Hydrogen atom and orthogonal polynomials

AZA Preprint: 1990/7C. Hydrogen atom and orthogonal polynomials J.S. Dehesa, F. Domínguez Adame, E.R. Arriola, A. Zarzo Updated preprint version: October 1992 of Ref.: C. Brezinski, L. Gori and A. Ronveaux eds., Orthogonal Polynomials and Their Applications, Proc. III International Symposium on Orthogonal Polynomials and Their Applications (Erice, Italy, June 1990), IMACS Ann. Comput. Appl. Math. vol. 9 (1991) pp.223 229.

Hydrogen atom and orthogonal polynomials J.S. Dehesa Depto. de Física Moderna, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain. e-mail address: dehesa@ugr.es F. Domínguez Adame Depto. Física de Materiales, Universidad Complutense, E-28040 Madrid, Spain E.R. Arriola Depto. de Física Moderna, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain. A. Zarzo Depto. de Matemática Aplicada, E.T.S. de Ingenieros Industriales, Universidad Politécnica de Madrid, E-28006 Madrid, Spain. e-mail address: azarzo@etsii.upm.es Abstract. Firstly we argue that the quantum-mechanical study of natural systems (e.g. nuclei, atoms, molecules and solids) is a very rich source of orthogonal polynomials. Then, this is illustrated in the so-called Coulomb problem (i.e. the movement of a charged particle in a attractive or repulsive Coulomb potential) of which the hydrogen atom, the simplest realistic physical system, is a particular case. Connected with this problem there appear orthogonal polynomials associated to the names Laguerre, Gegenbauer, Hahn, Pollaczek and Dirac, in both non-relativistic and relativistic cases. Finally, the distribution of zeros of Laguerre and Pollaczek polynomials are chosen to be discussed in detail since they allow to determine the nodal structure of the wavefunctions of Coulomb physical states and the density of energy levels of the repulsive Coulomb problem, respectively. 1. Quantum Mechanics and Orthogonal Polynomials The quantum mechanical study of the internal structure of natural systems (e.g. nuclei, atoms, molecules, solids) is an almost inexhaustible source of different sets of orthogonal polynomials. To show that this statement is true, let us consider a natural system with a Hamiltonian operator H, which completely characterizes the system, given by H = K + V (1) where K and V are the total kinetic energy and potential energy operators, respectively. Then, the equation of motion of the system in the non-relativistic case is the Schrödinger equation (H E)Φ = 0 (2) Updated preprint version October 1992 of Ref.: C. Brezinski, L. Gori and A. Ronveaux eds., Orthogonal Polynomials and Their Applications, Proc. III International Symposium on Orthogonal Polynomials and Their Applications (Erice, Italy, June 1990), IMACS Ann. Comput. Appl. Math. vol. 9 (1991) pp.223 229. 1

2 J.S. Dehesa, F. Domínguez Adame, E.R. Arriola and A. Zarzo where E is the eigenvalue and Φ is the corresponding eigenvector, which represent the total energy and the wavefunction of the physical state of the system, respectively. Let us enumerate some of the ways in which different sets of orthogonal polynomials appear in solving the Schrödinger equation (2). 1.1. Tridiagonalization of the Hamiltonian operator Nowadays, one knows that the operator H is usually tridiagonalized by means of one of the three following methods: (i) Conventional method. One obtains a matrix representation of the Hamiltonian and then tridiagonalizes the resulting fully matrix by means of Householder- or Givens-like algorithms [39]. (ii) Lanczos method. Here, one uses a Lanczos-like algorithm [25, 32, 33, 38] which directly put the Hamiltonian in a tridiagonal form. (iii) Tight-Binding Approach. Often, one directly finds a tridiagonal form for the Hamiltonian of a physical many-body system by means of the so-called tight-binding approximation [24, 28]. It is based on the assumption that each constituent of the system only intaracts with its two nearest-neighbors. This approximation works very well for numerous physical systems, specially solids. Any of these three methods alows to write the Hamiltonian operator H in the form a 1 b 0 b 0 a 2 b 1 0 b 1 a 3 b 2 H J =.......... 0.............. which is usually called the Jacobi Hamiltonian of the system, which is a real and symmetric tridiagonal matrix [17]. Now one uses the following property of a Jacobi matrix: the characteristic polynomials of its principal submatrices, denoted by {P n (x), n = 1, 2,...}, satisfy the three term recurrence relation P n (x) = (x a n )P n 1 (x) b 2 n 1P n 2 (x) P 1 (x) = 0 ; P 0 (x) = 1 ; n = 1, 2,... (3) In doing so, one has converted the problem of finding the energies and wavefunctions of the physical state of the system as solutions of the associated Schrödinger equation into a mathematical

Hydrogen atom and orthogonal polynomials 3 question which entirely lies in the field of orthogonal polynomials. Indeed, the energies are given by the zeros {x m,n ; m = 1, 2,..., n} of a polynomial P n (x) and the corresponding wavefunctions are vectors whose components are the same polynomials evaluated at the corresponding zero [39]. 1.2. Discretization of the Schrödinger equation From now on, for the sake of clarity in the presentation, we will restrict ourselves to consider the non-relativistic movement of a single particle of mass m in the one dimensional space. Then, the equation of motion (2) reduces as where h is the universal Planck s constant. Since h2 d 2 Φ(x) 2m dx 2 + [V (x) E] Φ(x) = 0 (4) d 2 Φ(x) Φ(x + ɛ) 2Φ(x) + Φ(x ɛ) dx 2 = lim ɛ ɛ 2 we find a recurrence relation of the type (3) with coefficients ) 2 ( h a n = m + v n 1 ; b n = h2 2m where {v n ; n = 0, 1, 2,...} corresponds to the discretized potential of the particle. There are other ways of discretizing the Schrödinger equation (4) giving rise to a three term recurrence relation of type (3) and consequently producing new sets of orthogonal polynomials (see e.g. [37]. 1.3. Schrödinger s equation for solvable potentials Sometimes, for specific potentials V (x), it is possible to show that the physically admissible (i.e. single-valued, finite and continuous everywhere) solutions Φ(x) of the Schrödinger equation (4) are of the form [26, 27] Φ n (x) = P n (x) e Q m(x) (5) where P n (x) and Q m (x) are polynomials. The exponential part takes care of the singularities of the potential. Usually, {P n (x), n = 0, 1, 2,... } forms a set of orthogonal polynomials of an unknown nature in the general case. From (5) one notices that the nodes of the physical wavefunctions are given by the zeros of the orthogonal polynomials P n (x). 1.4. Momentum Space Equally well we may face the physical problem in momentum representation. Here we may also make considerations similar to those already done in position representation. Consequently,

4 J.S. Dehesa, F. Domínguez Adame, E.R. Arriola and A. Zarzo new sets of orthogonal polynomials characterizing the energies and momentum wavefunctions are encountered. Summarizing, two different sets of orthogonal polynomials may be associated to any natural system in both position and momentum representation. One is related to the wavefunctions and another one is connected with the energies of the system. In general these sets of orthogonal polynomials are of unknown nature, certainly not the classical orthogonal polynomials except for simple systems (e.g. harmonic oscilator, hydrogen atom). In spite of that, sometimes many of their properties can be found from the equation of motion itself. In particular this is so for the nodal structure of the wavefunction and the density of energy levels either rigurously or in an approximate way [2, 3, 19, 21]. 2. The Coulomb Problem and Orthogonal Polynomials Here we illustrate the general observations done in the previous section for a specific physical case. We have chosen the Coulomb problem, that is the study of the movement of a particle of mass m in the potential V (r) = C r where r = (x, y, z) and r = (x 2 + y 2 + z 2 ) 1/2. C is a parameter with positive or negative sign, in which case we talk about repulsive or attractive Coulomb problem, respectively. The hydrogenic atoms are typical attractive Coulomb problems with C = Ze 2, where e is the charge of the electron and +Ze is the charge of the nucleus. Let us describe the different sets of orthogonal polynomials which appears in the solution of the Coulomb problems. 2.1. Laguerre polynomials L α n(x) The equation of motion of this problem is the Schrödinger equation ( 1 2m 2 + E + C ) Φ(r) = 0 (6) r in the non-relativistic case, and the Klein-Gordon equation [ 2 + m 2 (E + C ] r )2 Ψ(r) = 0 (7) in the relativistic case, where the spin degree of freedom has been omitted. In spherical polar coordinates, the solutions of both equations are [34] of the form R(r) Y lm (θ, φ), where the angular functions Y lm (θ, φ) are the well known spherical harmonics and the radial part is given as solution of the differential equation d 2 R dρ 2 + 2 dr ρ dρ + [ λ ρ 1 4 ] a(a + 1) ρ 2 R = 0

Hydrogen atom and orthogonal polynomials 5 being in the non-relativistic case, and ρ = 2 2mE r ; λ = n + l + 1 ; a = l ρ = 2 m 2 E 2 r ; λ = n l + a a l = (l + 1 2 )2 C 2 1 2 in the relativistic case with spin zero. The positive integer numbers n and l are called principal and orbital quantum numbers. It can be shown [34] that the radial wavefunction is R(ρ) = ρ a e ρ/2 L 2a+1 λ a 1 (ρ) (8) Then, the nodal structure of the Coulomb wavefunction is fully determined by the distribution of zeros of Laguerre polynomials in both relativistic and vanishing-spin-relativistic cases. For a hydrogenic atom, which is essentially a particle of spin 1/2 in an attractive Coulomb potential, one should use in the relativistic formulation, not the Klein-Gordon equation (7) but the Dirac equation [34]. Then the corresponding radial wavefunctions do not involve any longer a single Laguerre polynomial but a combination of two Laguerre polynomials (see also [6], which receive the name of Dirac polynomials. 2.2. Gegenbauer polynomials G (α) n (r) These polynomials characterize the momentum wavefunctions of the Coulomb problem in a similar way as the Laguerre polynomials do so in the position representation as we have just described [8]. 2.3. Hahn polynomials The Schrödinger s equation of the Coulomb problem (6) has an exact solution in both spherical and parabolic coordinates [8]. For the discrete spectrum, the wavefunction in both coordinate systems are connected one to another by Clebsh-Gordan coefficients [35, 36]. These coefficients can be expressed in terms of the Hahn polynomials given by [4, 29] Q n (x; α; β, N) = 3 F 2 [ n, n + α + β + 1, x α + 1, N ; 1 which are orthogonal over a finite point set in the following sense N Q n (x; α; β, N)Q m (x; α; β, N) x=0 [ ] [ ] x + α N x + β = 0 ; m n N x N x ]

6 J.S. Dehesa, F. Domínguez Adame, E.R. Arriola and A. Zarzo In the case of the continuous spectrum, a similar relationship between the spherical and parabolic wavefunctions exist. But here the corresponding transformation coefficients involve the continuous Hahn polynomials [5, 35] P n P n (x; a, b, c, d) given by [4] P n (x) = (i) n (a + c) n(a + d) n n! 3F 2 [ n, n + a + b + c + d 1, a ix a + c, a + d ; 1 ] which are orthogonal with respect to an absolutely continuous complex measure. 2.4. Pollaczek polynomials: P λ n (x; a, b) The radial Coulomb Hamiltonian can be transformed directly into a tridiagonal matrix by using a L 2 discretisation basis [10, 11, 40]. The orthogonal polynomials associated to this tridiagonal matrix are the Pollaczek polynomials [14] P n (x) = P λ n (x; a, b). For a b and λ > 0, the polynomials are orthogonal in [ 1, 1] with respect to an absolutely continuous measure and satisfy the recurrence relation P n (x) = ( ) b x + P n 1 (x) n + λ + a 1 n(n + 2λ 1) 4(n + λ + a)(n + λ + a 1) P n 2(x) P 1 (x) = 0 ; P 0 (x) = 1 ; n = 1, 2,... It happens that Pollaczek polynomials involved in the repulsive Coulomb problem do not have discrete spectral points, while those associated to the attractive Coulomb problem present an infinite number of discrete spectral points [7]. Also, it turns out that the level energies E of the repulsive Coulomb problem are the zeros of the polynomial P λ n (x; 2C, 2C) with λ = l + 1 ; x = E λ2 8 E + λ2 8 Moreover, one can show that the energy spectrum of the Coulomb systems may be determined from the spectrum of zeros of Pollaczek polynomials in both non-relativistic and spin-zerorelativistic cases. Then, one realizes that both the nodal structure of the physical wavefunctions in position and momentum representations and the density of energy levels of Coulomb systems are fully characterized by mean of the spectral properties of the above four systems of orthogonal polynomials. However, not so much is known about them. In particular, for Hahn polynomials and Pollaczek polynomials only the works of Levit [31] and Bank and Ismail [7] are known to the knowledge of the authors. (9)

Hydrogen atom and orthogonal polynomials 7 3. Zeros of Laguerre Polynomials and Nodes of Coulomb Wavefunctions Here we study the distribution of zeros of Laguerre polynomials by means of its moments around the origin µ r µ r (n, α) = 1 n x r i,n (10) n where {x i,n ; i = 1, 2,..., n} are the zeros of the polynomial L α n(x). Then, we use the resulting expressions to analyze how the nodal distribution of the Coulomb wavefunctions gets modified by non-spin relativistic effects. To calculate the moments (10) we use the differential equation satisfied by Laguerre polynomials [1] and the following THEOREM [12, 13, 15]: where x [L α n(x)] + (α + 1 x) [L α n(x)] + n [L α n(x)] = 0 (11) Let P n (x) be a polynomial satisfying the second order differential equation g 2 (x, n)p n (x) + g 1 (x, n)p n (x)+ g 0 (x, n)p n (x) = 0 (12) g i (x, n) = c i j=0 a (i) j x j ; i = 0, 1, 2. (13) Then it is fulfilled that r+c 2 3 m= 1 a (2) m+3 r J m+2 = n 2 c 1 j=0 where the J symbol denotes the spectral sum rules a (1) j µ r+j 1 ; r = 1, 2,... (14) J k = xk l 1,n = x l 1 l l1,n x l2,n 2 0 if k = 0 1 2n(n 1) if k = 1 n(n 1)µ 1 if k = 2 n(n k 2 )µ k 1 + n2 k 2 2 t=1 µ k 1+tµ t if k > 2

8 J.S. Dehesa, F. Domínguez Adame, E.R. Arriola and A. Zarzo Notice that (14) is the basic relation which allows us to calculate recurrently the spectral moments µ r of any polynomial P n (x) in terms of the coefficients a (i) j which characterize the differential equation (12) with polynomial coefficients (13) satisfied by P n (x). The application of this theorem to equation (11) gives in a recurrent way the moments of the distribution of zeros of the Laguerre polynomials L α n(x) as In particular, for r = 3, 4 one has µ 1 = n + α µ 2 = (n + α)(2n + α 1) (15) µ r+1 = r 1 (2n + α r)µ r + n µ r t µ t t=1 r = 2, 3,... µ 3 = (n + α)(5n 2 + 5nα 6n + α 2 3α + 2) [ µ 4 = (n + α) (5n 2 + 5nα 6n + α 2 3α + 2) (2n + α 3) + 2n(n + α)(2n + α 1)] For large n values it is possible to solve the associated inverse moment problem. One finds that the distribution of zeros ρ n (x) is given by (i) If x x x + : ρ n (x) = where x ± are the roots of the radicand. (ii) ρ n 0 otherwise. 1 [ x 2 + 2(2n + α 1)x (α 1) 2] 1/2 2πα This result may also be found by random-matrix techniques [9]. In the asymptotic limit (i.e. n ) one easily obtains 1 2π ( x 4 1)1/2 if 0 < x < 4 ρ (x) = 0 x 4 where ρ (x) = lim n ρ n (x/n) Other asymptotic properties of zeros of Laguerre polynomials have been recently reviewed [22, 23]. The application of these results to the Schrödinger and Klein-Gordon equations of the Coulomb problem, allow to study in an explicit way how the nodal structure of Coulomb wavefunctions gets modified due to the relativistic effects associated to the Einstenian mass variation with the velocity of the particle. In particular, equation (15) gives that µ r (R) < µ r (N R) ; r = 1, 2,...

Hydrogen atom and orthogonal polynomials 9 which shows that the relativistic values µ r (R) of the nodal moments are smaller than the nonrelativistic ones µ r (N R). From here, one concludes that relativity pushes down the centroide and decreases the spread of the nodal distribution as well as it makes more concentrated around the centroid such distribution. [2, 20, 21]. These and other properties are described in detail elsewhere 4. Zeros of Pollaczek Polynomials and Energy of Coulomb Systems Here the distribution of zeros of Pollaczek polynomials is studied by means of its moments around the origin defined by equation (10), where now {x i,n ; i = 1, 2,..., n} denote the zeros of the polynomial P λ n (x; a, b). To calculate these moments we use the three-term recurrence relation (9) satisfied by these polynomials and the following theorem [16, 18]: Let P n (x) be a polynomial satifying the recurrence relation given by (3). Then, the spectral moments µ r defined in (10) are as µ m = 1 F m (r n 1, r 1, r 2, r 2,..., r j, r j+1) (m) n t a r 1 i b2r 1 i a r 2 i+1 b2r 2 i+1... b2r j+1 i+j ; m = 1, 2,..., n. The (m) symbol denotes a sum over all the partitions (r 1, r 1, r 2, r 2,..., r j, r j+1 ) of the number m restricted as follows (i) (ii) j+1 j r i + 2 r i = m If r s = 0, 1 < s < j, then r k = r k, k > s and j = m 2 m 1 2 if m is even if m is odd The factorial coefficients F m are given by F m (r 1, r 1, r 2, r 2,..., r j, r j+1) = m j+1 (r i 1 + r i + r i 1)! (r i 1 1)! r i! r i! The first three moments supplied by this theorem are µ 1 = 1 n a i n

10 J.S. Dehesa, F. Domínguez Adame, E.R. Arriola and A. Zarzo { n µ 2 = 1 n { n µ 3 = 1 n a 2 i + 2 a 3 i + 3 n 1 b 2 i n 1 } b 2 i (a i + a i+1 ) The application of this theorem to the Pollaczek polynomials P λ n (x; a, b) produces all the moments of its distribution of zeros. In particular, one has µ 1 = b n Ψ 0(n) where the functions Ψ k (n) given by are tabulated. Since µ 2 = 1 n + 2a 1 n [b2 Ψ 1 (n) + aψ 0 (n)] + 2(n + λ + a 1) µ 3 = b { 2b 2 Ψ 2 (n) + 3 n 2 aψ 1(n) + 3 } 2 Ψ 0(n) + 3 4 3 [ n(2a + 1) + (λ + a 1) 2 ] 4(n + λ + a 1) 2 Ψ k (n) = ( 1)k+1 k! lim n n 1 j=0 Ψ k (n) n } (j + k + a) (k+1) = 0 one can show that the asymptotical (i.e. n ) values for the moments are µ 2k+1 = 0 ; µ 2k = 2 2k ( 2k k ) ; k = 0, 1, 2,... These moments corespond to the inverse semicircular distribution 1 π (1 x2 ) 1/2 if x < 1 ρ (x) = 0 otherwise which says that the asymptotical distribution of zeros of the Pollaczek polynomials is a regular distribution in the scaled variable x/n. From these results a number of properties about the level energies of the physical states of the Coulomb problem may be obtained but this will be shown elsewhere. 5. Conclusions We have argued that quantum mechanics of natural systems is an inexhaustible source of sets of orthogonal polynomials. This has been illustrated in a simple case: the Coulomb problem, where

Hydrogen atom and orthogonal polynomials 11 at least five different sets of orthogonal polynomials ( Laguerre, Gegenbauer, Hahn, Pollaczek, Dirac ) are shown to play an important role. Other orthogonal polynomials are also connected to these problem, e.g. the Hermite polynomials appears when one express the wave functionof the atoms in terms of the harmonic oscillator wave functions [30]. Finally, the distribution of zeros of the Laguerre and Pollaczek polynomials have been obtained by means of its moments around the origin in a detailed way.the resulting quantities for the Laguerre polynomials were used to analyze the influence of some relativistic effects (those non related to the spin of the particle) to the nodal structure of the Coulomb wavefunctions. References 1. Abramovitz, E.R. and Stegun, I.A., Handbook of Mathematical Functions (Dover, New York, 1972). 2. Arriola, E.R., Master Thesis (Univ. of Granada, Spain, 1985). 3. Arriola, E.R. and Dehesa, J.S., Nuovo Cimento 103 B (1989) 611. 4. Askey, R., J. Phys. A: Math. Gen. 18 (1985) L1017. 5. Atakisiyev, N.M. and Suslov, S.K., J. Phys. A: Math. Gen. 18 (1985) 1583. 6. Auvil, P.R. and Brown, L.M., Am. J. Phys. 46 (1978) 679. 7. Bank, E. and Ismail, M.E.H., Const. Approx. 1 (1985) 103. 8. Bethe, H.A. and Salpeter, E.E., Quantum Mechanics of the One and Two-electron Atoms (Academic Press, New York, 1957). 9. Bronk, B.V., J. Math. Phys. 5 (1964) 215 and 1664. 10. Broad, J.T., Phys. Rev. A18 (1978) 1012. 11. Broad, J.T., Phys. Rev. A31 (1985) 1494. 12. Buendía, E., Dehesa, J.S. and Sánchez-Buendía, M.A., J.Math. Phys.26 (1985) 2729. 13. Buendía, E., Dehesa, J.S. and Gálvez, F.J., Lect. Notes in Math. 1329 (1988) 222. 14. Chihara, T.S., An Introduction to Orthogonal Polynomials (Gordon and Breach, New York, 1978). 15. Case, K.M., J. Math. Phys. 21 (1980) 702 and 709. 16. Dehesa, J.S., J. CAM. 2 (1976) 249. 17. Dehesa, J.S., J. CAM. 7 (1981) 249. 18. Dehesa, J.S. and Gálvez, F.J., Global properties of zeros of orthogonal polynomials in Marcellán (ed.), Proc. Third Spanish Symposium on Orthogonal Polynomials and its Appl. (Segovia, 1985). 19. Dehesa, J.S., Arriola, E.R. and Zarzo, A., J. CAM. (1990). 20. Dominguez Adame, F., Master Thesis (Univ. of Granada, Spain, 1985). 21. Domínguez Adame, F. and Dehesa, J.S., Preprint (Univ. of Granada, Spain, 1990). 22. Gawronski, W., J. Approx. Theory 50 (1987) 214. 23. Gawronski, W., Preprint (Univ. of Trier, RFA, 1990). 24. Haydock, R., Heine, V. and Kelly, M.J., J. Phys. C 5(1972) 2845.

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