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A generalized jwkb approximation for multichannel scattering
CiiEMICAL PHYSICS 2 (1973) 381-399. Q NORM-HOLLAND FUBLISHIN~ COMPANY
A GENERALIZED
JWKB ~PROX~~AT~~N
FOR MULTICHANNEL
SCATTERING
t:
Received 30 July 1973 Revised manuscript received 17 September 1973
The we&known JWCB npproximstian is gcnenlized to handle multichannel suttering problems_ fnduded is s discussion of how the JWKB connection fotrnukts can be applied to the rnu~t~pi~&ssical ~min5 po~tswhi~h occur in the m~tichannel case. A new and useful form of the distortedwavetra~forma~ion is dcrivcd andapplied to the two state curve crossing problem. The result is a useful factorization and paramctetiza:ion of the S-matrix. The parameters arc cvzduatcd approximatcfy by a method due originally to Zencr to obtain L;m&u-Zencr lypc formuks. It is aJso shown how the equations of the impact parameter method wiih stra$ht line trajectories an be derived from our generalized JWKB equations.
The JWKB(designated WILE!in many texts) approximation is discussed in most quantum mechanics text books. It is a very useful methad for generating ap proximate solutions of the one dimensional SchrBdinger equation. As is well known, this approximation is valid only in the limit of a slowly varying potential such that the fractional change in the deBroglie wavelength over a waveleng~ is small compared to unity. The JWKB salutian not only requires less effort fo compute than the exact solution, but it can also add insights in&othe physics, enabling inferences to be drawn which would not be possible otherwise. Applied to the problem of scattering from a central potential, the JWKB approximation is used to solve the radial SchrBdinger equation, thus obt~~g the * JWKB phase shift for each P value of the pattial wave expansion {11. In the more general case of inelastic scattering, the radial Schr~ger equation is replaced by a system of coupled differential equatioqs [2). This system of coupled differential equations is written most elegantly t W0t.k sup&ted
by rhe Air Force &ice of Scientic Research, (AFSC), USAF, underGrant No. AFOSR-72-2212.
in matrix form. Various semi-classicalmethods have been proposed for obtaining approximate solutions to this radial matrix Schrcidingerequation l3-61. In this paper we propose. a new method which is a direct generalization of the ordinary JWKB approximation. la section 2, we introduce the reader to the method by deriving the ordinary JWKB expression for the S-matrix in a mannet which can be gener~~d to the multich~nei case. In section 3 a generalized multichannel JWKB approx~ation which includes a method for handling the multiple classical turning points is derived. In section 4 we discuss the distorted wave transformation and derive a new version which we cali the split-distorted wave transformation. Section 5 inciudes an application of the spiitdistorted wave traasformation to a tv+.state curve crossing prabkm to okin a useful factorization and ~a~ete~~on GZ the Smatrix, then the me&ad of Zener is used to ob. tain approximate Landau-Zoner type formu%& Fir&&, in section 6, several approx~a~ons to oat generalized JWKE equations are discussed.
2. Potential scat&&g by the. JWKB methad
!.
As ti aid ~unde~~~g the general muir&&armef in&hod to. be developed, we wiH font ymstier sjr$Ie :.
:
382
B.R. Johnson. A gckxalked
J WKB
approxindon
channel potential scattering from a spherically symmetric potential, V(x), and derive the ordinary rwKB solution for the S-matrix in a way that can be easily generalized to the multichannel case. The derivation is based on physically intuitive ideas of the scattering process. Assume that the total potential, V(x) plus the centrifugal potential which we designate Vs(x), is negligibly small in the asymptotic region x > x2 _The asymptotic farm of the wavefunction can be written in the form Jl(x).tzxs = exp [-ik(xz) + exp[W+)
(x -x2)] (x - x2)1 R(x2),
(9
where R(+) is, by our definition, the reflection coefficient at the position x2, and k(x)=tl-92/&5-
mulrichnnnel
~(xL<,,=exp]ik(xt)
smrrering
(x- xt)]b
+exp[-Wxt)(x-xr)]ti,
(3)
and in region C the wavefunction is the transmitted wave +(&X,
= exp [ik(x2) (x - x2)] 7% .
(4)
In a similar manner g and ? are reflection and transmission coefficients which characterize region B when the incident wave, exp[-ik(x2) (x -x$11, is detined in region C. This is a wave of amplitude I, traveling in the negative direction. The total wavefunction in region C is tbc sum of this incident wave and a reflected wave WX>X,
v(x)-82(P+a)2/2Eu2])‘/2.(2)
Here E is the total energy, ~1is the reduced mass and Q the angular momentum quantum number. (Note that Kanger’s prescription [7] has been used, i.e., (Q+$)? replaces Q(Q+ 1) in the centrifugal potential term.) Conservation of probability requiresR(x2) to be a complex number of unit amplitude. Letx, denote the classical turning point. Divide the problem at some division point xt where xr < x,
for
= exp[-i&) + exp[ik(x$
(x - x2)]f (x -x1)]&
(5)
and the wavcfunction in the region x G x1 is the ~~smitted wave ~(xkx,
=exp[-ik(xt)(x-xt)]
71
.
(6)
Eqs. (3-6) define the four parameters 6l, T, z and ? which characterize region B. Region A (0 =Gxxt. Using this potential, consider a scattering problem in which the incident wave is defmed in the region x 2xt to be exp[-ik(xl) (x - xt)] u. This is a wave of amplitude u traveling in the negative direction. The total wavefunction is the sum of the incident and reflected waves W;crr,
= exp[-&(x1)(x--xl)] + cxp[ik(xl)(x-xl)]
u &T&r.
(7)
Thus, using electrical engineering jargon, we can think of the problem as three “black boxes”, shown ~hematic~y in fig. 2, which are interconnected as indicated by the lines. Eqs. (8-10) should be ‘obvious from this figure.
383
B.R. Johnson. A gcncralized JWKB approximation for mulrichannel scnmring
substitute this into eq. (10) to obtain R(x2)
= z + T(1 -R(x,)tR]-’
R(q)?.
(11)
Thus,
the overall reflection coefficient, R(x2). has ken egressed in terms of the four puameters hr, r, I~Iand T which characterize the potential in region B and the parameter R(x,) which characterizes the potential in region A. WC now wish to obtain the JWKB expressions for these parameters. Since, within the JWKB approximation, reflections can only occur at a classic_alturning point, it is immediately evident that di = 6I= 0. Thus, eq. (I 1) reduces to the much simpler cxpressian
R(xZ) = TR(x,)?.
(12)
The JWKE! expression in the region xt Qx
$(x) = [k(x)] -112exp
Fig. 1. Potential is plotted vertically 2nd radkd dktancc horizontally. In iig. la the total radial potcntixl. VQ(~). is shown schcmatimlly divided into three regions. The ConStNction of the potentials in figs. lb and lc arc described in the text E is the total energy and xl the classical turning point.
LZ=3+Rb
(8)
b =R(+)P
(9
R(x2)I=%+
7%.
Substitute eq. (8) into eq. (9), solve for 6, then
(10)
[i[
k(x’) clx’]
The definition of the transmission coefficient, T, is (see eq. (4) with $(x,) = 6) (14)
IL(x2)=TNq).
Comparing cqs. (13) and (14). it is evident that T = [I+)]
-II2
exp[il
k(r)dr]
[k(xl)jt/2.(15)
Similarly, the JWKB expression for a wave traveling in the negative direction is
$(x) = [k(x)] -l/2 exp [-i J k(x’) dx’] x2
(9). X tk(xz)l 1’2rc,
(16)
Making use of eq. (6) it is apparent that
The JWTCBexpression for the wavefunction in the Fii 2. “Black box” schematic of tie scaltering problem.
region xt
B.R. Johnson, A gmemlized JWKB appmxinwtin /or mulrichannelsarttaLng
384
WI
$(xj=C[k(r)]+*sin
where Cis an arbitrary constant. Using eqs. (7) and (18) it is easy io show that
R(xl)=exp
2ij’,k(*)br-irri2] Xt
.
w
Substiitutingeqs.(19),(17)and(IS)intoeq.(12)we obtain the obvious result
R (x2) = exp
k(x)dx-in/2
- exp[i(kx
1,
- PIT/~)] - IZIT/;?)&-~/*S kl/* ,
(21)
where k =k(+.
= -exp[-2i(kx2
- h/2)]
R(x2).
which is the usual JWKB expression for the phase shift [Il.
3. The multichannel JWKB approximation
[d*/dr*
+tr-* p*(x)]
where p(x)=
[2p{EI-V(x)-
~-[fi*Q(Q+J)/2~*]1~]
II2
(28)
In this equation: E is the total energy, ~1is the reduced mass, V(X) is the symmetric potential matrix which approaches zero as x approaches infmity, P is the angular momentum quantum number, I is the unit matrix and c is a diagonal matrix whose elements are the threshold energies of the various channels. The wavefunction Jl(r)‘is a square matrix, each column being a linearly independent solution of eq. (27). The logderivative matrix is defined to be
w
v’(x) +I+
The usual way to express a one izhanncl scatter& matrix is in terms of th& phase shift. R. defined by _
eqs. (25).and (24)
(27)
where the prime means differentiation with respect to x. Differentiating eq. (29) and using eq. (27) to eliminate the second derivative term, we obtain the matrix Ricatti equation
SX,-r_
Compa+g
+(x)=0,
Y(~)=JI’(xN-U,
Substituting eq. (20) into (23)
Sqexp[2ig].
(26)
(22)
Comparing eqs. (21) and (1) s*p,
dr-,loc,+(P+$)n/Z,
A generalized JWKB approximation can be derived for the multichannel (matrix) Schrtiinger equation. The derivation is analogous to the derivation given in section 2 for the ordinary JWKB approximation. The matrix SchrCdinger equation can be written in the form:
At this point it seems that we have used a very complicated rezoning process to obtain eq. (20) when it cdd have been obtained as directly as eq. (19). We remind the reader that the main purpose of this section is to establish a method of derivafion that can be generalized to the multichannel cae. The reflection coefficientR(x2) can easily be related to the S-matrix. The S-matrix has the following defmition in terms of the asymptotic form of the wavefunction ,J,(x)x- OD= exp[-i(kx
Q= ‘- [k(x)-k] I =1
(29)
p* (2) + v*(i) = 0.
(30)
In a manner entirely analogous to the derivation of the JWKB expansion in powers of h, we write theexpansion [9] ...I . A311 Substituting into eq. 00) and equating like powe of fi, the following expressions are obtained for the fmt two terms of the expansion V(X) = (i/ti)[oo(x) +(rl/i) al(x)
+ (fi#g2(x)+
,:
385
k3.R Johns&. A generalized JlVKBqqraximalion for mdlichanne~ scatterirlg
should be recalled that exactly the same difficuIty occurs in the ordinary JWKB case and that thy?SO~&~OQ isto retain the second term, ut(x), of the JWKB expansion (see eq. 3 1). In the multich~e1 case this is also a solution, i.e., we can solve From eq. (29) we obtain
e’(x) =I [* i k(x) + s1 IX)] \ii (x) . (34)
$‘@I = YW#W.
Substituting the first term of the expansion for v(x) into eq. (34) gives J;fx)S-tik(x)Jt(x),
(35)
where k(x) = n-l p(x).
(36)
It is convenient at this point to defme the unitary matrix U(x, x3 ) by the equation U’(x,xl)=i
k(x)U(x,xl)
(37)
(the prime means differentiation with respect ~0x1 with the initial boundary condition U(x,, x1) = I #
(38)
Then it is obvious that the solution to eq. (35) (with the plus sign) can be written Jt@)= U~X,~~)~(~~),
(39)
and the solution to this equation with the minus sign is JICx)= UYXX~)WX,),
(401
where * means complex conjugate. The unitary matrix, U(X, x1), can be written in exponential form [lO-121 U(~,~l)=explA(xP~t)I, where the anti-hermitian ~ferenti~ equation A’(x,xlI = i(exp(C+I]-l
(411 matrix A(x, .x1) saWTies the C,, k(x),
A(xl,xli)=O.
(421 (43)
The linear operator C, is defined by CAL-At-LA:
(44)
(45)
Numerical studies hsve shoarn the solution to eq. (45) to be an cxcellenl approximation to the wavefunction. However, ~~0~~ this is a possible approach, it is not sat~factory, from the point of view either of computations efficiency or of theory. ~ompu~tio~ly, a long calculation which involves matrix diagonal& zation is required to solve eq. (33) for si (x); also. the quantity p’(x) is required, which means that we must know the fist derivative of the potential matrix. Finally, even if we do solve for u1 (x), jt is more dif!G cult to solve eq. (45) than eq. (35). TXe theoretical objection to eq. (45) is that the solution, al;iough an excellent approximation, does not conserve probab~~ exactly. An expression which is simple to calculate, which conserves probability exactly and which is an excellent approximate sotution (see appendix 1) is the following; Kx) = [kW-“2
WW$
tklxl)lV2
‘%x,).
(4)
or (if we choose the ninus sign in eq. (32)) Jr(x) = [k(xWy2
U*(x, xi) Ik(x,)l u2 +&xl) - (47)
Functions (46) and (47) are our choices for the generalized JWKB solutions of the multichannel Schr~inger equation. The fmt Magnus appro~mation to the matrix A@, x1) [IO-12) (see eq. (42)) is X A(x, x1) ?5:i
I xi
kfx’) dr’ .
(481
~ornb~~g this approximation with eqs (4 1). (46j and (47) we obtain 4(x) = [k(x)]-*f* exp [2i[k(x’)dx’] x W~W2
wg.
,’ ‘.
‘.. ” (49)
where L is some arbitrary ware
m&ix. The function q(x), defied in eq. (39); is not a satisfactory approtiation to the wavefunction. It
which is f&maUy identical to the ordi&ry J’Wl&~tipression, the only difference b&g that the quantities
386.
B.R. Johson,
generalizedJWKB
A
approxinwrbn
involved are matrices, instead of ordinary numbers. Since in general matrices do not commute,‘the order in which the ma&x factors appear is essential. Unfortunately, in man9 cases, the,first Magnus approximation is not accurate enough and so eq. (37) must be solved more accurately. A variety of methods for solving eq. (37) were
h]
exp [i “/-’
sca~rering
the appropriate qu+i ties in these equations as matrices. The factors in these equations have been written in proper order (the order is important since lhe~matrices do not commute).
Written
in matrix
form, eq. (12) is R(x2)=TR(x1)7.
tried. Some of them were: exact numerical solution (by Runga Kutta method) of eq. (37), exact numerical solution of eq. (42), second Magnus approximation [IO-121 and repeated firstMagnus [13]. Although there may be special cases when some other method is better, it was found in most cases that the repeated first Magnus was the most efficient method of solving eq. (37). The repeated Magnus method consists in dividing an interval over which a solulion is desired into subintervals. solving each subinterval by the first MagnuS approximation, and then multiplying these solutions. For example, take the case in which the interval from x1 to xN is divided into N- I subintervah withx,
/or multichannel
Wh]
(51)
The JWKB e3ression for the transmission matices T and T are easily evaluated. By definition, the transmission matrix 1 satisfies the equation +(x2)=
(52)
TJl(+),
where the wavefunction is traveling in the positive direction. Therefore cq. (46) (which corresponds to a positive sign in front of k(x)) applies. Comparing eqs. (52) and (46) it is evident that T = [k(x2)]-‘I2
U(x2,x,)
[k(x1)]li2.
(53)
Similarly 7 satisfies the equation (54)
Nx,)=TJI(x2).
where the wavefunction is traveling in the negative direction. Therefore eq. (47) (which corresponds to the minus sign in eq. (32)) applies. If we invert eq. (47) we obtain
xN-2
9!x,) X... exp [i[
k(x)dx]
.
(50)
This method converges to the exact solution as the
number of subintervals is hicreased. The S-matrix is computed by the method developed in sectidn 2 for the single channel case. The potential is divided into two parts at some division point, x1. It is important that x1 be located to the right of all the ‘classical turning points. By this it is meant that the matrix [k(x)12 has all pcsilivc eigenvalucs for x >xl. Let x2 be a point located in the asymptotic region of the potential, i.e., the lotal potential, V(x) plus the centrifugal potential, is negligibly small in the region x 3 x2. For convenience we designate the region x x2 as region C. The analysis proceeds exactly as in the one channel case. -In fat!, equations (1) through (12) and the analysis -which accompanies these equations can-y over intact and will not be repeated here. we need only to gnterpret
= [k(x,)l-1’2
+(x,x,)
[k(x)11’2q(xI,
(59
where lhe superscript T means transpose and we have made use bf the fact that U(x, xl) is a unitary matrix. Comparing eqs (55) and (54) it is evitit that 7 = [k(Q]-
‘I2 UT(x2, x1) [k(x2)] l/2 .
(56)
Comparing the expressions (53) and (56) for T and 7 it is seen that they are simply related. Each is the in-verse of the complex conjugate of the other, i.e., 7 = (T*)-l .
(57)
The relation of the S-matrix to the reflection matrix is easily derived by comparing the two expressions, (2 I) and (l), for the asymptotic form of the wavefunction Sx2-c- = -[k(x2)]l/*
exp{-i[k(x3x2-@r/2)1]
X exp{-i[k(x2)x2-(P2)l]}
}R(x2)
[\(x~)]-~/~. (58)
Compute R(x2) from eqs. (56), (S?j and (5 1) and substitute thiz.into eq. (58) to obtain
Sx,-.w
=-exp
[-iIk(x2)xz-(Qnl2)i1)
X U(x2,x,)[k(xl)l’12 R(q)
W(X,)~--“~
X UT(x,,xt)exp{-i[k(x2)x2-011).
8(x,)= j (59)
[k(x)-k(m)]&-k(~)xf+(fM!)i.
(70)
Xf This expression for&(x& can be evaluated analyt-. .
It is useful at this point to define the matrix g(x,)=
[k(xl)]tj2
I?@)
[k&t)]-“‘,
idly (see appendix 2 or appendix 3 if coulomb po@o)
which is somewhat analogous to an S-matrix. It has the desirable properties of being both unitary and symmetric whereas R(xt) is neither. Substitutingeq. (60) into eq. (59) gives the following expression for the S-matrix: s x,-L- = -exp{-i[k(x7)x2-(P~/2)tj) X U(+x*):(xt)
UT(x2. x1)
X exp{-i[k(x2)xz-(n~l~)il).
(61)
The potential V(r) will, in many czes, be short range, which means it is negligibly small beyond some cutoff point xf- Beyond point XT, only the centrifugal potently is inlportant. The unitary matrix U(x2, xl) can be written as a product: U(x?_. “1) = 0(x2. Xf) lJiX[, xt ) .
(621
In the region x > xT, the matrix k(x) is diagonal; thcrefore U(x,, x,-) can be expressed in exponential form:
U(x2,xr)=exp
if t
k(x)&]
_
(63)
xi?
The matrix elements of k(x) in the region x 2 xf are [k(X)]@
=ki[l -bF/x2]1’26g,
(64)
where ki are the matrix elements of k(x -+~0). Kc., [k(==‘)]ii=~i$,
(65)
ki=A’“[2$l(EW&i))t’2,
0351
and bj J [P(P+ I)] “2/kf *
(67)
Eq. (61) can be rewritten in the form s = -5 F&t) where
S ,
(4
tent&Is are involved).
The only remaining task is to compute the matrix c(xt). This matrix characterizes the region of the potential x dxt which contains the classical turning points. In the single channel case WCmade use of the JWKB connection formulas to obtain an expression for R(x,), but iq the multichannel case the problem is considerably more difficult. One solution, of course, is to use the exact value of !@xt ), obtained by numerically solving the SchrBdingcr eq. (27) over the intervai x < x1. This is a reasonable prop~sat since in most problems the interval over which the Schrijdir&r equation would have to be numerically solved is small. Thus the proposal is to solve the problem by integralin~ the Schr~~ngcr equation nume~c~ly out to xt and then solving the rest of the problem by the more efficie$ multichannel 3WKB method to generate the matrix G and compute the S-matrix by eq. (68). This is an especially good technique to use for problems with long-rsngc, slowly varying potentials. If we compute the S-matrix by this method, then the Langer technique (71 of replacing the quantity Q(Pf 1) by (Q + i)* in the cent~fuga~ potential term should not be used. As an exampIe of the use of this technique Cosolve a problem, we have chosen the two channel curve crossing problem which Olson and Smith [ 141 CMsidered in their analysis of He+-Ne scattering. The potential used is given in reference [ 14) by eq. (45) and table 1. Atomic units were used throu&out the calculations. The reduced masi used was 6039 au and the energy Was 2.60566 au (70.9 eV). The end point itf the ~te~ation (xf in our notation) was 12 au.-The point x1 was chosen to be about 0.35 au greater than the most positive classical turning point; thus nt ranged in vaIue from 1.6 au for P = 0 to 26 au for P z.360. The results of U&e calculations for select& values of P are giveriin table I in the columns fabcled
__
0
Isle,‘1
of exact and IWKH
1.000 0.925 0.894 0.952 0.989 0.998 1.000 1,000
1.ooo
1.000 0.924 0.892 0.952 0.989 0.998 1.000 I .ooo
1.000 I.000 0.924 0.893 0.952 0.989, 0.998 1.000 1.000
a) JWKB(B) b)
4.513 3.269 1.614 5.933 1.183 s.951 3,685 0,296 5.984 5.129 3.840 2.142 0.111 4.060 1.439
--
.
.--.-....
_ -
-
-
-
-
JWKB(A)‘)
v~~~~~ of P
- - ._ -_.-
-
o.w/(-l)d)o.L71(-1) 0.105 0.106 0.262 0.262 0.147 0.147 0.289 0.289 0.300 0.300 0.195 0.195 0.220 (-1) 0.223(-I) 0.4 14 (-2) 0.4 I9(-2) 0.382 0.383 0.450 0.451 0.307 0.307 0.150 0.150 0.577 (-1) 0*577(-l) 0.182(-l) 0.1%2(-1) 0.4861-2) 0.485 (-2)
tklct
I Slp,l
for dccted
b) cnla~latcd by eq. (89) and related equations. C) McnsuredIn radkna d) Number In patenfhesesis power of ten by which the table entzy is to be multiplied.
4.515 3.273 1.618 5.936 1.785 5.952 3.684 0.299 5.986 5.128 3,841 2.147 0.107 4.065 1.435
0.847 0.849 0.847 4.513 3.270 1.615 5.933 I.703 5.951 3.684 0.294 5.984 5.12% 3.841 2.142 0.111 4.060 1.439
.A)’ Calculated by q. (68) end r&ted equalions,
340. 350 360
.330
1.000
0.981
0.981
0.994 0,965 0,989 0.957 0.954 0.981
0.994 0.965 0.9@9 0.957 0.954
1.000 1.000 f.OOO
0.994 0.965 0.989 0.957 0.954
c)
Exact JWKB(A)
ArgQ(%
values ol the &matrix elements Sf{) rnd S[[)
‘, Exact JWKB(A) a) JWKB(B) b)
20 40. 200 204 2ofi 210 214 290 300 310 320
P’
Table 1 Camp&an
-
-.
a.171(-1) 0.106 0.262 0.148 0.290 0.301 0.196 0.209 (-1) 0.797 (-2) 0.379 0.448 0.307 0.151 0.598 (-1) 0.223 (-I) 0.107 (-1)
.-
lWKB(B) b)
c)
5.666 2.650 0.217 5.427 2.381 3.956 0.738 0.552 4.805 4.826 1.369 3.889 6.132 1.844 3.616 5.188
_
5.668 2.651 0.21 e 5.428 2.383 3.958 0.739 0.551 4.808 4.826 1.371 3,886 6.136 I:%45 3,622 5.169
5.661 2.650 0.216 5.428 2.381 3.956 0,739 0.553 4.805 4.826 1.369 3.888 6.132 1.844 3.616 5.188
Exact JWKMA) a) JWKB(B) b)
At&)‘,
389
B.R. Johncon. A generdizcd JWKB approrimarion for mdrichannd rarlefing
IWKB(A). The exact values were obtained by numerically solving ihe two channel matrix Schradinger equation. The exact results were computed to fourfigure accuracy and rounded to three figures for the table. An examination of the table shows that the JWKB(A) results are quite good over the entire range of P values for St I and S,,, in both magnitude and argument. 3.2. Method B: JWKB connection forrnuh Special caseyarise in which we can devise methods for computing S(xr) based on the JWKB connection formulas. This occurs when the problem can be ap. proximately decoupled in the region x G x, . In the most trivial case, the problem is decoupled because the off-diagonal matrix elements of k(x) are zero in this region. Let Xi be the class&l turning point in the ith channel. The matrix S(xl) will be diagonal with matrix elements (see cqs. (19) and (60)) [2i(
[S~)ljj=eXp
J
ki(X) dx--n/4)]
&ii,
Let x0 be a point eq$
to or less than the “left-mast” turning point. Then S(r,) in this diagonal representation is &(x1) = exp
2i i’ I[
G,)=!3(+lT(xt).
(78)
The orthogo@ matrix C which diagonalizes k(x) also diagonrdiics S(xt). Therefore g&r)=
[CTll(xt)CI
P%l(x,)CIT.
(79)
The matrix g(xt) is not uniquely defined by eq. (78); however, we can make it unique by requiring that g(x,) is also diagonalized by C. Thus, comparing eqs. (79) and (77),
(71)
11
(
q(x)dv-(a/4)1
. (80) II
To
where (72)
[k(x)Iir*
In some cases it is possible to diagonalize the problem by an orthogonal transformation in the region x 4x1. Let C be the orthogonal matrix which diagonalizes the potential matrix (for the present, we assume C to be a constanf independent of x) cV(x)=CT[
v(x)+tT]c.
(73)
The matrix V(r) is diagonal. Define the diagonal matrix Q(x) to have matrix elements Qi(x)=h-‘{2~[E-
clli(x)-h2(P+~)2/2~2~~1’2
Let xl be the classical turning point of the ith channel, i.e., xI is the point where Q;(q) = 0 -
(74)
(75)
For values of x < Xi, Q&r) is imaginary. Defme the diagonal matrix q(x) with matrix elements
Thus
J’ k(x) bx -(n/4)
,
x
I
-
(81)
*o where k(x)=Cq(x)CT,
xg 4-X =Gxt .
(821
Eq. (82) is a very important defmition. It redefines the matrix k(x) in the region x
=o
(77)
We must now transform the diagonal matrix cd(xt) b_ack to the non-diagonal representation. The matrix S(xr). is unitary and symmetric and,= such, it can be expressed in the form
-3
ki(x)=
.
q(x) dx - (514) I II
x0
I
(83)
B.R. Johnson.
-390
A tyenetalized JWKB app~orindon
Eq. (84) is true ordy because we have assumed the orthogonal~matrix C to be constant. More generally, the matrix U(x, x0) is a solution of the differential equation U’(x.x,)=ik(x).U(x,xo),.
(89
with the boundary condition U(r,, x0) = I.
(86)
We now lift the restriction that C is constant and assume that cq. (83) is stilt a good solution for g(xr )
with U(xt , x0) defined by eqs. (85) and (86) and k(x) defined by eq. (82) where the matrix C in eq. (83) is now a function of x. Substitute eq. (78) into eq. (68) s = -G
e(q)1
rG dq)l=.
(87)
Define G = E g(xj) exp[irr/2! t
(88)
Substituting eqs. (69) and (83) into (88) we obtain the following elegant and simple expression for the Smatrix: S=GG’,
(89)
where G = exp[i&(xr) t (in14) I] U(X~.X,-,).
(90)
and U(x, x0) is the solution of eq. (85). If we compute the S-matrix by this method, in which we have made use of the JWKB connection formulas, then Langer’s technique (71 of replacing the quantity f!(n t 1) by (P + i)* in the ten trifugal potential term should be used. This statement applies in particular to eqs. (28) and (67), which will be changed by this replacement. -As an example of the use of this method, we have solved ihc same problem considered previously. The results arc given in table 1 in the columns labeled JWKB(B). An examination of the table shows that the results are quite good over almost the entire range of P values. The exceptions occur when the magnitude, t@t, is very small. Note !I particular I,$,)] for P= 214 290,340,X0 and.360. Note also that although I&( is in error me values of Arg(S#) are still quite good for, these P values. The consequences of these errors in ISg] are not very great, however, because although the rela’tive error is quite large, the absolute error is very small. To demonstrate this, we have
for mulrichannel
scattering
calculated the total inelastic cross section, Qt2, for excitation from the ground state. It was found that the exact results had converged to four figures by the time Qtz had been summed to P = 360. The result, rounded to three figures, is Q,* = 0.693 au. The JWKB(B) results, also summed to P = 360 and rounded to three figures, is exactly the same. In each case the convergence was checked by summing to P=400. Method B was derived by assuming that the potcntial in the region containing the turning points, 0
RR. Johnson. A generalist-d
IWKB
approxi%atiim
for multicimnel
rmrlcring
4. Distorted wave bansformatians In this section we will derive various forms of the distorted wave transformation. They are of interest in their own right_ and also as a necessary prelude to deriving Landau-Zencr type formulas for curve crossing problems. The basic first-order differential equation derived previously is (c.f. eq. (85)) U’(x, x0) = i k(x) tJ(x, x0),
ft is possible to derive a more symmetrical form of the distorted wave transformation, which we call the “split-distorted wave transformation”. Let xc be some point betwcenx,, andxt, i.e.,xO
ww
(91)
Using the distorted wave transformation we can write
(92)
U(xt.x,)=exp
with boundary condition U(x,.xo)
= I.
The first distorted wave transformation derived is well known [ 15,161. Let k(x) = kotx) + k, (x1 z
[-[I
h,(x)d+]
fJ,(q.+
002)
k&3&]
UL(xo.xc).
(103)
to be and also (93)
where k,,(x) is the diagonal part of k(x) and kt (x) is the non-diagonal part; also let
U(x,, Xc) = cxp [-r1 4
U(x,x~)=U~(x,xwg)u~(x‘x~),
Take the hermitian conjugate of eq. (103) and use the relation
(94)
where
U(X, xf) = Ut
Ub(x, x0) = t ko(x> U,+, x0) -
(95)
(Xfs Xl)
(W
to obtain
Eq. (95) can be solved since ko(x) is a diagonal matrix: U, (x. xr,) = exp
[
i _f kg (x’) dx’] .
(96)
=0
Substituting eqs. (105) and (102) into (101) gives
Substitu~g eqs. (93) and (94) into (91) and using eq. (95), we obtain
U(xl, x0) = exp
i j’
ko(x) ti]
XC
U(x, x0) = exp [_i 1
k,G’Nx’]
U,(x,xo)
(97)
and where Ut (x1, x0) is a solution of eq. (98) with Ui (x,x0) = i K(x) tJt (X x0),
(981
where
r(i)=er.[-il.,(;)a.‘]
tC(x)=exp[-!
j ka(x’)dx’] (107)
xk,(,)rrp[ijk*(r)ar] x0 with boundary condition
(99)
This procedure can be generalized to more than a single splitting. For examp!e, Iei
392.
BR. Johnwn,
A generalized JWKB appmximdon
for multidannel
xattering
when computing U1 (x1, x0) with eq. (98). It is obvious how this procedure can be generalized to even more division points
u(x,,x~)=u(~*.~~)u(x~.x*)-
w@)
‘, The matrix
U(x,,,x,) can be transformed as in eq. (I(K) with xb as the division point: U(x,,x,)
= exp
i j?
1 XI 1
XU1(x2,xt)exp
irQ(=)dx
iu the .qditdistorkd wave picture
5. I. Ckrve crossing
ko(x) dx
xb
5. Smahix
Substituting the split-distorted wave expression, eq. (106), for U(xf, xO).tito eq. (90) gives (110)
Likewise, for the matrix U(x, ,x,-J with xp Y the di-
G = exp [illl tJl&,
~0)
expb~l,
(119
where the diagonal matrices ‘1 and G have matrix elements
vision point, U(x,, x0) = exp
ii’ E 5
vi = [6(Xf)]ii + xc [k(X)]ii _I-
ka(x) dx]
[fkob)&].(111) 1
Substituting eqC (111) and (110) into (109) gives
[
(116)
*C
and
X U1bl,q-& exp
U(x2.xO)=exp
dx + n/4
k,(x)&
ii’
3
XL$(x,~~,)exp[i[
k&)dx]
ii” kg(x) dx] , (112) x0
;;;: =i’ [k(X)] ii ~. X0
(117)
The S-matrix is then obtained by arbstituting eq. (115)into(89). So far our treatment has been completely general. ‘Lzt us now specialize to the case of a two channel problem in which the potential cumes cross. The matrix U1 (x,, x0) is a unitary 2 X 2 matrix which, in general, can be expressed in terms of three real parameters. Pl/* exp [-io]
-(l
-P)ll*exp[-i@i
U 1(Xf. x0) = ( (1 -P)*lz
exp[i@]
P1j2 exp[ioi (11’3)
Xkl(x)exp
i ~ko(cc’)d_r’
, I
013)
4
when computing U1 (x2. xt) with eq. (98) and . K(x)=eyp
[-i[kbb?h’]
Xll(.)erp[ijk~~x~)&~,
(114)
where 0
c;(x) = i k1*(X)
terization of the S-matrix S=exp[id
Ur(xf,xo)exp[2i;i]
U~(xr,xu)exp[irl)
. X exp
(11%
i z [k,,(x’)-&(x’)] 1 152
dx’
C,(x). I
am
This is similar to the factorized form of the Smatrix given by eq. (2) in reference [ 171. It should be apparent how this factorization scheme can be gen. erahzed by making use of the multiple split-distorted wave expression given by eq. (112) or its gener~ation to obtain a factorized S-matrix similar to eq. (8) in reference [ 17) _ Doing the matrix multiplications indicnted in eq. (119) results in the following pararneterized form for the S-matrix elements:
Expand the matrix k(x) about the crossing point x, (it is easily verified that k,, (xx) and k22(x) cross at the same point, xe, that VI1 (x) f L, and Vzr (x)+ L, cross) and retain only the first ~on-~~~~g irrm
Sll = fP expPi(;i; - aI1
k,*(X)==+,
kll(x)-k2z(X)~Q(x--XCj’
02%
+ U-P) exp[W?~ - 0113expI2iql ,
(120)
$2 = fPexp[2it;i2 + 011 +0-P) s12
= 821
(121)
expl2iG~ + tW expf2if721, 1-P)) 1/2{exp[i(2;i; +# - 0))
= [P(
where the notational change is C, 0) = Qj(x* x0) >
j= 1.2,
‘(126)
C2(x)=~~&Jn).
i= 1,2.
027)
where by our conventions, E and R are both positive quantities. With these approximations, eqs. (124) and (125) are C;(x)=
C;(x) = -i e exp [iax*
It is easy to show from this that \s,,i*
= 4P(1 -P) sirG(;j; -;j;-$++u).
(123)
The physical interpretation of the oscillatingprobability expressed in eq. (123). resuking from the interference of two amplitudes, is weIl known from the Landau-Zener theory of curve crossing. 5.2. Zenef upproximtion ihe parame ten P, u and 4 can be computed by an approximate analytic technique due originally to Zener [18] for-solving eq. (98). Written out in component form, with some change in notation, eq. (98) reads
Ci(x) = i k,z (x) -i 7 [k,,(x’)-kz(X[)l
Xexp (
I
=C
-i (sexp[-iax
C,(x),
(130)
(122)
-exp[i(2;j;-~+al)exp[i(tll+82)1.
h’
C26) i
024)
C,(x) .
030
Zener has protided a method for obtaining the asymptotic solution to these equations. His method is to eliminate C,(x) by substituting eq, (131) into (130). The resulting second-order differential equation for Cr (x) can be transformed to the Standard form of the Webcr equation, the soh~tions of which are the parabolic cylinder functions which have a known asymptotic form. From this risymptotic solution it is possible to obtain two sets of asymptotic values of C,(x) and C,(x) corresponding to the two sets of initial conditions implied in eqs. (126) and (127) when i = 1,2. From this we obtain our approximation to the matrix U, (x,x& which has the form given in eq. (1 I8), with the following values for tfte parameters of P, u and #: P= expf-27ry],
(132)
u=o,
(133).
Cp= n/4 + Argl%)
+ 7 NY) - 27 I@),
.(f34)
.?. .. .
.( ..,
-’
:
where’, ‘.
:
‘.
T 5 &&I I
(135)
and p is sdme characteri#ic bngt@ pammeter far the prcibtem which is not exactly determined by the anal:
‘Substituting from eq. (70) into kq. (90), we obtain
ysis. The exact value of p .isnot very impartant since it OC~WS in a ~oga~~rnic term. .A reasonable value whkh we have found, based on a per~rbat~on theory am&h, ii J3”&2E-
*
(e = base natural lo&
(136)
The ~aramele~ Q and e (see eqs. f 128) and (129)) are the exp~sion coefficients of the matrix k(a) about the crossing point. In order to compare our formulas with the formulas of landau and Zener, it is essentirdto relate.&and F to the equivalent ~~f~~jenf~ far the expansion of the potential matrbr V(x). ExpandirigV(x) about the crossing point, xc, and retaming oniy the first terms, 037)
v&al=a,
p&(x)-
v;t(x)++--
8, =I(.=-~J*
(138)
where by our conventions, a and fare both positive qu&ies. Usingthe relation between V(x) and k(x) provjded by cqs. (28) and (36), it is not hard to show the fo~ow~S relations are true e =OpV,,
.
(145)
Substituting thii into eq. (144) produces an expression which ir the product of hvo matrix exponents functians. Strictly speaking, the exponents cannoi be added; however, it is the essence of the fu;st Magnus ~~proxirnat~onto assume that they can. Thus G=exp
i = B(x) - k(-)] dx - k(a) x0 [iJ XQ
+ {@+Qnlz]I
*
(1%).
II
039)
is lhe first Magnusapproximate to G.
(14Q)
The matrix in the exponent is symmetric; therefore G = GT and the first Magnusapproximate of the S-matrix is
and 4 =fhu,.
u(x,x,)=erp[Ihfk(x)b;]
where u, ij the velocity at the crossingpoint given by p,=@lri)[EIn
Yll(x,)-ti*(p+4)2121u~--LI]3112. de&& these relations we have assumed (IQ0
. that
2i j.[k(~~)-k(m)] II
This last ~mption
breaks down, of course, when the crq&ng p&t and brig points are very near to ‘each other. ‘. Substituting eqs. (139) and (140) into (13S)‘gives 4 ,=‘i&E& )
(143)
which is the usual Landau-Zener form for this : patieter
[a, IS]. ” ).
: :
dx-k(o)xg
-%
f [(~~~~~~~~I
043
~~*2(x~~~‘~~ u, -
..
S=exp
11
-
(147)
This can also be writ&en s = expr2i4 ) where qis the gener~ed phase&& matrix
(148)
ate folly identical, the only difference being that in eq. (149) the quantities are matrices.
It is an inrcmst~g exercise to derive one of tbc more familiar semi~classical expressions of theSmatrix by using the SWKB expressions derived earlier as our starting point In particular, it is easy to derive an expression for the S-matrix which corresponds to solving the time-dcpcndent Schradinger equation for a time-varying potential p;enerated by the incident particle following a straight line trajectory with impact parameter b. The S-matrix is given by eqs. (89) and (90) with the matrix U(x,, x0) obtained by solvingeq. (85). The matrix k(x), defined by eqs. (28) and (36) (replacing P(P + 1) by (2 + i)2) can be rewritten in the factored form:
Then substitute eq;( 154) into eq. (90) for G and use eq. (157) to obbiri
The exponent in the first exponential factor is just the phase-shit for the centrifugal Potential which is zero; therefore GXf--
= exp Wfi) [J+JC~)~ c:) Uk @f. ~0).
t x(r) = j-c+‘) 0
dc’ + 6.
The d~ferenti~s
(16%
are related by
ti=u(C)dr.
I - [ llrtu(~)]
[V(x) +c 1 ,
0521
where 0531
is the radial veiocity of the incident particle. Using the distorted wave ~~sformation
U,(X,,Xo).
(154)
dU, (x, xo)fdr = i K(X) U, (x, x0),
G,A,
= expHil@ rtl
Ul(f.0) ,
063)
where [email protected])/dt=
--(ii&) f V[xtt)l
+ &~~~~f,O),(l6~)
Equation (164) can be transformed to the retorted waveform U;W)=
Here, U1 (of, x0) is the soluticm of the d~fcrenti~ equation
where
(162)
x0)i -Mm-u(+t. Thus, eq. ( 159) is transformed to
Nx) = ~~0~~)~~ ,
U(Xf, x0) = CXP[-i’WW] 1
060
For the particular case of the straight line trajectory it can be shown that
first two terms of its binomial expansion k(x)=+,(x)
(W
Change the independent variable from the radial distance, x, to the time, t. These variables are related by the equation
expf-WcErl
U&O),
where WW
dU~{r,O)/dr = -(i/ti)
G(r) U,(t,O) s
Th matrix element
of a(f) are given by
065)
B.R. Johnron. A geneml&edJWKBapproxrnnrion /or
396
‘cand, as mentioned previously, is the threshold energy level of the ith channel. Substitutingeq. (165) into eq. (16% G= U,(-,O).
‘(169)
Using eqs (167) and (168) it is easy to show that the matrix a(t) is hermitian, i.e., a(r) = at (I).
(170)
Since the time variable, t, was chosen to be zero at the distance of closest Ipproach, the radial distance, x(l), is symmetric about I= 0, i.e., x(r)= x(-r).
(171)
From eqs. (171) and (167) it is easiIy shown that Q(r) = V(4).
(172)
Using eqs. (172), (170) and (166) it can be verified that u;o,o>
= U*(O, -0 I
(173)
mulrichannel s~ttering
and the first Magous approximation. Undoubtedly other approximation metbods can also be devised. In this section, we remind the reader that we can always break the region x0 Gx Gxf up into subregions and write U&f, xO)= u(r[9rn) U(x,,x,)
*.. U(x,rXO)
‘( 178)
where XO
(179)
Then we can treat each subregion separately and use the appropriate approximation in each region. For example: In one region we may have to solve cq. (85) exactly; in another in which curve crossing occurs we might make the split-distorted wave transformation with the Zener approximation, and in another region the first Magnus approximation might be appropriate. These various approximations, each valid only over a limited region, can then be combined by eq. (178) to obtain the overall solution.
and therefore G-t = tJ2 (0, --)
.
(174)
7. sumnlary
Thus, we obtain fmally S=GG*=U+,-=).
(175)
If we assume that eq. (166) can be evaluated by the first Magnus approximation, we obtain S = exp[2iri) ,
(176)
where the matrix Q has matrix elements given by q’il = -ffj-t
/ exp[i wiir] YVir(r)] dr . -0D
(177)
This should be compared to the similar formula derived by Cross by a different metbocl [4].
6.3. Combined approximtions The major numerical effort in solving for the Smatrix as expressed in eqr (89) and (90) is to solve’ . for the matrix U&. x0). This matrix is defmed by the differential equation (85) with the boundary condition (86). Several approximation techniques have been men-tioped in previous’sections, i.e., Landau-Zener type approximation, which applies in curve crossing cases,
A multichannel approximation method that is a direct generalization of the JWKB approximation has been presented. The turning point problem was handled in two ways. In method A, the Scluijdinger equation was solved exactly in the turning point region and then JWKB solutions were joined to this exact solution. In method El, the problem was assumed to be exactly cliagonalizable in the turning point region, i.e., the unitary matrix, C(x), which diagonaltzed the matrix k(x) was assumed constant in the turning point region. Then the ordinary JWKB connection formulas were used to solve the diagonalized problem and the solution was retransformed back to the original non-diagonal representation. By this technique we derived formulas which were assumed to remain valid even when the unitary matrix C(x) was not strictly constant in the turning point region. An example problem was worked out and the results presented using both methods A and B. Next, a new form of the distorted wave transformation which we designated the split-distorted wave transformation was derived. This allowed us to factorizc the S-matrix and when applied to the two stale curve crossing problem led us directly to a derivation
of Landau-Zener tVpe formulas for the S-matrix. Finally, two approx~ations to our generalized JWKB formulas for the S.matrix were presented. The first was simply an app~c~tion of the Magnus approximation to our formulas. In the second, by making certain further appro~m3tians, we were able to give a new derivation of the straight line t~jecto~imp~ct parameter formut3tian ofscatte~ng theory.
Two matrix functions have been proposed as the generalized JWKB solution to the matrix Schradinger equation, One was the solution to eq. (45) (which we repeat here), * 01 WI $64,
(Al.1)
where k(x) and at(x) are defined in section 3, This equation has the adv~~age that it was derived using a systematic expansion in powers of& {see eq. (3 I)); it is also ;i very good approximation. It has two disadvantages, however: It is c~m~crsome to cvalustc numcric~~, and it does not conserve probability exactly. The Dther solution proposed was eq. (46) (reseated here) GExI- Ik’k)l-‘”
~[k(~)]-‘~*~’ =a&) fk(x)]-“2. Substi~t~g
(AL%}
the easily proven relation
{[k(x)]-‘12}’
= -[k(r))-f~2
into eq. (A1.4), we obtain a(w) = -[k(x)]-tit
{[k(x)] J~2)‘[k(r)]-1~2 (A1.5)
{[k(x)] ‘D )’ ,
(Al.@
thus dete~inin~ the function a(x) which appears in eq” (A1.3). From eq. (A1.6) it is easy to prove &e relation
Appendix I
Jr’(x) = f* i W
the resuIt
Ukx,)
[ktxlIf1/2
JtCfxl),(AI.2)
where iJ(x, x1) is a solution of eq. (37). This solution is relati-leiy simple to compute numerically, it conserves Frobability exactly and is also an excellent ap proximation. Obviously, with ail these advantages, we chose ii: as the generalized JWKB solution of the matrix Schrijdinger equation. The only disadvantage (if it can be considered a disadvantage) is that it was not derived by any systematic expansion procedure, but merely (a not very occults intuitive guess, based.on the form of the single channel solution. It is the purpose of this appendix to relate as much as possible these two solutions. Assume that the differentj~ equation which generates Jr(x), given by eq. (Al .2), is Jr”(x) = fi klx) + ~G.91 IJr69,
(A1.3)
i.e., the same form as eq. (Al. 1) except that in this case U(X) is an’as yet undeterred function of x. Substituting the wavefunctian IA1.Z) into eo. (A1.3) produces
which shouId be compared to the equation for eI (x) (eq. (33) in text),
In #is appendix an analytic expression is derived for the elements of the diagonal phase shift mat&, 6&j, defined by eq. (70): &&x~)= f
[k&x) -k&=)1
dx - ki(+xf
+ &r/2,
(AZ* 11
-9” where k&x) is (see cq. (64)) kj(x)=kj(l
-L&X*]~f*
.
(A2.2)
Therefore
l%e integral in eq. (A2.3) is easily evaluated by rn~~g the substrfu~on x = “f/Y ,
(AM)
and the finaf result is &i&f) = &/2-kjxf
[(J -~~~~~)I’~
B.R. Johnson. A generalized JWKB approximari?n
398
6j(Xf) :.pn/2 - kibj +
[(X:/b;- I)“*
/VT mulrichannelscrrrrering
Sx,-r-
= -exp[-i{k(x2)x2-@n/2)
(A2.6)
Sill-’ (bi/Xf)] ,
x U(x2.q)~(q)
Xexp[-i
which can also’be written .ti the form ‘ai(rf) = PIr/:!-_jbi
{($/bf -
-+ tall-~ [($/#Appendix
3:
I)“’
I)-‘/*]).
(A2.7)
W)x+ca
= exp E-i [kx - @n/2) I]) - exp{i[kr-
@r/Z) I]},k-t/*S kf/* .(A3.1)
In the case where some or all of the channel potentials include a Coulomb term, this expression is generalized to [19,20] Jl(x)-exp{-i[lc-y
In (2kx)-(&r/2)
+up 1
[k(x) - k(x2)l k - k(x+c
“(Xf)*, -t_ = j *r
t (&r/2) I -up + y ln[2k(x2)x2]
. (A3.7)
This expression CM be written in component form and evaluated analytically, just as it was in appendix 2 for the non-Coulomb case. Analogous to eq. (A2.2) tie expression for ki(X) for the Coulomb case is ki(X)
=
ki( 1 - 27i/kiX
- b3/X2)
(A3.8)
_
Therefore
S kl/*, (A3.2)
where
[( 1 - 27i/kiX - bf/X2)“2-
l] dx
Xc
-kixftPfl/2.
[oQ]i+ Ti1n(xix2)’
(M.3)
Here e is the electronic charge and Zt and Z2 a;e diagonal matrices whose matrix elements measure the charge of atom 1 and atom 2 in the various channek. The matrix elements of Uie diagonal matrix [Op]i = Arg f’(P + 1 + iyi)
(A3.4)
are the Coulomb phase shift for each state. The relation of the S-matrix to the reflection matrix, analogous to eq. (58), is SI, --
UT(+.q)
I +an])
yIr1(2kx)-(Pn/2)l+a~])k-~/*
7= ~1e2V2 Z&k-r
+
(~3.6) Eqs. (68) and (69) remain unchanged in form; however, eq. (70), which dcfmes the quantity &i(xf). is changed to
s,(Xr)x, __ = ki _/ -exp{+i[kx-
yln[2k(x&]
(k(x2)x2-(lllr/2)1-rIn[2k(x2)xl]
Coulomb potentials
Recall that the definition of the&matrix, for the case of no Coulomb interactions, is given by the asymptotic expression for the radial wavefunction
I -
(A3.9)
The integral in eq. (A3.9) is easily evaluated by making the substitution x = x-r/y ,
(A3.10)
and the result is 6i(x,)=Pn/2+ri
In(2ki)-
[o,]i-kix,l-ri,(A3.11)
where I=(1 tp-a)‘/*
= -[k(x2)]1~2exp[-i{k(x2)x~-(&r/2)I - r ln [2k(x2)+1
X R(x2) exp[-i(k(x&-(&r/2) - Tln[2k(x&2]
+ al/* {sin-q(2Q~)/(p*+4a)q
tsin-‘fp/Bl(p*+4c#q)
t@/2)On[2(1
-h1[4/~~]),
+apH
+op)]
The new form of eq. (61) is
+~7-#~+2+13]
I
[k(x2)]“”
(A3.12) (A3.5)
4 = -%J(kix~) a 7 b /xf . f
3
(A3.13) (A3.14)
If 7i = 0, the above expression reduces to eq. (A2.5) for the non-Coulomb case, as it should.
B.R. Johnson, A
gcnernlized JWKB upproxltition for mul~chunnel sastrering
Referems
[ I] R.B. Bernstein, in: hfolccuku beams, cd. J. Ross (triterscience, New York, 1966) p. 75. [2] B.H. Brands-en, Atomic colbion theory (W.A. Benfamirr; New York, 1970). 131 D.R. Batesand DS.F. Crothcn, Roe. Roy.Soc. A315 (1969) 465. [4] RJ. Cross, J. Chem. whys. 47 (1967) 3724~48 (1968) 4838. [Sl E.C.G. St%kcIberg. Hclv. F’hys. Acta 5 (1932) 370. f6l J.B. Debs, W.R. Thorson and SK. Knudson, whys. Rev. A6 (1972) 709; J.B. Delos and W.R. Thorson. flryr Rev. A6 (1972) 720. [7] R. Langer, F%ys. Rev. 51 (1937) 669. f8l L.D. Landau and EM, Lifshitz, Quantum mechanics (Add~on-Wc~cy, Reading, Mass., 1965). 191 J. Heading An introduction to phase-integral melhods (hiethuen. London, 1962).
399
[ 101 W. Magnus, Commun. Rue Appt. Math. 7 (!9.54) 649.. (111 D.W. Robiin, He&. J’Jtys.,Acta 36 (1963) 140. [ 121 P. L’echukasand J.C. Light, J; ‘Ckem. phyi. 44 (1966) 3097. [ 131 S. Chart. J.C. Light and J. tin. J. Chcm. whys. 49 (1968) 1141 EE. Ofsan and F.T. Smith, Phys. Rev. A3 (1971) I&?? (Err&, J’hys. Rev. A6 (t972) 526). [ 151 D.R. Bates, Roe. ehya. Sac fLondon) 73 (19X& 227. 1161 R.D. Leviie, Chem. whys. Letters 2 (1968) 76. [I?] JE. Bayfield, E.E. Nikirin aJId A.[. Reznikov. Chcm. Phys. Letters 19 (1973) 471. [It31 C. Zener, hoc. Roy. Sot. Al37 (1932) 696. (191 N.F. Nott and H.S.W. Massey, The theory of atomic coflisions. 3rd Ed. (Clarendon Press, Oxford, 1965). 1201 A. hfcssiah, Quantum mechanics, Vol. 1 snort-Hoed, Amsterdam, 1965).
A GENERALIZED
JWKB ~PROX~~AT~~N
FOR MULTICHANNEL
SCATTERING
t:
Received 30 July 1973 Revised manuscript received 17 September 1973
The we&known JWCB npproximstian is gcnenlized to handle multichannel suttering problems_ fnduded is s discussion of how the JWKB connection fotrnukts can be applied to the rnu~t~pi~&ssical ~min5 po~tswhi~h occur in the m~tichannel case. A new and useful form of the distortedwavetra~forma~ion is dcrivcd andapplied to the two state curve crossing problem. The result is a useful factorization and paramctetiza:ion of the S-matrix. The parameters arc cvzduatcd approximatcfy by a method due originally to Zencr to obtain L;m&u-Zencr lypc formuks. It is aJso shown how the equations of the impact parameter method wiih stra$ht line trajectories an be derived from our generalized JWKB equations.
The JWKB(designated WILE!in many texts) approximation is discussed in most quantum mechanics text books. It is a very useful methad for generating ap proximate solutions of the one dimensional SchrBdinger equation. As is well known, this approximation is valid only in the limit of a slowly varying potential such that the fractional change in the deBroglie wavelength over a waveleng~ is small compared to unity. The JWKB salutian not only requires less effort fo compute than the exact solution, but it can also add insights in&othe physics, enabling inferences to be drawn which would not be possible otherwise. Applied to the problem of scattering from a central potential, the JWKB approximation is used to solve the radial SchrBdinger equation, thus obt~~g the * JWKB phase shift for each P value of the pattial wave expansion {11. In the more general case of inelastic scattering, the radial Schr~ger equation is replaced by a system of coupled differential equatioqs [2). This system of coupled differential equations is written most elegantly t W0t.k sup&ted
by rhe Air Force &ice of Scientic Research, (AFSC), USAF, underGrant No. AFOSR-72-2212.
in matrix form. Various semi-classicalmethods have been proposed for obtaining approximate solutions to this radial matrix Schrcidingerequation l3-61. In this paper we propose. a new method which is a direct generalization of the ordinary JWKB approximation. la section 2, we introduce the reader to the method by deriving the ordinary JWKB expression for the S-matrix in a mannet which can be gener~~d to the multich~nei case. In section 3 a generalized multichannel JWKB approx~ation which includes a method for handling the multiple classical turning points is derived. In section 4 we discuss the distorted wave transformation and derive a new version which we cali the split-distorted wave transformation. Section 5 inciudes an application of the spiitdistorted wave traasformation to a tv+.state curve crossing prabkm to okin a useful factorization and ~a~ete~~on GZ the Smatrix, then the me&ad of Zener is used to ob. tain approximate Landau-Zoner type formu%& Fir&&, in section 6, several approx~a~ons to oat generalized JWKE equations are discussed.
2. Potential scat&&g by the. JWKB methad
!.
As ti aid ~unde~~~g the general muir&&armef in&hod to. be developed, we wiH font ymstier sjr$Ie :.
:
382
B.R. Johnson. A gckxalked
J WKB
approxindon
channel potential scattering from a spherically symmetric potential, V(x), and derive the ordinary rwKB solution for the S-matrix in a way that can be easily generalized to the multichannel case. The derivation is based on physically intuitive ideas of the scattering process. Assume that the total potential, V(x) plus the centrifugal potential which we designate Vs(x), is negligibly small in the asymptotic region x > x2 _The asymptotic farm of the wavefunction can be written in the form Jl(x).tzxs = exp [-ik(xz) + exp[W+)
(x -x2)] (x - x2)1 R(x2),
(9
where R(+) is, by our definition, the reflection coefficient at the position x2, and k(x)=tl-92/&5-
mulrichnnnel
~(xL<,,=exp]ik(xt)
smrrering
(x- xt)]b
+exp[-Wxt)(x-xr)]ti,
(3)
and in region C the wavefunction is the transmitted wave +(&X,
= exp [ik(x2) (x - x2)] 7% .
(4)
In a similar manner g and ? are reflection and transmission coefficients which characterize region B when the incident wave, exp[-ik(x2) (x -x$11, is detined in region C. This is a wave of amplitude I, traveling in the negative direction. The total wavefunction in region C is tbc sum of this incident wave and a reflected wave WX>X,
v(x)-82(P+a)2/2Eu2])‘/2.(2)
Here E is the total energy, ~1is the reduced mass and Q the angular momentum quantum number. (Note that Kanger’s prescription [7] has been used, i.e., (Q+$)? replaces Q(Q+ 1) in the centrifugal potential term.) Conservation of probability requiresR(x2) to be a complex number of unit amplitude. Letx, denote the classical turning point. Divide the problem at some division point xt where xr < x,
for
= exp[-i&) + exp[ik(x$
(x - x2)]f (x -x1)]&
(5)
and the wavcfunction in the region x G x1 is the ~~smitted wave ~(xkx,
=exp[-ik(xt)(x-xt)]
71
.
(6)
Eqs. (3-6) define the four parameters 6l, T, z and ? which characterize region B. Region A (0 =Gx
= exp[-&(x1)(x--xl)] + cxp[ik(xl)(x-xl)]
u &T&r.
(7)
Thus, using electrical engineering jargon, we can think of the problem as three “black boxes”, shown ~hematic~y in fig. 2, which are interconnected as indicated by the lines. Eqs. (8-10) should be ‘obvious from this figure.
383
B.R. Johnson. A gcncralized JWKB approximation for mulrichannel scnmring
substitute this into eq. (10) to obtain R(x2)
= z + T(1 -R(x,)tR]-’
R(q)?.
(11)
Thus,
the overall reflection coefficient, R(x2). has ken egressed in terms of the four puameters hr, r, I~Iand T which characterize the potential in region B and the parameter R(x,) which characterizes the potential in region A. WC now wish to obtain the JWKB expressions for these parameters. Since, within the JWKB approximation, reflections can only occur at a classic_alturning point, it is immediately evident that di = 6I= 0. Thus, eq. (I 1) reduces to the much simpler cxpressian
R(xZ) = TR(x,)?.
(12)
The JWKE! expression in the region xt Qx
$(x) = [k(x)] -112exp
Fig. 1. Potential is plotted vertically 2nd radkd dktancc horizontally. In iig. la the total radial potcntixl. VQ(~). is shown schcmatimlly divided into three regions. The ConStNction of the potentials in figs. lb and lc arc described in the text E is the total energy and xl the classical turning point.
LZ=3+Rb
(8)
b =R(+)P
(9
R(x2)I=%+
7%.
Substitute eq. (8) into eq. (9), solve for 6, then
(10)
[i[
k(x’) clx’]
The definition of the transmission coefficient, T, is (see eq. (4) with $(x,) = 6) (14)
IL(x2)=TNq).
Comparing cqs. (13) and (14). it is evident that T = [I+)]
-II2
exp[il
k(r)dr]
[k(xl)jt/2.(15)
Similarly, the JWKB expression for a wave traveling in the negative direction is
$(x) = [k(x)] -l/2 exp [-i J k(x’) dx’] x2
(9). X tk(xz)l 1’2rc,
(16)
Making use of eq. (6) it is apparent that
The JWTCBexpression for the wavefunction in the Fii 2. “Black box” schematic of tie scaltering problem.
region xt
B.R. Johnson, A gmemlized JWKB appmxinwtin /or mulrichannelsarttaLng
384
WI
$(xj=C[k(r)]+*sin
where Cis an arbitrary constant. Using eqs. (7) and (18) it is easy io show that
R(xl)=exp
2ij’,k(*)br-irri2] Xt
.
w
Substiitutingeqs.(19),(17)and(IS)intoeq.(12)we obtain the obvious result
R (x2) = exp
k(x)dx-in/2
- exp[i(kx
1,
- PIT/~)] - IZIT/;?)&-~/*S kl/* ,
(21)
where k =k(+.
= -exp[-2i(kx2
- h/2)]
R(x2).
which is the usual JWKB expression for the phase shift [Il.
3. The multichannel JWKB approximation
[d*/dr*
+tr-* p*(x)]
where p(x)=
[2p{EI-V(x)-
~-[fi*Q(Q+J)/2~*]1~]
II2
(28)
In this equation: E is the total energy, ~1is the reduced mass, V(X) is the symmetric potential matrix which approaches zero as x approaches infmity, P is the angular momentum quantum number, I is the unit matrix and c is a diagonal matrix whose elements are the threshold energies of the various channels. The wavefunction Jl(r)‘is a square matrix, each column being a linearly independent solution of eq. (27). The logderivative matrix is defined to be
w
v’(x) +I+
The usual way to express a one izhanncl scatter& matrix is in terms of th& phase shift. R. defined by _
eqs. (25).and (24)
(27)
where the prime means differentiation with respect to x. Differentiating eq. (29) and using eq. (27) to eliminate the second derivative term, we obtain the matrix Ricatti equation
SX,-r_
Compa+g
+(x)=0,
Y(~)=JI’(xN-U,
Substituting eq. (20) into (23)
Sqexp[2ig].
(26)
(22)
Comparing eqs. (21) and (1) s*p,
dr-,loc,+(P+$)n/Z,
A generalized JWKB approximation can be derived for the multichannel (matrix) Schrtiinger equation. The derivation is analogous to the derivation given in section 2 for the ordinary JWKB approximation. The matrix SchrCdinger equation can be written in the form:
At this point it seems that we have used a very complicated rezoning process to obtain eq. (20) when it cdd have been obtained as directly as eq. (19). We remind the reader that the main purpose of this section is to establish a method of derivafion that can be generalized to the multichannel cae. The reflection coefficientR(x2) can easily be related to the S-matrix. The S-matrix has the following defmition in terms of the asymptotic form of the wavefunction ,J,(x)x- OD= exp[-i(kx
Q= ‘- [k(x)-k] I =1
(29)
p* (2) + v*(i) = 0.
(30)
In a manner entirely analogous to the derivation of the JWKB expansion in powers of h, we write theexpansion [9] ...I . A311 Substituting into eq. 00) and equating like powe of fi, the following expressions are obtained for the fmt two terms of the expansion V(X) = (i/ti)[oo(x) +(rl/i) al(x)
+ (fi#g2(x)+
,:
385
k3.R Johns&. A generalized JlVKBqqraximalion for mdlichanne~ scatterirlg
should be recalled that exactly the same difficuIty occurs in the ordinary JWKB case and that thy?SO~&~OQ isto retain the second term, ut(x), of the JWKB expansion (see eq. 3 1). In the multich~e1 case this is also a solution, i.e., we can solve From eq. (29) we obtain
e’(x) =I [* i k(x) + s1 IX)] \ii (x) . (34)
$‘@I = YW#W.
Substituting the first term of the expansion for v(x) into eq. (34) gives J;fx)S-tik(x)Jt(x),
(35)
where k(x) = n-l p(x).
(36)
It is convenient at this point to defme the unitary matrix U(x, x3 ) by the equation U’(x,xl)=i
k(x)U(x,xl)
(37)
(the prime means differentiation with respect ~0x1 with the initial boundary condition U(x,, x1) = I #
(38)
Then it is obvious that the solution to eq. (35) (with the plus sign) can be written Jt@)= U~X,~~)~(~~),
(39)
and the solution to this equation with the minus sign is JICx)= UYXX~)WX,),
(401
where * means complex conjugate. The unitary matrix, U(X, x1), can be written in exponential form [lO-121 U(~,~l)=explA(xP~t)I, where the anti-hermitian ~ferenti~ equation A’(x,xlI = i(exp(C+I]-l
(411 matrix A(x, .x1) saWTies the C,, k(x),
A(xl,xli)=O.
(421 (43)
The linear operator C, is defined by CAL-At-LA:
(44)
(45)
Numerical studies hsve shoarn the solution to eq. (45) to be an cxcellenl approximation to the wavefunction. However, ~~0~~ this is a possible approach, it is not sat~factory, from the point of view either of computations efficiency or of theory. ~ompu~tio~ly, a long calculation which involves matrix diagonal& zation is required to solve eq. (33) for si (x); also. the quantity p’(x) is required, which means that we must know the fist derivative of the potential matrix. Finally, even if we do solve for u1 (x), jt is more dif!G cult to solve eq. (45) than eq. (35). TXe theoretical objection to eq. (45) is that the solution, al;iough an excellent approximation, does not conserve probab~~ exactly. An expression which is simple to calculate, which conserves probability exactly and which is an excellent approximate sotution (see appendix 1) is the following; Kx) = [kW-“2
WW$
tklxl)lV2
‘%x,).
(4)
or (if we choose the ninus sign in eq. (32)) Jr(x) = [k(xWy2
U*(x, xi) Ik(x,)l u2 +&xl) - (47)
Functions (46) and (47) are our choices for the generalized JWKB solutions of the multichannel Schr~inger equation. The fmt Magnus appro~mation to the matrix A@, x1) [IO-12) (see eq. (42)) is X A(x, x1) ?5:i
I xi
kfx’) dr’ .
(481
~ornb~~g this approximation with eqs (4 1). (46j and (47) we obtain 4(x) = [k(x)]-*f* exp [2i[k(x’)dx’] x W~W2
wg.
,’ ‘.
‘.. ” (49)
where L is some arbitrary ware
m&ix. The function q(x), defied in eq. (39); is not a satisfactory approtiation to the wavefunction. It
which is f&maUy identical to the ordi&ry J’Wl&~tipression, the only difference b&g that the quantities
386.
B.R. Johson,
generalizedJWKB
A
approxinwrbn
involved are matrices, instead of ordinary numbers. Since in general matrices do not commute,‘the order in which the ma&x factors appear is essential. Unfortunately, in man9 cases, the,first Magnus approximation is not accurate enough and so eq. (37) must be solved more accurately. A variety of methods for solving eq. (37) were
h]
exp [i “/-’
sca~rering
the appropriate qu+i ties in these equations as matrices. The factors in these equations have been written in proper order (the order is important since lhe~matrices do not commute).
Written
in matrix
form, eq. (12) is R(x2)=TR(x1)7.
tried. Some of them were: exact numerical solution (by Runga Kutta method) of eq. (37), exact numerical solution of eq. (42), second Magnus approximation [IO-121 and repeated firstMagnus [13]. Although there may be special cases when some other method is better, it was found in most cases that the repeated first Magnus was the most efficient method of solving eq. (37). The repeated Magnus method consists in dividing an interval over which a solulion is desired into subintervals. solving each subinterval by the first MagnuS approximation, and then multiplying these solutions. For example, take the case in which the interval from x1 to xN is divided into N- I subintervah withx,
/or multichannel
Wh]
(51)
The JWKB e3ression for the transmission matices T and T are easily evaluated. By definition, the transmission matrix 1 satisfies the equation +(x2)=
(52)
TJl(+),
where the wavefunction is traveling in the positive direction. Therefore cq. (46) (which corresponds to a positive sign in front of k(x)) applies. Comparing eqs. (52) and (46) it is evident that T = [k(x2)]-‘I2
U(x2,x,)
[k(x1)]li2.
(53)
Similarly 7 satisfies the equation (54)
Nx,)=TJI(x2).
where the wavefunction is traveling in the negative direction. Therefore eq. (47) (which corresponds to the minus sign in eq. (32)) applies. If we invert eq. (47) we obtain
xN-2
9!x,) X... exp [i[
k(x)dx]
.
(50)
This method converges to the exact solution as the
number of subintervals is hicreased. The S-matrix is computed by the method developed in sectidn 2 for the single channel case. The potential is divided into two parts at some division point, x1. It is important that x1 be located to the right of all the ‘classical turning points. By this it is meant that the matrix [k(x)12 has all pcsilivc eigenvalucs for x >xl. Let x2 be a point located in the asymptotic region of the potential, i.e., the lotal potential, V(x) plus the centrifugal potential, is negligibly small in the region x 3 x2. For convenience we designate the region x
= [k(x,)l-1’2
+(x,x,)
[k(x)11’2q(xI,
(59
where lhe superscript T means transpose and we have made use bf the fact that U(x, xl) is a unitary matrix. Comparing eqs (55) and (54) it is evitit that 7 = [k(Q]-
‘I2 UT(x2, x1) [k(x2)] l/2 .
(56)
Comparing the expressions (53) and (56) for T and 7 it is seen that they are simply related. Each is the in-verse of the complex conjugate of the other, i.e., 7 = (T*)-l .
(57)
The relation of the S-matrix to the reflection matrix is easily derived by comparing the two expressions, (2 I) and (l), for the asymptotic form of the wavefunction Sx2-c- = -[k(x2)]l/*
exp{-i[k(x3x2-@r/2)1]
X exp{-i[k(x2)x2-(P2)l]}
}R(x2)
[\(x~)]-~/~. (58)
Compute R(x2) from eqs. (56), (S?j and (5 1) and substitute thiz.into eq. (58) to obtain
Sx,-.w
=-exp
[-iIk(x2)xz-(Qnl2)i1)
X U(x2,x,)[k(xl)l’12 R(q)
W(X,)~--“~
X UT(x,,xt)exp{-i[k(x2)x2-011).
8(x,)= j (59)
[k(x)-k(m)]&-k(~)xf+(fM!)i.
(70)
Xf This expression for&(x& can be evaluated analyt-. .
It is useful at this point to define the matrix g(x,)=
[k(xl)]tj2
I?@)
[k&t)]-“‘,
idly (see appendix 2 or appendix 3 if coulomb po@o)
which is somewhat analogous to an S-matrix. It has the desirable properties of being both unitary and symmetric whereas R(xt) is neither. Substitutingeq. (60) into eq. (59) gives the following expression for the S-matrix: s x,-L- = -exp{-i[k(x7)x2-(P~/2)tj) X U(+x*):(xt)
UT(x2. x1)
X exp{-i[k(x2)xz-(n~l~)il).
(61)
The potential V(r) will, in many czes, be short range, which means it is negligibly small beyond some cutoff point xf- Beyond point XT, only the centrifugal potently is inlportant. The unitary matrix U(x2, xl) can be written as a product: U(x?_. “1) = 0(x2. Xf) lJiX[, xt ) .
(621
In the region x > xT, the matrix k(x) is diagonal; thcrefore U(x,, x,-) can be expressed in exponential form:
U(x2,xr)=exp
if t
k(x)&]
_
(63)
xi?
The matrix elements of k(x) in the region x 2 xf are [k(X)]@
=ki[l -bF/x2]1’26g,
(64)
where ki are the matrix elements of k(x -+~0). Kc., [k(==‘)]ii=~i$,
(65)
ki=A’“[2$l(EW&i))t’2,
0351
and bj J [P(P+ I)] “2/kf *
(67)
Eq. (61) can be rewritten in the form s = -5 F&t) where
S ,
(4
tent&Is are involved).
The only remaining task is to compute the matrix c(xt). This matrix characterizes the region of the potential x dxt which contains the classical turning points. In the single channel case WCmade use of the JWKB connection formulas to obtain an expression for R(x,), but iq the multichannel case the problem is considerably more difficult. One solution, of course, is to use the exact value of !@xt ), obtained by numerically solving the SchrBdingcr eq. (27) over the intervai x < x1. This is a reasonable prop~sat since in most problems the interval over which the Schrijdir&r equation would have to be numerically solved is small. Thus the proposal is to solve the problem by integralin~ the Schr~~ngcr equation nume~c~ly out to xt and then solving the rest of the problem by the more efficie$ multichannel 3WKB method to generate the matrix G and compute the S-matrix by eq. (68). This is an especially good technique to use for problems with long-rsngc, slowly varying potentials. If we compute the S-matrix by this method, then the Langer technique (71 of replacing the quantity Q(Pf 1) by (Q + i)* in the cent~fuga~ potential term should not be used. As an exampIe of the use of this technique Cosolve a problem, we have chosen the two channel curve crossing problem which Olson and Smith [ 141 CMsidered in their analysis of He+-Ne scattering. The potential used is given in reference [ 14) by eq. (45) and table 1. Atomic units were used throu&out the calculations. The reduced masi used was 6039 au and the energy Was 2.60566 au (70.9 eV). The end point itf the ~te~ation (xf in our notation) was 12 au.-The point x1 was chosen to be about 0.35 au greater than the most positive classical turning point; thus nt ranged in vaIue from 1.6 au for P = 0 to 26 au for P z.360. The results of U&e calculations for select& values of P are giveriin table I in the columns fabcled
__
0
Isle,‘1
of exact and IWKH
1.000 0.925 0.894 0.952 0.989 0.998 1.000 1,000
1.ooo
1.000 0.924 0.892 0.952 0.989 0.998 1.000 I .ooo
1.000 I.000 0.924 0.893 0.952 0.989, 0.998 1.000 1.000
a) JWKB(B) b)
4.513 3.269 1.614 5.933 1.183 s.951 3,685 0,296 5.984 5.129 3.840 2.142 0.111 4.060 1.439
--
.
.--.-....
_ -
-
-
-
-
JWKB(A)‘)
v~~~~~ of P
- - ._ -_.-
-
o.w/(-l)d)o.L71(-1) 0.105 0.106 0.262 0.262 0.147 0.147 0.289 0.289 0.300 0.300 0.195 0.195 0.220 (-1) 0.223(-I) 0.4 14 (-2) 0.4 I9(-2) 0.382 0.383 0.450 0.451 0.307 0.307 0.150 0.150 0.577 (-1) 0*577(-l) 0.182(-l) 0.1%2(-1) 0.4861-2) 0.485 (-2)
tklct
I Slp,l
for dccted
b) cnla~latcd by eq. (89) and related equations. C) McnsuredIn radkna d) Number In patenfhesesis power of ten by which the table entzy is to be multiplied.
4.515 3.273 1.618 5.936 1.785 5.952 3.684 0.299 5.986 5.128 3,841 2.147 0.107 4.065 1.435
0.847 0.849 0.847 4.513 3.270 1.615 5.933 I.703 5.951 3.684 0.294 5.984 5.12% 3.841 2.142 0.111 4.060 1.439
.A)’ Calculated by q. (68) end r&ted equalions,
340. 350 360
.330
1.000
0.981
0.981
0.994 0,965 0,989 0.957 0.954 0.981
0.994 0.965 0.9@9 0.957 0.954
1.000 1.000 f.OOO
0.994 0.965 0.989 0.957 0.954
c)
Exact JWKB(A)
ArgQ(%
values ol the &matrix elements Sf{) rnd S[[)
‘, Exact JWKB(A) a) JWKB(B) b)
20 40. 200 204 2ofi 210 214 290 300 310 320
P’
Table 1 Camp&an
-
-.
a.171(-1) 0.106 0.262 0.148 0.290 0.301 0.196 0.209 (-1) 0.797 (-2) 0.379 0.448 0.307 0.151 0.598 (-1) 0.223 (-I) 0.107 (-1)
.-
lWKB(B) b)
c)
5.666 2.650 0.217 5.427 2.381 3.956 0.738 0.552 4.805 4.826 1.369 3.889 6.132 1.844 3.616 5.188
_
5.668 2.651 0.21 e 5.428 2.383 3.958 0.739 0.551 4.808 4.826 1.371 3,886 6.136 I:%45 3,622 5.169
5.661 2.650 0.216 5.428 2.381 3.956 0,739 0.553 4.805 4.826 1.369 3.888 6.132 1.844 3.616 5.188
Exact JWKMA) a) JWKB(B) b)
At&)‘,
389
B.R. Johncon. A generdizcd JWKB approrimarion for mdrichannd rarlefing
IWKB(A). The exact values were obtained by numerically solving ihe two channel matrix Schradinger equation. The exact results were computed to fourfigure accuracy and rounded to three figures for the table. An examination of the table shows that the JWKB(A) results are quite good over the entire range of P values for St I and S,,, in both magnitude and argument. 3.2. Method B: JWKB connection forrnuh Special caseyarise in which we can devise methods for computing S(xr) based on the JWKB connection formulas. This occurs when the problem can be ap. proximately decoupled in the region x G x, . In the most trivial case, the problem is decoupled because the off-diagonal matrix elements of k(x) are zero in this region. Let Xi be the class&l turning point in the ith channel. The matrix S(xl) will be diagonal with matrix elements (see cqs. (19) and (60)) [2i(
[S~)ljj=eXp
J
ki(X) dx--n/4)]
&ii,
Let x0 be a point eq$
to or less than the “left-mast” turning point. Then S(r,) in this diagonal representation is &(x1) = exp
2i i’ I[
G,)=!3(+lT(xt).
(78)
The orthogo@ matrix C which diagonalizes k(x) also diagonrdiics S(xt). Therefore g&r)=
[CTll(xt)CI
P%l(x,)CIT.
(79)
The matrix g(xt) is not uniquely defined by eq. (78); however, we can make it unique by requiring that g(x,) is also diagonalized by C. Thus, comparing eqs. (79) and (77),
(71)
11
(
q(x)dv-(a/4)1
. (80) II
To
where (72)
[k(x)Iir*
In some cases it is possible to diagonalize the problem by an orthogonal transformation in the region x 4x1. Let C be the orthogonal matrix which diagonalizes the potential matrix (for the present, we assume C to be a constanf independent of x) cV(x)=CT[
v(x)+tT]c.
(73)
The matrix V(r) is diagonal. Define the diagonal matrix Q(x) to have matrix elements Qi(x)=h-‘{2~[E-
clli(x)-h2(P+~)2/2~2~~1’2
Let xl be the classical turning point of the ith channel, i.e., xI is the point where Q;(q) = 0 -
(74)
(75)
For values of x < Xi, Q&r) is imaginary. Defme the diagonal matrix q(x) with matrix elements
Thus
J’ k(x) bx -(n/4)
,
x
I
-
(81)
*o where k(x)=Cq(x)CT,
xg 4-X =Gxt .
(821
Eq. (82) is a very important defmition. It redefines the matrix k(x) in the region x
=o
(77)
We must now transform the diagonal matrix cd(xt) b_ack to the non-diagonal representation. The matrix S(xr). is unitary and symmetric and,= such, it can be expressed in the form
-3
ki(x)=
.
q(x) dx - (514) I II
x0
I
(83)
B.R. Johnson.
-390
A tyenetalized JWKB app~orindon
Eq. (84) is true ordy because we have assumed the orthogonal~matrix C to be constant. More generally, the matrix U(x, x0) is a solution of the differential equation U’(x.x,)=ik(x).U(x,xo),.
(89
with the boundary condition U(r,, x0) = I.
(86)
We now lift the restriction that C is constant and assume that cq. (83) is stilt a good solution for g(xr )
with U(xt , x0) defined by eqs. (85) and (86) and k(x) defined by eq. (82) where the matrix C in eq. (83) is now a function of x. Substitute eq. (78) into eq. (68) s = -G
e(q)1
rG dq)l=.
(87)
Define G = E g(xj) exp[irr/2! t
(88)
Substituting eqs. (69) and (83) into (88) we obtain the following elegant and simple expression for the Smatrix: S=GG’,
(89)
where G = exp[i&(xr) t (in14) I] U(X~.X,-,).
(90)
and U(x, x0) is the solution of eq. (85). If we compute the S-matrix by this method, in which we have made use of the JWKB connection formulas, then Langer’s technique (71 of replacing the quantity f!(n t 1) by (P + i)* in the ten trifugal potential term should be used. This statement applies in particular to eqs. (28) and (67), which will be changed by this replacement. -As an example of the use of this method, we have solved ihc same problem considered previously. The results arc given in table 1 in the columns labeled JWKB(B). An examination of the table shows that the results are quite good over almost the entire range of P values. The exceptions occur when the magnitude, t@t, is very small. Note !I particular I,$,)] for P= 214 290,340,X0 and.360. Note also that although I&( is in error me values of Arg(S#) are still quite good for, these P values. The consequences of these errors in ISg] are not very great, however, because although the rela’tive error is quite large, the absolute error is very small. To demonstrate this, we have
for mulrichannel
scattering
calculated the total inelastic cross section, Qt2, for excitation from the ground state. It was found that the exact results had converged to four figures by the time Qtz had been summed to P = 360. The result, rounded to three figures, is Q,* = 0.693 au. The JWKB(B) results, also summed to P = 360 and rounded to three figures, is exactly the same. In each case the convergence was checked by summing to P=400. Method B was derived by assuming that the potcntial in the region containing the turning points, 0
RR. Johnson. A generalist-d
IWKB
approxi%atiim
for multicimnel
rmrlcring
4. Distorted wave bansformatians In this section we will derive various forms of the distorted wave transformation. They are of interest in their own right_ and also as a necessary prelude to deriving Landau-Zencr type formulas for curve crossing problems. The basic first-order differential equation derived previously is (c.f. eq. (85)) U’(x, x0) = i k(x) tJ(x, x0),
ft is possible to derive a more symmetrical form of the distorted wave transformation, which we call the “split-distorted wave transformation”. Let xc be some point betwcenx,, andxt, i.e.,xO
ww
(91)
Using the distorted wave transformation we can write
(92)
U(xt.x,)=exp
with boundary condition U(x,.xo)
= I.
The first distorted wave transformation derived is well known [ 15,161. Let k(x) = kotx) + k, (x1 z
[-[I
h,(x)d+]
fJ,(q.+
002)
k&3&]
UL(xo.xc).
(103)
to be and also (93)
where k,,(x) is the diagonal part of k(x) and kt (x) is the non-diagonal part; also let
U(x,, Xc) = cxp [-r1 4
U(x,x~)=U~(x,xwg)u~(x‘x~),
Take the hermitian conjugate of eq. (103) and use the relation
(94)
where
U(X, xf) = Ut
Ub(x, x0) = t ko(x> U,+, x0) -
(95)
(Xfs Xl)
(W
to obtain
Eq. (95) can be solved since ko(x) is a diagonal matrix: U, (x. xr,) = exp
[
i _f kg (x’) dx’] .
(96)
=0
Substituting eqs. (105) and (102) into (101) gives
Substitu~g eqs. (93) and (94) into (91) and using eq. (95), we obtain
U(xl, x0) = exp
i j’
ko(x) ti]
XC
U(x, x0) = exp [_i 1
k,G’Nx’]
U,(x,xo)
(97)
and where Ut (x1, x0) is a solution of eq. (98) with Ui (x,x0) = i K(x) tJt (X x0),
(981
where
r(i)=er.[-il.,(;)a.‘]
tC(x)=exp[-!
j ka(x’)dx’] (107)
xk,(,)rrp[ijk*(r)ar] x0 with boundary condition
(99)
This procedure can be generalized to more than a single splitting. For examp!e, Iei
392.
BR. Johnwn,
A generalized JWKB appmximdon
for multidannel
xattering
when computing U1 (x1, x0) with eq. (98). It is obvious how this procedure can be generalized to even more division points
u(x,,x~)=u(~*.~~)u(x~.x*)-
w@)
‘, The matrix
U(x,,,x,) can be transformed as in eq. (I(K) with xb as the division point: U(x,,x,)
= exp
i j?
1 XI 1
XU1(x2,xt)exp
irQ(=)dx
iu the .qditdistorkd wave picture
5. I. Ckrve crossing
ko(x) dx
xb
5. Smahix
Substituting the split-distorted wave expression, eq. (106), for U(xf, xO).tito eq. (90) gives (110)
Likewise, for the matrix U(x, ,x,-J with xp Y the di-
G = exp [illl tJl&,
~0)
expb~l,
(119
where the diagonal matrices ‘1 and G have matrix elements
vision point, U(x,, x0) = exp
ii’ E 5
vi = [6(Xf)]ii + xc [k(X)]ii _I-
ka(x) dx]
[fkob)&].(111) 1
Substituting eqC (111) and (110) into (109) gives
[
(116)
*C
and
X U1bl,q-& exp
U(x2.xO)=exp
dx + n/4
k,(x)&
ii’
3
XL$(x,~~,)exp[i[
k&)dx]
ii” kg(x) dx] , (112) x0
;;;: =i’ [k(X)] ii ~. X0
(117)
The S-matrix is then obtained by arbstituting eq. (115)into(89). So far our treatment has been completely general. ‘Lzt us now specialize to the case of a two channel problem in which the potential cumes cross. The matrix U1 (x,, x0) is a unitary 2 X 2 matrix which, in general, can be expressed in terms of three real parameters. Pl/* exp [-io]
-(l
-P)ll*exp[-i@i
U 1(Xf. x0) = ( (1 -P)*lz
exp[i@]
P1j2 exp[ioi (11’3)
Xkl(x)exp
i ~ko(cc’)d_r’
, I
013)
4
when computing U1 (x2. xt) with eq. (98) and . K(x)=eyp
[-i[kbb?h’]
Xll(.)erp[ijk~~x~)&~,
(114)
where 0
c;(x) = i k1*(X)
terization of the S-matrix S=exp[id
Ur(xf,xo)exp[2i;i]
U~(xr,xu)exp[irl)
. X exp
(11%
i z [k,,(x’)-&(x’)] 1 152
dx’
C,(x). I
am
This is similar to the factorized form of the Smatrix given by eq. (2) in reference [ 171. It should be apparent how this factorization scheme can be gen. erahzed by making use of the multiple split-distorted wave expression given by eq. (112) or its gener~ation to obtain a factorized S-matrix similar to eq. (8) in reference [ 17) _ Doing the matrix multiplications indicnted in eq. (119) results in the following pararneterized form for the S-matrix elements:
Expand the matrix k(x) about the crossing point x, (it is easily verified that k,, (xx) and k22(x) cross at the same point, xe, that VI1 (x) f L, and Vzr (x)+ L, cross) and retain only the first ~on-~~~~g irrm
Sll = fP expPi(;i; - aI1
k,*(X)==+,
kll(x)-k2z(X)~Q(x--XCj’
02%
+ U-P) exp[W?~ - 0113expI2iql ,
(120)
$2 = fPexp[2it;i2 + 011 +0-P) s12
= 821
(121)
expl2iG~ + tW expf2if721, 1-P)) 1/2{exp[i(2;i; +# - 0))
= [P(
where the notational change is C, 0) = Qj(x* x0) >
j= 1.2,
‘(126)
C2(x)=~~&Jn).
i= 1,2.
027)
where by our conventions, E and R are both positive quantities. With these approximations, eqs. (124) and (125) are C;(x)=
C;(x) = -i e exp [iax*
It is easy to show from this that \s,,i*
= 4P(1 -P) sirG(;j; -;j;-$++u).
(123)
The physical interpretation of the oscillatingprobability expressed in eq. (123). resuking from the interference of two amplitudes, is weIl known from the Landau-Zener theory of curve crossing. 5.2. Zenef upproximtion ihe parame ten P, u and 4 can be computed by an approximate analytic technique due originally to Zener [18] for-solving eq. (98). Written out in component form, with some change in notation, eq. (98) reads
Ci(x) = i k,z (x) -i 7 [k,,(x’)-kz(X[)l
Xexp (
I
=C
-i (sexp[-iax
C,(x),
(130)
(122)
-exp[i(2;j;-~+al)exp[i(tll+82)1.
h’
C26) i
024)
C,(x) .
030
Zener has protided a method for obtaining the asymptotic solution to these equations. His method is to eliminate C,(x) by substituting eq, (131) into (130). The resulting second-order differential equation for Cr (x) can be transformed to the Standard form of the Webcr equation, the soh~tions of which are the parabolic cylinder functions which have a known asymptotic form. From this risymptotic solution it is possible to obtain two sets of asymptotic values of C,(x) and C,(x) corresponding to the two sets of initial conditions implied in eqs. (126) and (127) when i = 1,2. From this we obtain our approximation to the matrix U, (x,x& which has the form given in eq. (1 I8), with the following values for tfte parameters of P, u and #: P= expf-27ry],
(132)
u=o,
(133).
Cp= n/4 + Argl%)
+ 7 NY) - 27 I@),
.(f34)
.?. .. .
.( ..,
-’
:
where’, ‘.
:
‘.
T 5 &&I I
(135)
and p is sdme characteri#ic bngt@ pammeter far the prcibtem which is not exactly determined by the anal:
‘Substituting from eq. (70) into kq. (90), we obtain
ysis. The exact value of p .isnot very impartant since it OC~WS in a ~oga~~rnic term. .A reasonable value whkh we have found, based on a per~rbat~on theory am&h, ii J3”&2E-
*
(e = base natural lo&
(136)
The ~aramele~ Q and e (see eqs. f 128) and (129)) are the exp~sion coefficients of the matrix k(a) about the crossing point. In order to compare our formulas with the formulas of landau and Zener, it is essentirdto relate.&and F to the equivalent ~~f~~jenf~ far the expansion of the potential matrbr V(x). ExpandirigV(x) about the crossing point, xc, and retaming oniy the first terms, 037)
v&al=a,
p&(x)-
v;t(x)++--
8, =I(.=-~J*
(138)
where by our conventions, a and fare both positive qu&ies. Usingthe relation between V(x) and k(x) provjded by cqs. (28) and (36), it is not hard to show the fo~ow~S relations are true e =OpV,,
.
(145)
Substituting thii into eq. (144) produces an expression which ir the product of hvo matrix exponents functians. Strictly speaking, the exponents cannoi be added; however, it is the essence of the fu;st Magnus ~~proxirnat~onto assume that they can. Thus G=exp
i = B(x) - k(-)] dx - k(a) x0 [iJ XQ
+ {@+Qnlz]I
*
(1%).
II
039)
is lhe first Magnusapproximate to G.
(14Q)
The matrix in the exponent is symmetric; therefore G = GT and the first Magnusapproximate of the S-matrix is
and 4 =fhu,.
u(x,x,)=erp[Ihfk(x)b;]
where u, ij the velocity at the crossingpoint given by p,=@lri)[EIn
Yll(x,)-ti*(p+4)2121u~--LI]3112. de&& these relations we have assumed (IQ0
. that
2i j.[k(~~)-k(m)] II
This last ~mption
breaks down, of course, when the crq&ng p&t and brig points are very near to ‘each other. ‘. Substituting eqs. (139) and (140) into (13S)‘gives 4 ,=‘i&E& )
(143)
which is the usual Landau-Zener form for this : patieter
[a, IS]. ” ).
: :
dx-k(o)xg
-%
f [(~~~~~~~~I
043
~~*2(x~~~‘~~ u, -
..
S=exp
11
-
(147)
This can also be writ&en s = expr2i4 ) where qis the gener~ed phase&& matrix
(148)
ate folly identical, the only difference being that in eq. (149) the quantities are matrices.
It is an inrcmst~g exercise to derive one of tbc more familiar semi~classical expressions of theSmatrix by using the SWKB expressions derived earlier as our starting point In particular, it is easy to derive an expression for the S-matrix which corresponds to solving the time-dcpcndent Schradinger equation for a time-varying potential p;enerated by the incident particle following a straight line trajectory with impact parameter b. The S-matrix is given by eqs. (89) and (90) with the matrix U(x,, x0) obtained by solvingeq. (85). The matrix k(x), defined by eqs. (28) and (36) (replacing P(P + 1) by (2 + i)2) can be rewritten in the factored form:
Then substitute eq;( 154) into eq. (90) for G and use eq. (157) to obbiri
The exponent in the first exponential factor is just the phase-shit for the centrifugal Potential which is zero; therefore GXf--
= exp Wfi) [J+JC~)~ c:) Uk @f. ~0).
t x(r) = j-c+‘) 0
dc’ + 6.
The d~ferenti~s
(16%
are related by
ti=u(C)dr.
I - [ llrtu(~)]
[V(x) +c 1 ,
0521
where 0531
is the radial veiocity of the incident particle. Using the distorted wave ~~sformation
U,(X,,Xo).
(154)
dU, (x, xo)fdr = i K(X) U, (x, x0),
G,A,
= expHil@ rtl
Ul(f.0) ,
063)
where [email protected])/dt=
--(ii&) f V[xtt)l
+ &~~~~f,O),(l6~)
Equation (164) can be transformed to the retorted waveform U;W)=
Here, U1 (of, x0) is the soluticm of the d~fcrenti~ equation
where
(162)
x0)i -Mm-u(+t. Thus, eq. ( 159) is transformed to
Nx) = ~~0~~)~~ ,
U(Xf, x0) = CXP[-i’WW] 1
060
For the particular case of the straight line trajectory it can be shown that
first two terms of its binomial expansion k(x)=+,(x)
(W
Change the independent variable from the radial distance, x, to the time, t. These variables are related by the equation
expf-WcErl
U&O),
where WW
dU~{r,O)/dr = -(i/ti)
G(r) U,(t,O) s
Th matrix element
of a(f) are given by
065)
B.R. Johnron. A geneml&edJWKBapproxrnnrion /or
396
‘cand, as mentioned previously, is the threshold energy level of the ith channel. Substitutingeq. (165) into eq. (16% G= U,(-,O).
‘(169)
Using eqs (167) and (168) it is easy to show that the matrix a(t) is hermitian, i.e., a(r) = at (I).
(170)
Since the time variable, t, was chosen to be zero at the distance of closest Ipproach, the radial distance, x(l), is symmetric about I= 0, i.e., x(r)= x(-r).
(171)
From eqs. (171) and (167) it is easiIy shown that Q(r) = V(4).
(172)
Using eqs. (172), (170) and (166) it can be verified that u;o,o>
= U*(O, -0 I
(173)
mulrichannel s~ttering
and the first Magous approximation. Undoubtedly other approximation metbods can also be devised. In this section, we remind the reader that we can always break the region x0 Gx Gxf up into subregions and write U&f, xO)= u(r[9rn) U(x,,x,)
*.. U(x,rXO)
‘( 178)
where XO
(179)
Then we can treat each subregion separately and use the appropriate approximation in each region. For example: In one region we may have to solve cq. (85) exactly; in another in which curve crossing occurs we might make the split-distorted wave transformation with the Zener approximation, and in another region the first Magnus approximation might be appropriate. These various approximations, each valid only over a limited region, can then be combined by eq. (178) to obtain the overall solution.
and therefore G-t = tJ2 (0, --)
.
(174)
7. sumnlary
Thus, we obtain fmally S=GG*=U+,-=).
(175)
If we assume that eq. (166) can be evaluated by the first Magnus approximation, we obtain S = exp[2iri) ,
(176)
where the matrix Q has matrix elements given by q’il = -ffj-t
/ exp[i wiir] YVir(r)] dr . -0D
(177)
This should be compared to the similar formula derived by Cross by a different metbocl [4].
6.3. Combined approximtions The major numerical effort in solving for the Smatrix as expressed in eqr (89) and (90) is to solve’ . for the matrix U&. x0). This matrix is defmed by the differential equation (85) with the boundary condition (86). Several approximation techniques have been men-tioped in previous’sections, i.e., Landau-Zener type approximation, which applies in curve crossing cases,
A multichannel approximation method that is a direct generalization of the JWKB approximation has been presented. The turning point problem was handled in two ways. In method A, the Scluijdinger equation was solved exactly in the turning point region and then JWKB solutions were joined to this exact solution. In method El, the problem was assumed to be exactly cliagonalizable in the turning point region, i.e., the unitary matrix, C(x), which diagonaltzed the matrix k(x) was assumed constant in the turning point region. Then the ordinary JWKB connection formulas were used to solve the diagonalized problem and the solution was retransformed back to the original non-diagonal representation. By this technique we derived formulas which were assumed to remain valid even when the unitary matrix C(x) was not strictly constant in the turning point region. An example problem was worked out and the results presented using both methods A and B. Next, a new form of the distorted wave transformation which we designated the split-distorted wave transformation was derived. This allowed us to factorizc the S-matrix and when applied to the two stale curve crossing problem led us directly to a derivation
of Landau-Zener tVpe formulas for the S-matrix. Finally, two approx~ations to our generalized JWKB formulas for the S.matrix were presented. The first was simply an app~c~tion of the Magnus approximation to our formulas. In the second, by making certain further appro~m3tians, we were able to give a new derivation of the straight line t~jecto~imp~ct parameter formut3tian ofscatte~ng theory.
Two matrix functions have been proposed as the generalized JWKB solution to the matrix Schradinger equation, One was the solution to eq. (45) (which we repeat here), * 01 WI $64,
(Al.1)
where k(x) and at(x) are defined in section 3, This equation has the adv~~age that it was derived using a systematic expansion in powers of& {see eq. (3 I)); it is also ;i very good approximation. It has two disadvantages, however: It is c~m~crsome to cvalustc numcric~~, and it does not conserve probability exactly. The Dther solution proposed was eq. (46) (reseated here) GExI- Ik’k)l-‘”
~[k(~)]-‘~*~’ =a&) fk(x)]-“2. Substi~t~g
(AL%}
the easily proven relation
{[k(x)]-‘12}’
= -[k(r))-f~2
into eq. (A1.4), we obtain a(w) = -[k(x)]-tit
{[k(x)] J~2)‘[k(r)]-1~2 (A1.5)
{[k(x)] ‘D )’ ,
(Al.@
thus dete~inin~ the function a(x) which appears in eq” (A1.3). From eq. (A1.6) it is easy to prove &e relation
Appendix I
Jr’(x) = f* i W
the resuIt
Ukx,)
[ktxlIf1/2
JtCfxl),(AI.2)
where iJ(x, x1) is a solution of eq. (37). This solution is relati-leiy simple to compute numerically, it conserves Frobability exactly and is also an excellent ap proximation. Obviously, with ail these advantages, we chose ii: as the generalized JWKB solution of the matrix Schrijdinger equation. The only disadvantage (if it can be considered a disadvantage) is that it was not derived by any systematic expansion procedure, but merely (a not very occults intuitive guess, based.on the form of the single channel solution. It is the purpose of this appendix to relate as much as possible these two solutions. Assume that the differentj~ equation which generates Jr(x), given by eq. (Al .2), is Jr”(x) = fi klx) + ~G.91 IJr69,
(A1.3)
i.e., the same form as eq. (Al. 1) except that in this case U(X) is an’as yet undeterred function of x. Substituting the wavefunctian IA1.Z) into eo. (A1.3) produces
which shouId be compared to the equation for eI (x) (eq. (33) in text),
In #is appendix an analytic expression is derived for the elements of the diagonal phase shift mat&, 6&j, defined by eq. (70): &&x~)= f
[k&x) -k&=)1
dx - ki(+xf
+ &r/2,
(AZ* 11
-9” where k&x) is (see cq. (64)) kj(x)=kj(l
-L&X*]~f*
.
(A2.2)
Therefore
l%e integral in eq. (A2.3) is easily evaluated by rn~~g the substrfu~on x = “f/Y ,
(AM)
and the finaf result is &i&f) = &/2-kjxf
[(J -~~~~~)I’~
B.R. Johnson. A generalized JWKB approximari?n
398
6j(Xf) :.pn/2 - kibj +
[(X:/b;- I)“*
/VT mulrichannelscrrrrering
Sx,-r-
= -exp[-i{k(x2)x2-@n/2)
(A2.6)
Sill-’ (bi/Xf)] ,
x U(x2.q)~(q)
Xexp[-i
which can also’be written .ti the form ‘ai(rf) = PIr/:!-_jbi
{($/bf -
-+ tall-~ [($/#Appendix
3:
I)“’
I)-‘/*]).
(A2.7)
W)x+ca
= exp E-i [kx - @n/2) I]) - exp{i[kr-
@r/Z) I]},k-t/*S kf/* .(A3.1)
In the case where some or all of the channel potentials include a Coulomb term, this expression is generalized to [19,20] Jl(x)-exp{-i[lc-y
In (2kx)-(&r/2)
+up 1
[k(x) - k(x2)l k - k(x+c
“(Xf)*, -t_ = j *r
t (&r/2) I -up + y ln[2k(x2)x2]
. (A3.7)
This expression CM be written in component form and evaluated analytically, just as it was in appendix 2 for the non-Coulomb case. Analogous to eq. (A2.2) tie expression for ki(X) for the Coulomb case is ki(X)
=
ki( 1 - 27i/kiX
- b3/X2)
(A3.8)
_
Therefore
S kl/*, (A3.2)
where
[( 1 - 27i/kiX - bf/X2)“2-
l] dx
Xc
-kixftPfl/2.
[oQ]i+ Ti1n(xix2)’
(M.3)
Here e is the electronic charge and Zt and Z2 a;e diagonal matrices whose matrix elements measure the charge of atom 1 and atom 2 in the various channek. The matrix elements of Uie diagonal matrix [Op]i = Arg f’(P + 1 + iyi)
(A3.4)
are the Coulomb phase shift for each state. The relation of the S-matrix to the reflection matrix, analogous to eq. (58), is SI, --
UT(+.q)
I +an])
yIr1(2kx)-(Pn/2)l+a~])k-~/*
7= ~1e2V2 Z&k-r
+
(~3.6) Eqs. (68) and (69) remain unchanged in form; however, eq. (70), which dcfmes the quantity &i(xf). is changed to
s,(Xr)x, __ = ki _/ -exp{+i[kx-
yln[2k(x&]
(k(x2)x2-(lllr/2)1-rIn[2k(x2)xl]
Coulomb potentials
Recall that the definition of the&matrix, for the case of no Coulomb interactions, is given by the asymptotic expression for the radial wavefunction
I -
(A3.9)
The integral in eq. (A3.9) is easily evaluated by making the substitution x = x-r/y ,
(A3.10)
and the result is 6i(x,)=Pn/2+ri
In(2ki)-
[o,]i-kix,l-ri,(A3.11)
where I=(1 tp-a)‘/*
= -[k(x2)]1~2exp[-i{k(x2)x~-(&r/2)I - r ln [2k(x2)+1
X R(x2) exp[-i(k(x&-(&r/2) - Tln[2k(x&2]
+ al/* {sin-q(2Q~)/(p*+4a)q
tsin-‘fp/Bl(p*+4c#q)
t@/2)On[2(1
-h1[4/~~]),
+apH
+op)]
The new form of eq. (61) is
+~7-#~+2+13]
I
[k(x2)]“”
(A3.12) (A3.5)
4 = -%J(kix~) a 7 b /xf . f
3
(A3.13) (A3.14)
If 7i = 0, the above expression reduces to eq. (A2.5) for the non-Coulomb case, as it should.
B.R. Johnson, A
gcnernlized JWKB upproxltition for mul~chunnel sastrering
Referems
[ I] R.B. Bernstein, in: hfolccuku beams, cd. J. Ross (triterscience, New York, 1966) p. 75. [2] B.H. Brands-en, Atomic colbion theory (W.A. Benfamirr; New York, 1970). 131 D.R. Batesand DS.F. Crothcn, Roe. Roy.Soc. A315 (1969) 465. [4] RJ. Cross, J. Chem. whys. 47 (1967) 3724~48 (1968) 4838. [Sl E.C.G. St%kcIberg. Hclv. F’hys. Acta 5 (1932) 370. f6l J.B. Debs, W.R. Thorson and SK. Knudson, whys. Rev. A6 (1972) 709; J.B. Delos and W.R. Thorson. flryr Rev. A6 (1972) 720. [7] R. Langer, F%ys. Rev. 51 (1937) 669. f8l L.D. Landau and EM, Lifshitz, Quantum mechanics (Add~on-Wc~cy, Reading, Mass., 1965). 191 J. Heading An introduction to phase-integral melhods (hiethuen. London, 1962).
399
[ 101 W. Magnus, Commun. Rue Appt. Math. 7 (!9.54) 649.. (111 D.W. Robiin, He&. J’Jtys.,Acta 36 (1963) 140. [ 121 P. L’echukasand J.C. Light, J; ‘Ckem. phyi. 44 (1966) 3097. [ 131 S. Chart. J.C. Light and J. tin. J. Chcm. whys. 49 (1968) 1141 EE. Ofsan and F.T. Smith, Phys. Rev. A3 (1971) I&?? (Err&, J’hys. Rev. A6 (t972) 526). [ 151 D.R. Bates, Roe. ehya. Sac fLondon) 73 (19X& 227. 1161 R.D. Leviie, Chem. whys. Letters 2 (1968) 76. [I?] JE. Bayfield, E.E. Nikirin aJId A.[. Reznikov. Chcm. Phys. Letters 19 (1973) 471. [It31 C. Zener, hoc. Roy. Sot. Al37 (1932) 696. (191 N.F. Nott and H.S.W. Massey, The theory of atomic coflisions. 3rd Ed. (Clarendon Press, Oxford, 1965). 1201 A. hfcssiah, Quantum mechanics, Vol. 1 snort-Hoed, Amsterdam, 1965).