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NANOSYSTEMS:
PHYSICS, CHEMISTRY, MATHEMATICS
C. Parkash, et al. Nanosystems: Phys. Chem. Math., 2023,14 (1), 28-43.
http://nanojournal.ifmo.ru
Original article
DOI 10.17586/2220-8054-2023-14-1-28-43
Exact irregular solutions to radial Schrodinger equation for the case of hydrogen-like atoms
Chander Parkash1, William C. Parke2, Parvinder Singh3
1 Department Of Mathematics, Rayat Bahra University, Mohali, Punjab, 140104, India
2 Department Of Physics, The George Washington University, Washington D.C., USA 3Department of Chemistry, Rayat Bahra University, Mohali, Punjab, 140104, India
Corresponding author: William C. Parke, wparke@email.gwe.edu
Abstract This study propounds a novel methodology for obtaining the explicit/closed representation of the two linearly independent solutions of a large class of second order ordinary linear differential equation with special polynomial coefficients. The proposed approach is applied for obtaining the closed forms of regular and irregular solutions of the Coulombic Schrodinger equation for an electron experiencing the Coulomb force, and examples are displayed. The methodology is totally distinguished from getting these solutions either by means of associated Laguerre polynomials or confluent hypergeometric functions. Analytically, both the regular and irregular solutions spread in their radial distributions as the system energy increases from strongly negative values to values closer to zero. The threshold and asymptotic behavior indicate that the regular solutions have an re dependence near the origin, while the irregular solutions diverge as r-e-1. Also, the regular solutions drop exponentially in proportion to rn-1 exp (-r/n), in natural units, while the irregular solutions grow as r-n-1 exp (r/n). Knowing the closed form irregular solutions leads to study the analytic continuation of the complex energies, complex angular momentum, and solutions needed for studying bound state poles and Regge trajectories.
Keywords second exact solutions, irregular exact solutions, Coulombic Schrodinger equation, Frobenius method, Coulombic interaction
Acknowledgements The authors are grateful to Prof. Nyengeri Hippolyte, Department of Physics, Faculty of Science, University of Burundi, Bujumbura, Burundi, for providing his contribution towards the finalization of this manuscript.
For citation Parkash C, Parke W.C., Singh P. Exact irregular solutions to radial Schrodinger equation for the case of hydrogen-like atoms. Nanosystems: Phys. Chem. Math., 2023,14 (1), 28-43.
1. Introduction
Finding the closed-form of the first (regular) and the second (irregular) solutions to the quantum Coulomb problem for negative energies have not been well elucidated by the literature. Although, the regular solutions to the Coulombic Schrodinger equation have been more elucidated by an infinite power series representation, but the irregular solutions to the Schrodinger equation in the case of a bound electron experiencing a Coulomb force are being ignored for several reasons. Some of them indicate that irregular solutions are ill-behaved at the origin (due to the irregular singularity) and unbounded at infinity. But ignoring irregular solutions based on fancy reasons is worthless as they can be of great interest in problems describing hydrogenic bound electronic waves. The independent irregular solutions to the Coulomb Schrodinger equation for hydrogen like atoms can be effectively deployed for studying the complex poles for the bound, resonance and virtual states, hydrogenic wave functions, Regge trajectories, Pade extrapolation, and so on. If the electron potential energy is Coulombic only within a shell region around a central region, then irregular solutions can be accomplished in deriving shell region solutions on the boundaries of the shell. Geilhufe et al. [1] effectively constructed a Green function by taking into consideration the first (regular) and the second (irregular) solutions of the scattering (single site) problem and thereafter discussed the asymptotic behavior of the resulting spherical wave functions. Newton [2] studied the two linearly independent solutions of one dimensional Schrodinger equation (SE) and noted that the regular solution (in the form of Whittaker functions) that lies in the right half plane experiences analyticity near the origin whereas the irregular solution lying in the left half plane experiences logarithmic singularity and infinite derivative in the vicinity of the origin. Cantelaube [3] ruled out the necessity of the usual boundary condition imposed in extracting the radial part of the Coulombic SE in spherical polar coordinates and concluded that the first solutions are regular, but the second solutions are either singular or pseudo functions, the latter arises when the second solutions are derived by taking the Laplacian of the radial part. Khelashvili and Nadareishvili [4] exploited the singular behavior of the Laplacian operator and found that the radial wave function is less regular than 1/r, owing to the delta like singularity at the origin.
© Parkash C, Parke W.C., Singh P., 2023
Seaten [5] studied the asymptotic behavior of both regular and irregular solutions of radial SE with the Coulomb potential and remarked that the two solutions for attractive potentials decay exponentially whereas, for repulsive potentials, the regular (irregular) solutions grow (decay) exponentially. Khalilov and Mamsurov [6] constituted an expression for the radial Green's function after converting the regular and irregular solutions of radial Dirac equation into the form of Whittaker functions and found that the first (regular) solutions are R-integrable in the vicinity of the origin whereas the second (irregular) solutions are integrable at the point of infinity. Michel [7] compared the analytical and numerical computations of the first (regular) and the second (irregular) solutions of zero dimensional SE with the Coulomb potential. The authors concluded that, along the real axis, the numerical computation of the radial wave functions is quite difficult as they can vary significantly by many orders for absolute values of principal quantum numbers. Nevertheless, the analytic computation becomes more problematic when these wave functions are continued analytically in the complex plane. After solving coupled radial SE having regular singularity in the vicinity of the origin and irregular singularity at infinity, Galilev and Polupanov [8] rehabilitated irregular (logarithmic) solutions into the forms of asymptotic expansions and regular solutions as an algebraic combination of logarithmic function, power function, and power series.
Axel Schulze-Halberg [9] derived a finite normal series (order zero) solution of 1D radial SE with a large class of singular potentials having irregular singularity in the vicinity of infinity and/or the origin and thereafter computed energy eigenvalues corresponding to the SE. Gersten [10] rehabilitated the regular and irregular (logarithmic) solutions of the SE into the forms of spherical Bessel's functions and thereafter deployed backward recurrence relations and the Cauchy integral formula to achieve a 5-digit numerical accuracy of the resulting special functions. Based upon the numerical solutions of the SE with nuclear plus Coulomb potential, Mukhaamedzhanov et al. [11] established a novel procedure for obtaining the scattering pole parameters corresponding to the resonance(narrow and broad), virtual (anti bound) and bound states and utilized them to study asymptotic behavior of the Coulomb wave functions.
Cattapan and Maglione [12] numerically evaluated characteristic roots (eigenvalues) and corresponding characteristic vectors( eigenfunctions) of the SE and exploited them, by means of Pade extrapolation (approximant), to analytically continue the bound-states Coulomb wave function into the scattering region and finally achieved a larger numerical accuracy of the resonance parameters.
To overcome the limitations of the Milne-Thompson method, Midy et al. [13] deployed the enduring LT ( Lanczes Tau) method to numerically approximate the regular and irregular solutions of 1D SE, intended to extract the complex poles for the bound states, resonance and virtual states and finally achieved a 12-digit numerical accuracy of the resulting solutions. Thompson and Barnnet [14] made analytic continuation of the hypergeometric series and obtained some continued fractions for the logarithmic derivatives of first(regular) and second(irregular) Coulomb wave functions of the 1D SE and rehabilitated the resulting solutions into the forms of Whittaker and Bessel's functions. The explicit representation of the independent irregular solutions at hand can help us in studying the analytic behavior of Coulomb scattered-wave amplitudes as the energy of the scattered electron is extended into the complex plane, and as the electron angular momentum quantum number I takes on complex values, such as in Regge pole analysis(See Gaspard [16]). Furthermore, the closed-form expressions for the irregular solutions, rather than the usual Laurent series representation, provide explicit answers to how the irregular solutions behave for the electron waves at large distances from the nucleus, useful in presentations of the quantum Coulomb problem.Toli and Zou [17] obtained a Taylor series expansion of the regular Coulomb wave functions and concluded that the exact solutions of the SE, having the Coulomb potential for molecules consisting of more than two particles, cannot be achieved. Simos [19] developed multiderivative methods for comparing the numerical solution of the 1D SE with the existing exponentially- fitted Raptis-Allison method and Ixaru-Rizea method. Parke and Maximon [20] deployed the extended version of the Cauchy integral formula for obtaining the closed- form second independent solutions of the confluent hypergeometric difference equation for the degenerate case. Liu and Mei [21] applied the Laplace transform method as well as the transcendental integral function method for obtaining the second independent infinite series solution of the time dependent SE for hydrogen like atoms.
Many times the Frobenius coefficients, required for the explicit power series representation of a solution to a 2nd order differential equation, obey a three-term recursion relation which cannot be easily utilized. The difficultly comes from the fact that three-term recursion relations have two linearly independent solutions. While attempting to compute the first solution from the recursion, numerical contamination from the second solution can grow, destroying the accuracy of the first solution. We here consider a large class of ordinary second-order differential equations which yield, via the Frobenius method, two-term recursion relations that have explicit solutions. We derive simplifying results for the Forbenius coefficients. These are applied to find the irregular solution of the radial Schodinger equation for the case of hydrogen-like atoms. The irregular solution, having an 1 /rl+1 as well as a logarithmic singularity at the origin, is not ordinarily considered. However, it could be used, for instance, in a 'toy' problem wherein the nucleus is given a finite size, say radius ra, and there is a different potential energy for the electron at a radius rb > ra, such as a screened electron potential energy proportional to e-r/b/r. Between the radius ra and rb, both the first and the second Coulombic solutions would enter the steady bound-state wave function solution to match boundary conditions at ra and rb. (These boundary conditions are that the wave function, determining electron probabilities, and its derivative, determining the electron charge flux, must match on the boundaries.) However, under realistic conditions, ra is more than 10,000 times
smaller than the RMS radius r2 |R(r)|2 r2dr, so that the contribution of the second solution is also extremely
small. Moreover, theoretical quantum chemists have developed far more sophisticated techniques for getting good wave functions for inner electrons in atoms. This is why we call the problem a 'toy'.
The rest of this research paper is organized as follows: Section 2 addresses the concept of the Frobenius method required for the subsequent development of the proposed work. Section 3 discusses attempts to develop a novel procedure for obtaining the explicit representation of the two exact solutions of a large class of second order ordinary linear homogeneous differential equation with polynomials coefficients. Section 4 validates the applicability of the proposed procedure by solving the Coulombic Schrodinger equation. Some example expressions, plots and asymptotic behavior of both the regular and the irregular solutions to the Coulombic SE are given in Section 5. Finally, Section 6 summarizes the concrete conclusions and future scope of the work done in this paper.
2. Frobenius method
A large class of 2nd order homogeneous linear differential equations with polynomial coefficients and regular singularity at the origin can be expressed as:
LR = r2(ao + asrs)R" + r(£o + &rs)R' + (7q + 7srs)R = 0, (1)
where Oj, 7i (i = 0, s) are real constants with the additional provision that a0 = 0, and s needs not be positive nor any negative integer. The most commonly utilized procedure for obtaining the power series solutions to (1), under the situations, when the polynomial r2(a0 + asrs) and r(^0 + ^srs) possess regular singularity in the vicinity of x = 0, was first exploited by Frobenius [15]. Thus, following the work of Frobenius [15], equation (1) will admit at least one infinite series solution of the form:
w
R(r,m) = rm ^ bsk (m)rsk. (2)
k=0
Here, the factor rm reflects the threshold behavior of the resulting solutions to (1) and the exponent'm' is chosen so that the leading coefficient b0 (m) is a non-zero constant. The Frobenius series on the right of equation (2) can be differentiated term by term and converges on some interval (0, d) where d is the distance from the origin to the nearest zero of the polynomial (a0r + asrs) of arbitrary degree in the complex plane and has no zeros on (0, d). Further, the coefficients b0(m), bs(m), b2s(m), • • • and the exponent'm' are independent of r and the term rm may be complex for the negative powers of r or undefined at the regular singularity. Due to this reason, we shall consider only those solutions which are defined for positive values of r since solutions for negative powers of r can be similarly obtained by using the well-
w
known result which states that if rm ^ bsk (m)rsk is a power series solution of LR = 0 on the interval (0, d), then
k=0
w
|r|m bsk(m)rsk is also a solution on the intervals (—d, 0) and (0, d), respectively. Moreover, the coefficients bsk(mi)
k=0
can be determined recursively for k > 0 and for k = 0, an "indicial" equation must be satisfied. If the roots m1, m2 with m1 > m2, of the indicial equation differ by an integer, that is, when (m1 - m2)/s = t, t G Z+, then the two resulting solutions for y(x) will not be independent.
3. Procedure
Finding the closed-form of the first (regular) and the second (irregular) solutions to equation (1) have been well elucidated by developing a novel procedure. The proposed procedure is then applied for obtaining the explicit representation of the two independent solutions as well as some radial wave functions related to the Coulombic Schrodinger equation. The underlying procedure of getting these solutions is totally distinguished from getting them either by means of associated Laguerre polynomials or functions of hypergeometric nature. We observe that the series on the right of equation (2) is a positive term series and hence it is uniformly as well as absolutely convergent, which in turn, possesses first order partial derivatives with respect to the argument r. Thus, term by term partial differentiation of equation (2) w.r.t. r yields
w
R'(r, m) = ^(sk + m)bsk(m)rsk+m-i; (3)
k=0
w
R"(r, m) = ^(sk + m)(sk + m - 1)bsk(m)rsk+m-2, (4)
k=0
where the primes indicate partial derivatives of R with respect to r. Define
fi(m) = Ojm(m - 1) + ^¿m + (5)
where i = 0 or s and m is any number (real or complex) such that f0(sk + m) is defined for all positive integer values of k. A practical way to compute the coefficients bsk (m) Vk > 1, is through the following theorem.
œ
Theorem 1. If R = R(r, m) = bsk(m)rsk is a power series solution of LR = 0. Then
k=0
LR = fo(m)bo(m)rm. (6)
Proof: Insert expressions (3) and (4) into (1) to get
œ
LR = ^ [<ao(sk + m)(sk + m - 1) + Po(sk + m) + Yo bsk(m)rsk+m+
k=0
k=0
Plugging the notations defined by (5) in the forgoing equation (7) yields
œ
^ [as(sk + m)(sk + m - 1) + ¡3s(sk + m) + 7s] bsk(m)rs(k+1)+m. (7)
œ
LR = Y, fo(sk + m)bsk(m)rsk+m + ^ fs(sk + m)bsk(m)rs(k+1)+m. (8)
Is
k=o k=o
The index of the second summation in (8) is a dummy parameter and hence, without loss of generality, it can be shifted from k to k — 1 to obtain
LR = Y, fo(sk + m)bsk(m)rsk+m + fs(s(k - 1) + m)bs{k-i)(m)r
m)
k=o k=l
To obtain more clarity on the successive coefficients bsk (m) Vk > 1, extracting the first non zero term and combining the rest with the second summation yields
œ
LR = fo(m)bo(m)rm + ^ fo(sk + m)bsk(m) + fs (s(k - 1) + m)bs(k-i) (m) k = l
rsk+m. (9)
For LR = 0 to be satisfied, the coefficient of rm in the first term and of rsk+m in the bracket of the summation must be vanished and hence it becomes necessary to define
, t s bs(k-1)(m)fs(s(k - ^ + m)
bsk(m) =--^-' --- Vk > 1, (10)
fo(sk + m)
which completes the proof.
The equation f0(m) = 0, so-called indicial equation, is quadratic in m and determines possible values of'm' for a solution of the assumed Frobenius form to exist. The two-term recursion relation (10), with a given starting value for b0(m), will give one all the subsequent coefficients for larger values of k. We now see that the form of the original differential equation (1) allows for a two-term (rather than a three-term) recursion relation for the coefficients in a Frobenius series. Having this two-term recursion relation, it is straight forward to solve bsk(m) in terms of b0(m). Thus, giving k the values, 1, 2, 3,... to the recurrence relation (10) yields
bo(m)(-1)1fs(m) ^ bo(m)(-1)2fs(s + m)fs(m)
bs(m) = -Ti-i-^- , b2s(m) = -f to ,-w , ,-N- ,
fo(s + m) fo(2s + m)fo (s + m)
ba m = bo(m)(-1)3fs(2s + m)fs(s + m)fs(m) s fo(3s + m)fo(2s + m)fo(s + m)
and so on. Recursively, we have
h ( , bo(m)(-1)k j (fs(sj - ^ + m)
bsk(m) =- k \ , .-;- Vk > (11)
[1 = 1 fo(j + m)
k2
where 1 < j < k and ^Q bj = 1 if k2 < k1 for any expression bj. The result (11) has been furnished to compute the
j = ki
coefficient bsk (m) in our procedure.
To understand our procedure, let us denote the two possible solutions for'm' of the indicial equation fo(m) = 0 as m1, m2 with m1 > m2. Here, we briefly discuss the situation when the roots differ by an integer, that is, when
(m1 — m2)/s = t, t € Z+. Since
fo(sk + m{) = aosk(sk + st) = aos2k(k + t) = 0 Vao = 0, s,m,t € Z+,
it follows that bsk(mi) is defined for each k > 1 and therefore, the method of Frobenius permits us to obtain the first power series solution of (1) as
œ
Ri = R (r,mi) = £ bsfc(mi)rsk+mi. fc=0
But /0(sk + m2) = a0s2k(k - t) vanishes for k = t and hence the coefficient bsk(m2) can not be computed for k > t. In such situation, we construct the second solution as a linear combination of w1 and w2 as discussed in the following theorems. Theorem 2. Let
dR œ
wi = — = R (r, mi) log r + £ 6Sfc (mi)rsk+mi
m m=mi fc=0 with the additional provision that b0(mi) = 1. Then
= a0strmi. (12)
m=mi
Proof: The series on the right hand side of (2) can be differentiated partially with respect to the argument m to obtain
dR = ^ (^Z bsk (m)rsk ) = rm log r£bsfc (m)rsk+m + £ bSfc (m)rsk+m
oo oo
k=0 J fc=0 fc=0
œ
sfc+m
R (r, m) log r + &Sfc(m)r
fc=0
, , dR dR d2R d2R , Let the primes indicate partial differentiation with respect to r, such that —- = —-; ——r = -—tt. The resulting
dr dr dr2 dr2
Theorem 1 yields
d2 r d R
r2(a0 + asrs) ^ + r(A, + Asr8) ^ + (70 + Ysrs)R = /0(m)b0(m)rm. (13)
Both sides of equation (13) can be partially differentiated with respect to the argument m, which yields
2/ ^ d2 (dR\ 0 s. d (dR\ . s. dR
(a+a*rS)ddr2 [dmj + r(A)+^rS)dr (dmJ + (Yo+Y*rS)dm =
" dR"
L
Setting m = m1 yields the desired result.
dm
= /0(m)b0(m)rm + /0(m)b0(m)rm + /0(m)b0(m)rm log r.
t—1
Theorem 3. Assume the coefficients bs(m2), b2s (m2),... bs(i-1)(m2) are suitable chosen. Let w2 = £ bsk (m2)rsk+m2
fc=0
with the provision that b0(m2) = 1, then
L(w2) = fs(mi - s)6s(i_i)(m2)rmi. (14)
Proof: Since w2 is apolynomial of degree t - 1, we can, suitably, choose bst(m2) = bs(t+1)(m2) = ... = 0. Under these restricted conditions, we have
t—1 t—1 œ œ
W2 = £ bsfc(m2)rsk+m2 = £ bsfc(m2)rsk+m2 + £ bsfc(m2)rsk+m2 = £ bsfc(m2)rsk+m2. k=0 k=0 k=t k=0 With the aid of equation (9), the foregoing equation yields
œ
L(w2) = £(/)(sk + m2)6sfc(m2)+ /S(s (k - 1) + m2)bs(fc—1)(m2]) rsk+m2 k=1 t —1
= £ (f0(sk + m2)6sfc(m2) + /S(s (k - 1) + m2)bs(fc—1)(m2)) rsk+m2 k=1
+ (f0(st + m2)6st(m2) + /S(s (t - 1) + m2)bs(t—1)(m2)) rst+m2
œ
+ £ (/0(sk + m2)6sfc(m2)+ /S(s (k - 1) + m2)bs(fc—1)(m2)) rsk+m2. fc=t+1
Since fo(sk + m2) = aos2k(k —t) = 0 V1 < k < t — 1, we can use (10) to choose the coefficients bs(m2), b2s(m2),... bs(t-i){m2) to satisfy
bskm) = — fs(s(k — f)+mfs)-l)(m2) = {—i)k Q fsW- m2) V 1 < k < t — 1. (15) fo(sk + m2) = fo(sj + m2)
With the aid of newly introduced formula (15) and employing the conditions fo(st + m2) = fo(mi) = 0 and bst(m2) = bs(t+i)(m2) = ... = 0, we finally have
L(w2) = (fs(s (t — 1) + m2)bs{t-D(m2)) rst+m2 = fs(mi — s)bs(t-i) (m2)rmi.
We next move to compute the coefficients b'sk (mi) Vk > 1, which will strengthen the subsequent development of the proposed procedure.
3.1. Computation of b'sk (mj)
In finding the coefficient b'sk (mi), it is legitimate to take logarithm of each member of equation (11) and write
k k log \bsk(m) \ = J^log |fs (s(j — 1) + m) \ — ^log \fo(sj + m) \ + log\bo(m)\.
j=i j=i
Term by term logarithmic differentiation of each term of the foregoing series with respect to m yields
b'sk(m) =y f' (s(j — 1) + m) — * f0(sj + m) + b_0o{m). bsk (m) fs (s(j — 1) + m) ¿=1 fo(sj + m) bo(m)'
The last term on the right in above vanishes, since b0(mi) = 1. Thus, setting m = mi, we discover that
bsk (mi) = bsk (mi)Jsk (mi), (16)
where
T , n fs (s(j — ^ + mi) ^ fo(sj + mi)
Jsk(mi)= V f , -ry--r , -r. (17)
fs (s(j — 1) + mi) ¿=1 fo(sj + mi)
The expression, formed by pair of equations (16) and (17), will act as an important formula for obtaining the second exact power series solution of (1). Next, we turn to establish one more important formula involved in obtaining the linear combination L(Cwi + w2) which will put us in a better position to construct obtain the desired second exact solution to Coulombic Schrodinger equation. With the aid of equations (12) and (14), we finally have
L(Cwi + W2) = CL(wi) + L(w2) = Caostrmi + fs(mi — s)bs(t-i)(m2)rmi.
Lastly, We choose C such that
fs(mi — s)bs(t-i) (m2) tUtT/r, , N n
C =------so that L(Cwi + w2) = 0,
aost
which shows that R2 = R (r, m2) = Cwi + w2 is the second exact power series solution of (1). However, if C = 0, then there is no need to compute wi and hence, the second solution in this case becomes R2 = R (r, m2) = w2.
4. Two exact solutions of the Coulombic Schrodinger equation
To exemplify the procedure, we consider the case when s = 1. (The resulting differential equation is then of hypergeometric type.) We apply the proposed procedure for obtaining two exact solutions to the Coulombic Schrodinger equation:
^ d f ) +W (B — V (r) — f ^ ) R = 0, (18)
r2 dr \ dr J n2 \ 2m r2 J
where the non negative integer I represents the angular momentum quantum number, n = h/(2n), h is Planck's constant,
Ze2
B represents the total energy of the system and V(r) = — -- represents the potential energy. Incidentally, the
4neor
Coulomb potential energy, V(r), admits continuous states for B > 0, describing electron- nucleus scattering, and discrete
V—2mB
bound states for B < 0. We shall confine our discussion to the latter. Therefore, the values of k = -^- will be
taken as a positive real number with the units of inverse length.
Under these restrictions, equation (18) reduces to the following form:
1 d C 2 dR \ i / Zme2 \ 1 1(1 +1)\R_Q
r2 dr V dr / V V 4ne0h2 I r r2
h
Zme2
Dividing throughout by k2 and defining k0 =-—r in the foregoing equation, we obtain
4ne0ft2
1 d (r2dR^ + (-i + 2kQ * - r = 0, (19)
(«r)2 dr y dr J y k («r) («r)
which can be further simplified as
r2 + f - K2r2 + 2«0r - ^ + 1)^R = 0. (20)
dr2 dr
The parameter k0 is the inverse of the Bohr radius a. We first examine the asymptotic behavior of the solutions of (20)
for large value of r. For this, take p = «r so that dp = k. Thus,
dr
dR _ dR dp _ ^dR d / 2 dR 2 d2R + dR dr dp dr dp dr dr dp2 dp
Inserting these derivatives into (20) yields
d2R 2 dR / 2k0 1 ^ +1) dp2 p dp y k p p'
For large values of p, the coefficient in parenthesis can be approximated by -1, so equation (21) reduces to
d2R dp
The general solution of (22) is given by R = c1ep + c2e—but ep blows up for large value of p, which is an unbounded solution. Therefore, we must have c1 = 0. That leaves R = c2e—p. Matters are simplified if we extract the asymptotic behavior from R. Thus, we define anew function u(r) through
R(r) = e—pu(r) = e—Kr u(r). (23)
Next, we have
dR „„ / du \ d2 R „„ / d2u „du 2 "t" = e—'Kr — - ku ; — = e—— - 2k— + k2u dr dr dr2 dr2 dr
2 + ^ + -1 + ^ p )R = 0. (21)
2 = R. (22)
Inserting these derivatives and (23) into (20) yields
r2 ^ + 2r(1 - Kr) -^ + (-¿^ + 1) + 2 (ko - k) r) u = 0 (24)
dr2 dr
Here, we see that r = 0 is a regular singular point. In our notations, we have
s = 1,
ao = 1, ai =0; ft = 2, ft = -2k; 7Q = -¿^ + 1), 7i = 2(kQ - k). f0(m) = (m - ¿)(m + ¿ + 1), fQ(m) = 2m + 1; f1(m) = -2k (m - n +1), f1 (m) = -2k, where k0 = k^. 4.1. Comments and discussion
The roots of the indicial equation f0(m) =0 are m = ¿, -¿ - 1. Take m1 = ¿, m2 = -¿ - 1 so that m1 - m2 = 2¿ + 1 = t where t is a positive integer. Thus, by Frobenius method, equation (20) has at least one solution of the form
w
R(r,m) = e-Kr £ bk (m)rk+m. (25)
k=0
k2 r2
Since e-Kr = 1 - Kr +—-—+ • • •, the radial wave function R(r) behaves as b0 (m)rm for small r. Therefore, for m = ¿,
R(r) is not singular at the origin. But for m = -¿ - 1, R(r) <x 1/r£+1 for small r. Since ¿ = 0,1, 2, 3, • • •, the root m = -¿ - 1 makes the term 1/r£+1 infinite at the origin. But, for the bound state eigenfunction to be normalized, we should have
w a
R2r2dr « / —TT7:dr (X —7TT-
, (26)
0 0 0 where a is a small number. The m = - - 1 case gives states unnormalizable when ¿ > 0. When ¿ = 0, the divergence of the R(r) near the origin gives one a radial function which no longer satisfies the original SE, since the diverge at the origin is strong enough to make LR(r) = -(4n/r2)^(r), where £(r) is the Dirac delta function (see Messiah [18, p.352]). For this reason, standard references reject this root. Little effort is seen in the literature for obtaining the second exact solution of the Coulombic Schrodinger equation.
4.2. First exact solution of Coulombic Schrodinger equation
By the Frobenius method, equation (24) has at least one solution of the form
u(r,m) ^^ bk (m)r
k+m
(27)
k=0
Calculating the coefficients by successive differentiating a differential equation may be excellent in theory, but it is usually not a practical computational procedure. Rather, one might try to evaluate bk (m) recursively, one by one by writing the recurrence relation (10) first for k = 1, then k = 2, and so forth. Alternatively, we can convert the general formula for bk(m) in terms of b0(m). Thus, equation (10) gives one
bk (m) = -Giving k the values 1, 2,3,...,
bi(m)
b2(m)
fi(k - 1 + m)bk-i(m) 2k (k + m - r) bk-i(m)
fo(k + m)
(k + m - i)(k + m + i + 1)
Vk > 1.
(2k) bo(m) (1 + m - r)
jïTm-iïîïTmTi+ï),
(2k)2 bo(m) (2 + m - r) (1 + m - r)
(28)
bk(m) = (2k) bo(m) JJ
(j + m - r)
j=i
(j + m - i)(j + m + i +1)
V k > 1.
Setting m = mi = i in equation (28) yields
(2K)k bo(i) j + i - r
k!
j=i
j + 21 +1
Changing k to k +1 in the resulting equation (29) to obtain
bk+K
(2K)k+1 bo(i) kfpl j + i - r
(k +1) !
k+i
n
j=l
2K(k +1+ i - r) j + 2i + 1 = (k + 1)(k + 2i +2)
bk
(29)
(30)
Suppose Bk (m) represents the k term of the series represented by (27). Using (30), the ratio of (k + 1) h term to
kth term of this series for the case m = mi = I is represented as
Bk+i(£) bk+i(£) 2k (k + 1 + i - n)
Bk
bk (i)
(k + 1)(k + 2i +2)
for large k.
k
This means that the first solution u\(r) = u(r,i) is asymptotically equals to e . This implies R\(r) = R(r,i) = e-Krui(r) = eKr which blows up for large value of r. This means the series solution ui(r) = u(r, i) must terminate. This implies there must exist some maximum integer k such that bk = 0, and bk+1 = 0, bk+2 = 0 and so on. This is possible only if we can choose k + i +1 - r = 0 so that r = k + i +1 (so called principal quantum number) This suggests that R\ (r ) is a polynomial of maximum degree r - i - 1. Setting bo (mi ) = 1, and using r = ko/k, the first exact series solution represented by (25) is finally expressed as
R (r, mi) = e Kru(r
n-e-i (r) = e-^ ^ bk ( k=o
)rk+e = e re
'-T-i I TT (j + i - r) (2Zr_\'
^ k! H (j + 2i +1)\ ra J
(31)
k=o
j=i
The sum factor is the associated Laguerre polynomial L'ne-+e1-l(2Zr/(ra). 4.3. Second exact solution of the Coulombic Schrodinger equation
We first establish the fact that two solutions of the indicial equation give a set of linearly dependent solutions to equation (25). For this, changing k to k + 1 in (28) to obtain
(2K)k+1 (k + 1 + m - r) bo(m)
bk+1(m)
n
(j + m - r)
(k + 1 + m - i)(k + 1 + m + i + 1) ^ (j + m - i)(j + m + i + 1) '
Setting m = i, bo(i) = 1, the foregoing equation (32) yields
(2Kf+Uk+1+e-r) (k + 1)(k + 2i +2)
bk+1(
1 k!
Wk=i(j + i - r) Uk=i(j + 2i +1) '
b
k
But and
,=1 (^ +1>!
k
(j + ¿ - n) = (k + ¿ - n)! and vanishes for j < n - ¿.
j=1
Therefore
b m = (2K)k+1 (k +1+ I - n)!(2l +1)! (33)
bk+1(^ = ^ . (33)
^ + 1)!
Similarly, setting, m = - - 1, b0(- - 1) =-27+T in equation (32) to get
(2k) +1
b 1)= (2K)k+1 (k - I - n)(2l +1)! nk=1(j - I - n - 1)
k+1( ) (k +1)!(k - 2*)(2k)2'+1 nk=1 (j - 2* - 1) .
Changing k to k + 2¿ + 1, we get
(2к)к+2£+2 (k + , + 1 - n) (2, + 1>! Пк+12£+1(^ - , - П - 1)
(k + 1>!(k + 2, + 2)! (2k>2£+1 ^ nJfc+2£+1(j - 2, - 1) '
bfc+2£+1+1(-, - !-> "---"-—-' —-
But
fc+2£+1
^Q (j - 2, - 1) will vanish for j < 2, + 1, and otherwise
-=1 fc+2£+1
П (j - 2, - 1> = k!.
-=2£+2
fc+2£+1
Also, (j - , - 1 - n) vanishes for j < n + , + 1, and otherwise
-=1
fc+2£+1
П (j - , - 1 - n) = (k + , - n)(k + , - n - 1)...(1) = (k + , - n>!.
-=n+^+2
Therefore
b ( , 1> = (2«>fc+1 (k +, + 1 - n>!(^ +1>! (34)
- 1> =-(k + 1>!(k + 2, + 2)!-, (34)
which is exactly the same as (33). This suggests that the two solutions of the indicial equation yield a set of two linearly dependent solutions to the Coulombic Schrodinger equation (18). In other words, out of these two, only one of them is useful and the other one can be dropped. Interestingly, discarding one of the solution has nothing to do with the regular singularity at r = 0. Incidentally, we will construct another linearly independent solution of the Coulombic SE (24) by employing the general procedure proposed in Section 3.
4.3.1. Computation of 6'fc (,) V k > n -,. Employing the resulting equations (16) and (17) for the case m = ,, b0(,> = 1 Z
and к = — to get
na
b'fc (,) = 6fc (,)Jfc (,), (35)
where
M,) = А П /j + * - n> (2ZV Vk > 1;
fc( > k! -=1(j + 2, + 1>VnV > ;
7(,> = /1 (j-1 +,) /0(j + ,>
Jfc(,) = Z^ / (A 1+ ,) Z^ /„ (j + j
-=1 /1 (j - 1 + ,) /o(j + ,) Vj - n + , j j + 2, + 1
1
1
1
which becomes an indeterminate for j = n - ¿.To overcome this situation, we re-write equation (28) as
nLi(m - n + j)nL„-€+i(j + m - n) /2Z
6fc (m) = bo(m)^
nk=i(j + m - i)(j + m + i +1)
n«
= bo(m)
(m - n + 1)(m - n + 2) • • • (m - i - 1)(m - i) n^-m^' + m - n) / 2Z
n;=i(j + m - i)(j + m + i +1)
n«
which vanishes for m = ¿. Therefore, we re-write the foregoing equation as
bk(m) = (m - i) Ck(m),
where
(m - n + 1)(m - n + 2) • • • (m - i - 1) flL«-£+1(j + m - n) /2Z
Cfc(m) = bo(m)
nk=i(j + m - i)(j + m + i +1)
n«
Differentiating the resulting equation (36) with respect to m yields
b'k (m) = (m - ¿)Ck (m)+ Ck(m). Plugging the root m = ¿, b0(¿) = 1 into (37) yields
(¿ - n +1)^ - n + 2) ••• (¿ - ¿ - 2)^ - ¿ - 1) nk=n-,+1(j + m - n) ( 2Z
bk (¿) = Ck (¿) = nkj+^1 Ina
(-1)^ (n - ¿ - 1)! nk=n-m(j + ¿ - n) (2Z
k=i(j + 2i +1)
n«
But
Also
^(j + 2i +1) = (1 + 2i +1)(2 + 2i +1) ••• (k - 1 + 2i +1)(k + 2i +1) = j=i
(k + 2i +1)! (2i +1)! ■
(36)
(37)
n (i - n + j)
Therefore
' - n + n - i + 1)(i - n + n - i + 2)(i - n + n - i +3) ••• (i - n + k)
= 1 • 2 • 3 • • • (i - n + k) = (k - n + i)!.
(-1)n-^ (n - i - 1)!(k - n + i)!(2i +1)! ( 2Z A' " k!(k + 2i +1)! Vk " n ^
4.3.2. Computation of b'k (¿) V1 < k < n - ¿ - 1. Re-writing equation (28) as
bk (m)
bo(m)(m - n + k)(m - n + k - 1) • • • (m - n + 3)(m - n + 2)(m - n + 1) ^^a) (m - i + k)(m - i + k - 1) • • • (m - i + 1)(m + i + 1 + k)(m + i + 1 + k - 1) • • • (m + i +2) b0(m)(m - n + k)!(m - i)!(m + i + 1)! (2Z '
.n«
(m - ¿ + k)!(m + ¿ + 1 + k)(m - n)! Setting m = ¿, b0 (¿) = 1, in the foregoing equation, in this case, yields
d (m - n + k)!(m - ¿)!(m + ¿ + 1)!
b't(i) =
2Z
dm (m - i + k)!(m + i +1 + k)(m - n)! \ n«
(38)
The expression on the right hand side of (40) can be further simplified as shown in the following theorem. Theorem 4. The coefficient
d (m - n + k)! (m - i)! (m + i + 1)!
dm (m - n)! (m - i + k)! (m + i +1 + k)!
simplifies to
k+i
k!
y-r n - i - j \ 1 1 1
1=1 2i + 1 + j J n - i - j + 2i + 1 + j + j
k
k
k
k
b
k
m
m=
Proof: Define
c(r, i, k) =
d (m - r + k)! (m - i)! (m + i + 1)!
dm (m - r)! (m - i + k)! (m + i +1 + k)! k
n (m - i + j) (m + i + 1 + j)
d dm
m-r+j
——F (r, i, k, m) dm
(40)
where
F (r, i, k,m) = n
m - r + j
(m -
3=1
Define a logarithmic derivative of F(n, i, k, m) through
1 d d
G (n,i,k,m) = ———-- — F (n,i,k,m) = — (ln F (n,i,k,m)),
F (n,i,k,m) dm dm
with the advantage that products in F(n, i, k, m) become sums. We have
d
m - r + j
G (r, i, k, m) = ^ lnn 7-, , - r + j , ,
w ' dm ^ \(m + i +1+ j)(m - i + j)
d
—— (ln (m - r + j) - ln(m + i + 1 + j) - ln(m - i + j))
ri'm —^
dm
j=i
k
Ei •
m-r +j m
1
m - i - j
Therefore (41) gives one
c(r, i,k) = F (r, i, k, m) G (r, i, k, m)
n
m - r + j
U=i
(m - i + j) (m + i + 1 + j)
1
k
E ■
^—' \ m — r+j j=i v ' J
m- i- j
(-1)
k!
11 2t +1+ jl Vr - 1 - j + j 2i + 1+ j
3=1 / 3=1
With the aid of resulting equation (40), the undergoing equation (39) yields
b'k (i)
(-1)
k+i
k!
n
r - i - j
E
1
+
1
1\ 2Z\
+1+ jjUK r - i - j- 2, + 1+1 + j) UJ V1 È k È r - i - L <41)
Here, we assume b'k (i) =0 W1 < k < 0.
4.3.3. Computation of wi. For obtaining the second exact solution of (24), the resulting expressions for bk (i) represented by (38) and (42) and Theorem 2 suggest us to write
n-i-1 œ
wi = u(r,i)logr + re E bk(i)rk + re E bk(i)rk.
k=i k=n-e
4.3.4. Computation of C. Setting bo(m2) = bo(-i - 1) = 1 into the resulting equation (28) yields
2Z\k Uj=i(r + i +1 - j)
bk(-i - 1)= — ra
k! fik=i(2i +1 - j)
In particular, setting k = 2i into the undergoing equation yields
2e
bu-i -1) = [22Z)n2! i(r +i + 1 - j) b2e( i 1) V ra) (2i)! Hf= i(2i +1 - j)
But
2e
\\(2i +1 - j) = (2i)(2i - 1) ■■■ 3 ■ 2 ■ 1 = (2i)!.
j=i
m
m
1
1
m
1
1
m
m
Also
2£
n(n + ¿ +1 - j) = (n + ¿ +1 - 1)(n + ¿ +1 - 2)...(n + ¿ +1 - 2¿ +1)(n + ¿ +1 - 2¿)
j=1
= (n + ¿)(n + ¿ - 1)...(n - ¿ + 2)(n - ¿ + 1)(n - ¿)! = (n + ¿)! = (n - ¿)! = (n - ¿)!.
Plugging the resulting expressions into (43) yields
b ( ¿ 1)= (n + ¿)! (2Z
- 1)=(n - ¿)!(2№)! tvn^J .
Therefore by definition
f1 (m1 - 1)bt_1(m2^ fl(¿ - 1)b2"M - 1)
C = -
a0st&0(mi) 2i +1
2Z (n - i)M- - 1) _ (П + i)! ( 2Z_y2'+i
na 2i +1 (2i + 1)!(2i)!(n - i - 1)! V na У
4.3.5. Computation of w2. By definition,
w2=f bk (-i - 1)r*-'-i=r-'-i f n'=f+i+1 - j > ( у 2 k= ' k= k! nk=i(2i+1 -j) 1 w
Finally, the second exact solution to Coulombic Schrodinger equation (18) is
Sf'(r) = R(r, m2) = e u(r, m2) = e (w2 + Cwi)
Zr
e
-iv^ i A n + i +1 - j(yk
^k! j=ii 2i +1 - j Uaj
(П + i)!r' (2^\ 2'+i I/"-- ^ -Л (j + i - n) k, l (2i +1)!(2i)!(n - i - 1)4 naj |\ ¿0 k! 1=1 (j + 2i + 1) V na У ' °gr
k
(43)
+ Е-1 ^ ITT^^+ 1 + )
+ ¿1 k! j=l 2, + 1+ jj -^Vn - , - j + j +2, + 1+ jAnay
^ (-1>n-£-1(n - , - 1>!(k - n + ,)!(2, + 1)! (2ZrV
+ k!(k + 2, + 1)! UaJ
We are now in a position to provide some example expressions and plots of the Coulombic radial wave functions of the second kind (and of the first kind) in the following section 5.
5. Examples of radial wave functions of second kind
Case 1. If n = 1, then , = 0. In this case, the graphical representation of the radial wave function of the second kind ( and of first kind) are represented in Figs. 1 and 2 and mathematically, employing equation (45), is
„z,^ Zr 1 2Z / ^ (k - 1)! /2Zr\
S?0(r) = e-Zr { --— log r + V , —
k
r a k!(k + 1)! a
Case II. If n = 2, then ¿ = 0,1. In this case, the graphical representation is displayed in Figs. 3 and 4, and , mathematically, using equation (45), is
?Z0(r) = e-zr 1 - 2Zf (1 - log r +5Zr - f (k - 2)' fZrx k 20( ) [r a \\ 2a J g + 4a ^ k!(k +1)! V a
Zr I 1 3Z 3Z2 Z3r / ^ 6(k - 1)! /Zr\k
r) = e 2» ' 1 1 1 1 — ~ 1 4 1
SzZi(r) = e-Я + 20: + Ô02 - 003 Кr + E"
r2 2ar 2a2 2a3 l ь ^ k!(k + 3)! V a
r
FIG. 1. Radial wave function R10(r)
Fig. 2. Radial wave function S10(r)
Fig. 3. Radial wave function R40(r)
Fig. 4. Radial wave function S40(r)
Fig. 5. Radial wave function R41(r)
Fig. 6. Radial wave function S4i(r)
Case III. If n = 3, then i = 0,1,2, In this case, the graphical representation is displayed in Figs. 5 and 6, and mathematically,
N 1 2Z ( 2Zr -Z2r2\ 4Zr -3Z2r2 ^ 2(k - 3)! (2Zr\
Szo(r^ = e-3a 1- y1 - mr + ~Zü2r)logr+-3ZT - + £ kKirriT )
fcN
k=3
N ( 1 -Z -Z2 16Z3r (( Zr\ 3Zr ^ 6(k - -2)! (2Zr
Sz i(r) = e-3M ^ + 3- + - --^l l,1 - 6a) iOS r +
k=3
Z -zr I 1 5Z 10Z2 10Z3 10Z4r -Z5r2 / ^ 5!(k - 1)! f2Zr\'
S2(r) = e 3" jr3 + 6^ + 27^ + 81^ +243Ö4 - 729a5 ^ gr + f^ k!(k + 5)! [l^)
FIG. 7. Radial wave function R50(r) FIG. 8. Radial wave function S50(r)
SZ 20 (r)
Fig. 9. Radial wave functions Sn¿(r) with Z increases
Case IV. If n = 4, then i = 0,1,2, 3. In this case, the graphical representation is displayed in Figs. 7 and 8, and mathematically
SZo(r) =
Sfi(r) =
Zr
e 4a
Zr
e 4a
S^r)
sZ3(r) =
Zr
e 4a
Z 2~2
r
1 _2Z / a _ ^ +
r a \ V 4a 8a2
Z 3r3 192a3
log r +
llZr
8a
19Z2r2
48a2
+
19Z 3r3
Zr
e 4a
4 +
4 +
5Z 5Z2
+ 4a2
4ar
3Z
4ar2
7Z
5Z 3r
~8a3~
5Z 2 5Z 3
^ ^^ + ttt^ +
+
16a2 r 7Z 2
48a3 7Z3
+
Zr 4a Z 2r2\ 80a2 ) log r + 7Zr 16a
5Z 4r Z 5r2 ((1 - Zr \
128a4 128a5 12a )
69Z2r
768a3
2 c
- E
k=4
6(k - 4)! (Zr k!(k + 1)^ 2a
12(k - 3)! ( Zr
1600a2
k=3
k!(k + 3)! V2a
+
7Z4
+
7Z 5r
r4 12ar3 40a2r2 192a3r 1152a4 7680a5
+
7Z6r2 46080a6
Z 7r3
13Zr c 5!(k - 2)! (Zr\ 1
^k(kT5)! \2a)
72a
k=2
92160a7
\0g r+E 7-!Ck-1! (Z- v
k = l
k!(k + 7)!.\2a)
Case V. If n = 5, then i = 0,1, 2, 3,4. In this case, we have
7 / \ Zr 1 2Z
SZo(r) = e - ---
r a
4Zr 4Z2r2 4Z3r3 2Z4r4
1---1-----1--
, 3a 25a2 375a3 9375a4
j log
r+
7Zr 7Z2r2 16Z3r3
5a
+
15a2 375a3
109Z4r4 c 4!(k - 5)! (2Zr\' - 93750a,4 + ^ k!(kT1y\~5a)
S z(r)-e -Zr\ 1 +6Z + 6Z2 16Z3 r
51 I r2 5ar 5a2 25a3
( 3Zr 3Z2 r2 Z3r3 \ 19Zi
V - 100" + 125a2 - 1875a3) 0g r +
19 Zr
167Z 2 r2 2500a2
+
257Z 23r3 112500a3
c 36(k - 4)! (2Zr\'
1
r
Sn/(r)
Fig. 10. Radial wave functions SV£(r) with t increases
Sn,(r)
Fig. 11. Radial wave functions Sn^(r) with n increases
5.1. Properties of the Coulombic radial solutions
As expected, Figs. 9-11 reveal that both the regular and the irregular solutions to the Coulombic radial SE spread in their radial distribution as the system energy increases from strongly negative values to values closer to zero. Also, both distributions move away from the origin as the angular momentum of the electron increases. However, the threshold and asymptotic behavior are quite different. The regular solutions have an re dependence near the origin, while the irregular solutions diverge as r-e-1. Asymptotically, the regular solutions drop exponentially in proportion to rn-1 exp (-r/n), in natural units, while the irregular solutions grow as r-n-1 exp (r/n).
6. Conclusions and future scope
We have shown that a large class of second-order linear differential equations with polynomial coefficients determine two-term recursion relations which can be solved explicitly, and also yield second (irregular) solutions. The procedure is applied to find the regular and irregular (second) solutions to the Coulombic Schrodinger equation for an electron experiencing a Coulomb force, and examples are displayed. Even though second solutions are ordinarily rejected in the Coulombic case because of their unbound character, having explicit expressions for them has utility for several reasons. One is in the study of the analytic behavior of general solutions as a function of the energy and angular momentum of the electron-nucleus system (See Gaspard [16]). Another is in the study of numerical-solution techniques when three-term recurrence relations are used to solve 2nd order differential equations. In such cases, the second solution may invade the wanted first solution through truncation errors during iteration. Asymptotically, the regular solutions drop exponentially while the irregular solutions grow exponentially. Moreover, knowledge of the behavior of the second solution can be used to control the errors in the first. With these techniques at hand, exploration of second solutions to other important linear differential equations is possible.
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Submitted 10 December 2022; revised 18 January 2023; accepted 19 January 2023
Information about the authors:
Chander Parkash - Department Of Mathematics, Rayat Bahra University, Mohali, Punjab, 140104, India; ORCID 0000-0003-1746-5894; cchanderr@gmail.com
William C. Parke - Department Of Physics, The George Washington University, Washington D.C., USA; ORCID 0000-0002-3771-5371; wparke@email.gwe.edu
Parvinder Singh - Department of Chemistry, Rayat Bahra University, Mohali, Punjab, 140104, India; ORCID 0000-0003-3815-4534; drparvinder62@gmail.com
Conflict of interest: the authors declare no conflict of interest.
