the expansion analog to (12) obtained by analytic continuation[5], valid solely for u and 0 > 0,

I cos(Oy)J~y(u)dy-- iexp[iucoshO] exp [=.y/2]

(13) - , sinh (7~y)

-co

allow us to obtain the J2 integral: oc

Je = Idkl ~2 [exp[- ik l (b + a - /~)] +c .c . ] , J

so that the propagator can be writ ten as

(14) K~<(b, a; T) = 2~ \ z,~-~--~-] _~ dpexp[i~P2](J1 + J2),

where

(15) J l = _2~r sin(z0 - oJ) + 2=V~ J2(zo - co) Z 0 - - ~o Z 0 - -

16V~ ~ ( _ ) t

ZO - - 09 l = O

J2~ + 1 (Zo - ~o)

( 2 l - 1 ) ( 2 / + 3)

P R O P A G A T O R R E L A T I V E T O T H E S T E P P O T E N T I A L 8 8 3

and

2 ~-VY sin(z0 - ~) (16) J2 = + Z o - - O)

+ - - Z 0 - - 09

I - zo_) -~ (4 + y2)cosh ( 2 y )

dy , for co-Zo > O,

with Zo = V~ (b + a) and oJ = ~/-y/~. Moreover, we have

(17) J iy (oJ Zo) dy = dy =

Jiy (oJ - z o) ] = _+2irc~ Res (4 + y2)cosh((r:/2)y) '

where c stands for the contour of the semi-circle with radius R--~ oo located in the upper half-plane for (Zo - o~) > 0, and in the lower half-plane for (Zo - co) < 0. This distinction comes from the asymptotic behaviour [6]--for large ~--of the modified Bessel functions

(18) Iv(vZ)'~ (l + z2)-l/4 [ Z] (27~)1/e exp v ~ v / l + z 2 + ~ l n 1 + ~

the first- and second-kind Bessel functions being related by the standard relations,

(19) J~(x) = exp[i2v]Iv(- ix) .

Eventually, we find that the main contribution to the value of the integral (17), comes

from the factor exp [~ In (z/(1 + ~ + z2))]. Thus, it follows that

2 ~ sin (z o - oJ) + (20) J2 = + sign (Zo - co) 2r: V~ J2 (Zo - oJ)

Z 0 - - o ) Z 0 - - o )

+ P 16 V~ z_, ~ ( - )l J21 +1 (Zo - o~)

Zo - o~ t=o (2l - 1)(2l + 3)

Let us now replace p by (p - (m/hT) (b + a)) and we shall thus eventually obtain the


compact expression of the propagator, for a and b < 0:

(21) K<<(b, a; T) = K0(b, a; T) -

]] - / - ~ dp P exp -~m-m[p- ~-~(b + a) .

It is to be noticed that the part of the propagator (21), coming from the reflection on the potential, is a superposition of Gaussians which refer to the image of point b with respect to the origin.

Second case: b and a > O.

Calculations similar to those of the preceding case will lead us to

(22) K>> (b, a; T) = K0(b, a; T) + K~>(b, a; T),

where

] i __1 exp - V0 T dk2 exp [ - i[ak~ - k2 (b - a)]] = (23) Ko(b, a; T) = 2~

m

is the propagator of a free particle bearing a phase factor, and

(24) K~>(b, a; T ) =

1 exp - T - 2~ --/- 0

d k l [ ~ C ~ e x p [ i k 2 ( b + a ) ] + c . c . ] e x p [ - i a k ~ ] +

Id kl tT~ exp [ik2 (b + a)] + c.c. ] exp [i~k~] + kl k2

,/7

is the propagator resulting from the transmission and the reflection by the potential, obtained by the use of

(25) 1~12- [ ~ ] ~ +c.c.


We obtain this propagator by noticing that kl dkl--k2 dk2, and by performing the changes V0 o 0, i.e.

(26) K>>(b, a; T) = Ko(b, a; T) -

I L- o p -~m p + - ~ ( b + a ) .

This propagator proves to be a superposition of Gaussians as well, and the second part also refers to the image of point b with respect to the origin.

Third case: b < 0 and a > O.

We can show that the propagator is now

(27) K ~ ( b , a ; T ) = l [ f f dkl kl) 1/2 + c.c.] exp [-iak~] [(~-2 J-exp[i(k2a-klb)] +

[ t- ~1/2 | + [dkl | '~l | J-[exp[i(k~a - kib)] + c . c . ] e x p [ - i ~ k ~ ] l -

V~'

Let us linearize the exponent in kl 2, first:

(28) KX (b, a; T) =

_ " 57. exp [i[k2 a + kl (8 - b)]] + c.c. +

oo kl(kl/1/2 5r[exp[i[k2a + kl(/~ -- b)]] +c.c.] ] .

Set

t _ _ b - a b+a a f o r p < - o r p > - -

tghp = fl - b ' 2a 2:~

~ - a for b - a b+a - - , - - < p < ,

a 2~ 2a


and let us decompose the integration interval on the p variable as follows:

I - ~ ' ~ a[U]b-a2a ' b+a[u]b+a2a-- 2a ' + ~ [

and let us calculate the integral (28) in the last two intervals first. In the ](b + a)/2~, ~ [ interval, the calculation is similar to that of the first case;

the result obtained is nil. In the ](b - a)/2a, (b + a) /2a[ interval, we have to evaluate the expression

(29)

(b + a ) / 2 ~ ["

I dpexp[i~p ] [ 2

(b - a)/2:~

;2 ] 2i a ~ dOexp[-iO]exp cosh,z

Sb ~a ~ +

[ ; [ ]]] 2 ~ ~ d 0 e x p [ - 0 ] e x p i V ~ a s i n h ( 0 + ~ ) +c.c. . + ~ 3b ~a cosh~

0

Let us use the Fourier-series expansion formula[7]

exp[ ixs in r = Jo(x) + 2 ~ J2n(x)cos(2nr + 2i ~ J2n+l(x)sin[(2n + 1)r n=l n=0

to calculate the first integral, as well as the formulae [8]

() I = 7: cos (z sinh 0), cos(Oy) cosh ~ y ,~iy (z) dy = 2 0

() sin(0y) sinh ~ y ,~/dy (z) dy = --2 sin (z sinh 0), 0

together with the following relations between the , ~ (z) imaginary-argument Bessel functions, the H~l)(iz) first-kind H~inkel functions, and the Jr(z) first-kind Bessel functions [9]:

i7~ [ 7V 1 H (1) iCy(z ) = - - exp - (iz), 2 2 y] Y

( 1 ) " Hiy (~z)= exp [~y] Jiy (iz) - J-iy (iz)

sinh (=y)

J~ (exp [m=i] z) = exp [mvrd] J,, (z),

in order to calculate the second integral and eventually show that the expression (29) is nil.

Let us notice that there is another way of showing that the integral (28), within the interval ](b - a)/2~, + ~ [ of variation of the variable p, is nil. Indeed, the inte-


grals

(30)

f dkx(kl lj2

kl 1/2

I v~

J-exp[i[kea + k l ( fl - b)]] +c.c. = J3 + J ~ ,

,~exp[i[koa + kl(~ - b)]] +c.c. = J4 + J4*,

can be brought down to an integral on the contour f~ thanks to the following adequate variable changes:

i) kl = X/7 cos 0 in J3 + J3* followed by the replacing of 0 by 7: - 0 in J.i*,

ii) k 1 = ~g/~cosh r with r = i0, in J4 + J4* followed by the replacing of 0 by 0 - =, in Jd*, which leads to

(31) 2 ~ ] dOsinOcosOexp[-iO]exp[V~[i(~ - b)cos0 - as in0]] .

f7 The path of integration fB is shown in the following drawing:

Fig. 1.

0 ) ) X

- i ~ r : - i ~

The function

f(z) = V~ sin (2z) exp [ - iz] exp [ V~[i(/? - b) cos z - a sin z]],

is analytic inside and on the ABOr:A contour. Therefore, we can write

f dzf(z) = - f dzf(z) , V1 xe

wherein

z = x - i y , O<~x<~7: and y ---) ~

According to the Jordan lemma

I d i f (z) AB

= O, if Izf(z) l ~ o .


Now

(32) Izf(z)l y~ee y.

"exp[y]exp[ - V ~ s i n x [ ( f l - b 2 + a ) e x p [ y ] + ( a + b - f l ) e x p [ - y ] ] ]

i f f l > b - a , or a l s o p > ( b - a ) / 2 a . Consequently, the propagator (28) gets transformed into

- ~ 0 , y.--~ zc

(33) KX (b, a; T) =

_ 1 a dpexp[iap e] dkl kl J_exp[i[ke a _ kl(/? + b)]] +c.c. 2r.

(a - b ) /2~ 0

+ kl J [exp [i[k2a - kl (~ + b)]] + c.c.] .

There is even a more compact way to write this equation, by prolonging 5 r analytically into the complex plane (Re ~ is symmetric, whereas Im v]- is anti- symmetric). Thus

+

(34) KX (b, a; T) =

2r: ~ dpexp[iap 2] dkl ( a - b ) / 2 a - o~

exp [i[k2 a - kx (fl + b)]].

As we have

k~ ~Sr = 1 + co~,

the expression (31) can be decomposed in the following way:

(35) KX (b, a; T) =

1 - dpexp[izcp 2] dkl(1

27: ( a - b ) / 2 ~ - ee

By introducing the identity

+ cd~ )exp[i[k2a - kl( fl + b)]].

+ o o + z o -bee

I I f = 1 ,


into the second integral of eq. (35), we obtain, since ks = V k ~ - Y,

(36)

+ ~

f dkl - o r

c~ exp[i[k2a - k1(/3 + b)]] =

----i f dz f duexp[i[Vu -ya-z ]] f dkl 27:

~2 exp [ikl (y. - /3 - b)].

By using the results of the first case,

(37)

+ ~

1 ] dkl [2 exp[ ik l (Z- /3 b)] - 2 2r~

[ 2mVo ( _ fl _ b) I ]

X - / 3 - b 0( X -/3 - b),

we eventually come to the following result, for V0 > 0:

(38) K X ( b , a ; T ) = ~

(a- b)/2a

+ o o

dp exp [iap 2] ] dkl exp [i[k2 a - kl (/3 + b)]]. - o r

�9 1 - 2 dy J~ Y �9 o Y h

Fourth case: b > 0, a < 0.

With calculations similar to those of the third case, we come to the following propagator:

(39)

oo + z r

K g (b, a; T) = ~ dpexp[ iep 2] dkl ( b - a ) / 2 ~ - ~

exp [i[k2 b - kl (/3 + a)]].

[ �9 1 - 2 dy J2 Y �9 o Y h

3. - P a r t i c u l a r c a s e s .

3"1. Limi t i ng case where Vo--~ ~. - We have to check whether for Vo-+ ~ , we obtain the well-known propagator relative to a particle subjected to move along a semi-axis.

Noticing that

J2 (2x) J2 (2x) (40) ~ cr for x = O , and + O, for x ~ O ,

X ~--*~ )~ ~--*~

890 L. CHETOUANI~ L. GUECHI~ T. F. HAMMANN and Y. MERABET

it becomes obvious that J2 (~x)/x behaves like the Dirac distribution ~(x). Moreover, according to formula [10]

f J'~ dx if - l < R e q < R e v (~x) 1

(41) x v q ( ) 0 2"-q)~q-'~+lI" v + 1 -- q 2 2

we obtain

(42) f~ J2 (s dx 1 and consequently lim J2 (~x) ~(x) - - ~ - - - - �9

x 2 ' ~-~ x 0

Thus,

(43) lim K<<(b, a; T) = Ko(b, a; T) - K o ( - b , a; T) = V0 ---~ zc

= exp 2 - ~ ( b - a ) 2 - e x p 2 - ~ ( b + a ) 2 ,

(44) lim K>>(b, a; - i T ) = V0-o

= lim e x p [ - V o T / h ] {Ko(b, a; - i T ) - K o ( - b , a; - i T ) } = O, VO--> ~o

where we transformed T into ( - i T ) via the Wick rotation. Similarly,

(45) lim KX (b, a; T) = lira K< > (b, a; T) = 0. Vo-)~ Vo-~ :r

We thus obtain an adequate asymptotic behaviour, for Vo large.

3"2. Limi t ing case where T--)O. - Noticing that

] m exp Ax) 2 --~ ~(Ax), 2i=hT ToO

we obtain

(46) - --~ ~(b - a) , K<<(b, a; T)T_~O---) ~(b a) , K>>(b, a; T)T__~ ~

K~ (b, a; T) ~-~oO KX (b, a; T) --) O ' T - - ) O "

These equations obviously prove that the base is complete.


4. - L i n k w i t h t he F e y n m a n propagator .

The S action evaluated along the path linking points (a, 0) and (b, T), is given by

(47) S = f [ m22-2 - V(x)]dt,

and, according to the Feynman approach [11], the propagator is

(48) K ( b ' a ; T ) = lim ( ~ ( m 11/2N-j~ 1 [~ [~'r ]] N ~ J j=l \ 2--~h~ ] : dxjexp j=l~ (Sxj) e - ~V(xj) ,

which is a product of the infinitesimal propagators K(xj, xj_ 1; s) in which the infinitesimal action

S(x~, x~_ 1) = ~ (Sxj) 2 - W(x~),

is calculated along a segment of a line linking the points (xj_ 1, tj _ 2) and (xj, tj), with tj - tj_ 1 = ~ = TIN.

We then obtain the following infinitesimal propagators:

(49)

K << ( xj' xj -1; s ) -- 2-~=h~ ] exp ~ ( xj - xj _ l ) ,

K)) (x~, xj-1; ~) --

~ ( x ~ x 5 1) 2 - --- 2-~h~) exp - 1 -h-

K S (x 5, xr _ 1; ~) --- 0,

if Xj, Xj- I < O,

Y0), if xj, xj-1 > O,

i fxj>0, xj_l <O, orx j<0 , xj_~>O.

By putting all these cases together, the infinitesimal propagator can be written as

(50)

f m / 1/2 i K(Xj, xj_I; r = ( 2-~r ] exp[-~S(xj,

where

S(xj , x~_ l) = m (~x~)~ - WoO(Xj),

x~_l )],

which is the adequate form for the studied potential.

5. - C o n c l u s i o n .

By summation on all states, we have obtained the compact form of the propagator associated with the Heaviside potential V(x)= VoO(x).

Our results (eqs. (21), (26), (38), (39)) are different from those obtained in the


f ramework of the path integral approach. Fo r V0 ~ cr we effectively obtain the p ropaga tor of a particle subjected to move along a semi-axis and for T--* 0, we have demonst ra ted tha t the base of s ta tes is complete indeed.

R E F E R E N C E S

[1] L. H. HOSTLER: J. Math. Phys., 5, 185 (1979). [2] I. H. DURU and H. KLEINERT: Phys. Lett. B, 84, 185 (1979); Forts. Phys., 30, 401 (1982);

L. CHETOUANI and T. F. HAM~NN: Nuovo Cimento B, 98, i (1987); Phys. Rev. A, 34, 4737 (1986); J. Math. Phys., 28, 598 (1987); L. CHETOUANI, L. GUECHI and T. F. HAMMANN: Nuovo Cimento B, 101, 547 (1988).

[3] A. O. BARUT and I. H. DURU: Phys. Rev. A, 38, 5906 (1988). [4] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic

Press, New York, N.Y., 1965), p. 973, eq. (8.511.4). [5] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic

Press, New York, N.Y., 1965), p. 774, eq. (6.796.1). [6] M. ABRAMOWITZ and I. A. STEGUN: Pocketbook of Mathematical Functions (Harry

Deutsch, Frankfurt am Main, 1984), p. 122. [7] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic

Press, New York, N.Y., 1965), p. 973, eq. (8.511.3). [8] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic

Press, New York, N.Y., 1965), p. 774, eqs. (6.796.2), (6.796.3). [9] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic

Press, New York, N.Y., 1965), p. 954, eq. (8.407.1); p. 970, eq. (8.483.1); p. 968, eq. (8.476.1).

[10] I. S. GRADSHTEYN and I. M. RYZHIK: Table of Integrals, Series and Products (Academic Press, New York, N.Y., 1965), p. 684, eq. (6.561.17).

[11] R. P. FEYNMAN and A. R. HIBBS: Quantum Mechanics and Path Integrals (Mc Graw-Hill, New York, N.Y., 1965).

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Propagator relative to the step potential