The wave functions in Coulomb fields

Nuclear Physics, North-Holland Publishing Co., Amsterdam, 1 (1956)

THE WAVE F U N C T I O N S IN COULOMB FIELDS A. S. MELIGY

Department o/ Mathematics, University o/ Alexandria Received 29 March 1956 A b s t r a c t : I n this paper a detailed mathematical study of the confluent hypergeometric functions is carried out. Exact analytic expansions in series of Bessel functions are developed. These expansions axe found to be useful for the computation of the wave functions in Coulomb fields. They have the advantage over power series expansions in the clearance of imaginaxies from the wave functions for repulsive Coulomb fields.

1. Introduction The radial equation for an electron of energy E and mass m moving in the Coulomb field of a nucleus of charge Ze is dr---~ +

~-~ E +

ra

W = 0,

(1)

where W is r times the radial factor of the wave function and l~ is the angular m o m e n t u m of state considered. By the substitutions E=

2~9k~ ,

r-~

2 m e 2 Z Z and

l

m--½

(2)

this equation transforms into dzW

{

d,---f+

1

k+ t--re' /

jW=O

(3)

which is the confluent hypergeometric equation discussed by Whittaker and Watson 1). In their notation, two solutions of this equation near z ----- 0 are Mk, =(z) and M,. _,~(z). If Coulomb's law applies for all values of r, then only the regular solution M,, =(z) is needed. On the other hand, when this law is modified for small values of r both solutions are required. Since 2m is an integer, as inferred from eq. (2), M,. _~(z) is infinite. For this reason, a second particular solution of eq. (3) will be defined. In problems requiring numerical estimates, the radial wave functions are usually approximated to Bessel functions. This approximation is based on the fact that the zero energy radial equation 610

T H E WAVE FUNCTIONS IN COULOMB FIELDS

611

is a Bessel equation. Yost, Wheeler and Breit ~) tried to extend the range of approximation to cover small energies by developing an expansion of the function Mk, = into a series of Bessel functions, in which the coefficients are powers of l[k ~ i.e. powers of the energy. They obtained, after laborious work, a few terms in that expansion and the terms were found to grow cumbrously. In the present work, exact analytic expansions of the solutions of eq. (3) in series of Bessel functions are obtained. The a t t e m p t is based on a relation given by Whittaker and Watson between Bessel functions and the function Mo, ,~(z), in the special case k ---- 0, and expressed as follows:

Mo. ,,,(z) = 2 ~'' I'(m+ 1)z½i-"J,~(½iz).

(4)

2. A Second Particular Solution of the Confluent Hyper~,eometric Equation Let I~, ,~(z) denote the function

1 I~, ,~(z) = / ' ( l q - 2 m ) Mk, ,~(z).

(5)

The two functions I~, ~(z) and Ik, _=(z) obviously satisfy eq. (3). When 2m is an integer they are a constant multiple of each other i.e. F(½+m+k) I Ik • _ , ~ ( z ) = , c _ w, n I'(½--m+k) ~' ,,,(z).

In order to have two distinct solutions in this case, a function N~, ,~(z) defined by the equation r(½+m+k)

~ l k . N~

•

7?I(z)

.

,~(z) cos 2mu--I~, _,,,(z)

(6)

sin 2m~

is introduced. This function is a solution of eq. (3) when 2m is not an integer. When 2m is an integer, it is defined by the limiting form of this equation, viz.

1 0 {F(½+m+k).

Nk,,,,(z)=2z~O m ~ l k , , ~ ( z

} ~= 1 a ) - - ( - - ) ~-~m{I,,_,~(z)}, (7)

where 2m is made an integer after differentiation. On developing the ascending series in z for this function, the following expansion is obtained:

6'12

•

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MELIGY

r(½+m+k)T'(2m--n-4-1)

. (-)--~r(½+~+k)z--1

(8)

× {log z---~o(n)--~o(2m+n)+v?({--mWk--n)}, where ~o is the logarithmic derivative of the gamma function. The function I~, ,,(z) and N~, re(z) are fundamental solutions of eq. (3) when 2m is a n integer.

3. An Expansion of

Ik,,~(z ) in a Series of Bessel Functions

By suitably altering the contour integral for Whittaker's function Wk. =(z), the function Ik, ~(z) can be represented by the contour integral 1 f(o+) [ z\-½-ra+k

I~ • .(z) =

- - 2~i -

e-~" zt+-j®

(-t)-"-I

U+i)

e-'dt,

(9/

where the path of integration starts at infinity on the real axis, encircles the origin in the positive direction and returns to plus infinity; on this path - - a < ~ a r g ( - - t ) ~ and I z l < l t l . The factor (1-t-z]t)-½ 4+k in the integrand can then be expanded in powers of z]t and term-by-term integration can be performed, thus giving the power series for I~, ~(z). Substituting k = 0 in (9), 1 ,,0+, ( ~)-½-,. Io,,.(z)-=- 2nie-½,z½+~J~ (--t) -z'n-1 1 + e-'dt. (10) An attempt is now made to expand the integral for Ik ~(z) in a series of Io.,,,(z) integrals. The factor (l+z/t) ~ in the integrand of the integral (9) is expanded in a Maclaurin's series of (z/t)( l +z/t)-4, viz:

This power series expansion converges absolutely and uniformly along the contour, provided that

THE

WAVE

FUNCTIONS

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but since

it follows that

Thus on the contour of integration the condition that M < It[ is preserved. Further, along the contour le-tl ~ 1 for R(t) ~ 0 and "[e-tl ~ N for R(t) ~ O, where N is independent of t, so that by Weierstrass's test for uniform convergence the integrand of the integral for I~, =(z) has been expanded in a uniformly convergent series with regard to t, and so term-by-term integration is permlssible, thus: 1

tqO+)

kF(k+2)z"

~( _ 0 _x_9.,n_,.(1+~z) -½-'~- 'e-tdt

'."(z)=- ~-~e-V z~'~J® ~o(-)" !/.(k_~+1)

_

®

1 e-Vz~+'~(-)"

2xi

(.).

k/' k + ~ z

rlo+l

--=------./

r,i,(k_2 + 1)3®

l

z\-t,-,,,-[

,

t,

(-t)-~-"'-ql+-:/

by (10). But, from eqs. (4) and (5) and Legendre's duplication formula for the Gamma-function (~z)½ Xo..(z) = T(½+m) i - - J , , , (½iz)

(=z)~ - I,~ (½z), F(½+m) where I,~ is the modified Bessel function of the first kind, and so

I. .I.l= I..l,

e-'cu

614

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MELIGY

= (~rz)½{F(½~m)e.(½z) 1 !r(1kz' +m) I,~+½(½z) k~z

+ 2!l.,~+m)I,,,+l(½z)--... }.

(11)

This expansion of I~. ,~(z) in a series of Bessel functions is of quite a significant value. It is valid for all values of m and k. As seen, the convergence of the series is quite rapid since the I n functions decrease as their order increase and, moreover, each term in the expansion contains two factorial products in its denominator. For large values of k, the convergence m a y appear slow, b u t this increase in k is counterbalanced b y a decrease in z as inferred from eq. (2).

4. An Expansion of Nk.,~(z ) in Series of Ik, m(z) Functions An expansion of Nk, ,~(z) in series of Bessel functions m a y be obtained b y substituting-in eq. (6) the series in Bessel functions for I~, ,~(z) and L~, _,~(z) as given in eq. (11). B u t it seems difficult to express the limiting form of N~, ,~(z) when 2m is an integer, in series of Bessel functions. The reason for this difficulty arises from the fact that, in case of 2m being an integer, the series (11) does not show that Ik, re(z) is a constant multiple of Ik, _,~(z). However, one can express the limiting form of N~, ~(z) in series of Ik, ,~(z) functions; the latter functions have been already expressed in terms of Bessel functions. For convenience I~, ,~(z) is expanded as follows: I~, ~(z) = z m {aoI~, o(Z)+a½z½Ik, ½(z)+alzI~, l(z)+...}, where ao, al, a 1. . . . are independent of z. On equating coefficients 1 m 3+m 5n~oi z~+ , zi , z~T M , . . . on each side of this equation, the factors a o, a½, a x. . . . are determined uniquely in turn. T h e y are simply given b y

Thus

,__-

THE WAVE FUNCTIONS IN COULOMB FIELDS

~1~

This series converges absolutely and uniformly for all finite values of z, since the ratio test of convergence shows that uT+x/u, is of the order of --z]2r, where u, is the rth term. This series easily shows that Ik, ~(z) is a constant multiple of I~. _~(z) when 2m is an integer. It is therefore expected to lead ~o a simple limiting form for N~, ~(z) when 2m is an integer. Substituting from eq. (12) into eq. (7), the following expansion for this limiting form is obtained:

I'(~+m+k) P(½--m+k) (½~o( { + r e + k ) + ½~o(½--m+k) --~p(2an) --~o(2m+ 1) + l o g

z)Ik..,(z)

I'

1

r

where in the last series the term r = 2m is to be discarded. 3. T h e C o u l o m b Wave F u n c t i o n s for A t o m i c Electrons

If Coulomb's law applies for all values of r then the radial factor, multiplied by r, of the wave function of an atomic electron moving in the field of a nucleus of charge Ze is the regular solution I~, ~(z) of eq. (3), provided that k, m and z satisfy eqs. (2). Owing to the finite size of the nucleus one has to modify Coulomb's law in the nuclear domain. In this case the wave function beyond the nucleus is a linear combination of the functions I~, re(z) and Nk, re(z). For an atomic electron E < 0, the values of k and z are purely real as inferred from (2). The functions I~, re(z) and N~, ~,(z) behave at infinity accordingtothelawexp(z/2),sothatwhenzisreal t h e y both tend to infinity as z - + oo. It follows that, in general, there is no permissible combination of them which will remain finite at infinity. However, an exception arises when they are combined in the form of Whittaker's function Wk, re(z) which is

616

A.S.

MELIGY

known to remain finite at infinity. Whittaker and Watson introduced the following relation: F(--2m) M F(2m) Wk, ,,,(z) -- l " ~ k ) k, ,,,(z)+ F(½+m--k) i k ' - " ( z ) provided that 2m is not an integer. This restriction can be removed by eliminating Mk, _,~(z) by means of eqs. (5) and (6), thus: Wk. ,,,(z) = / ' ( ½ + r e + k ) cos n (k--m--½)Ik. ,,,(z) +F(½--m+k) sin z~ (k--m--½)Yk, re(z), (14) which is valid for all m. This is the form in which Ik, ,~(z) and N~, ,,(z) should be combined in order to give beyond the nucleus a well-behaved wave function for an atomic electron. 6. T h e C o u l o m b W a v e F u n c t i o n s in R e p u l s i v e F i e l d s The radial equation of a particle moving under the influence of a Coulomb potential Ze2/r is

.dr2 +

~

E--

~

(15)

W=0.

By the following substitutions m# Z s ?~k E - 27i9k~, r = 2megZ ~ and l = m--½

(16)

it transforms into - de -~

+

i - ~ +

w = o.

(17)

When 2m is not an integer, two independent solutions of this equation are l~k ' ,~(i¢) and l~k ' _,(i¢). Let ark, ,,(¢) denote the function ]k. =(¢) = i-(V"°I,k, ,,(i¢). (18) Substituting k = ik and z = i¢, in eq. (11) the following expansion is obtained:

Jk.~(¢) = (~¢)~ ~. (-)"

(~.)½i'I'(½+m) i "1 l"(½~) -F 1 IF(ik ~+ m ) J,n+½({~) -~ 2 t I ' (kS~(½¢) ~ + m ) J"+x +...

}.

(19)

"tHE

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When k and ¢ are real, this expansion involves no imaginaries. T h i s clearance of imaginaries i s the main advantage gained by this expansion over power series expansion: When 2m is an integer

A, -~g) = e-~= r(½+~+ik)

r(½--m+ik) ] ' ' ~(¢)

1

r(½+m+ik)r(½+m-ik)(e"+ (-)'=e -~') ],, ~(¢),

= ~

and therefore, in this case, a particular solution of eq. (17) has to be defined. Let Y,. re(C) 1

2--~l'(½+m+ik)F({+m--ik)(ea*+e

-'* cos

2~m)J,..~(¢)-],_..(¢)

(so)

sin 2~m which is a solution of eq. (17) when 2m is not an integer. When 2m is an integer, Y,, ,~(¢) is defined by the limiting form of this equation. In terms of I**, ,~(i¢) and Nik ' ,~(i¢), the function Yk, ,~(¢) is given by .* Yk.~(¢)=e'~(~-~)[~r(½+m+~k)r(½+m--~k)e ' (~-t--'klI,k,~(,¢)+N,k.~(~¢)]. 1

.

.

.~

.

.

.

Using eqs. (13) and (18), Y,. ,~(~) could be expanded, when 2m is an integer, as follows:

nY,, ~(¢) =

i *mr(i+m+/k) {~(½+m+/k)+V(½+m--/k) r(½--m+ik)

+log ¢-~(~m)-~(~m+ 1)}1,, re(C) T

--'

i"

~ F ( 2 m ) ~ ~ rl(2m+r)'

1

r

T

- r ( ~ m + 1)¢-~ ~

~=or! (2m--r) i"

r

" J* "(¢)'

618

A. S. MELIGY

where in the last series the term r = 2m is to be discarded. "When k and ~ are real, this expansion does not involve a n y imaginaries. It is therefore convenient for calculation. The functions J~, ,,(~) and Y~, ~,(~) form a complete system of solutions of eq. (17), when 2m is an integer. In the usual discussions of nuclear collisions the repulsive inverse square field is modified in the nuclear domain. In these cases the wave functions beyond the nucleus are linear combinations of f~, ,,(~) and Yk, ,,(~). The expansions (19) and (21) are hoped to be useful in the computation of these functions. References I)

E. T. Whittaker and G. N. Watson, Modern Analysis, Cambridge University Press (1927) 2) F. L. ¥ost, J. A. Wheeler and G. Breit, Phys. Rev. 49 (1936) 174