Completeness of the Dirac oscillator eigenfunctions

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1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 34 (001) PII: S (01)011-X Completeness of the Dirac oscillator eigenfunctions Radosław Szmytkowski and Marek Gruchowski Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, Narutowicza 11/1, PL Gdańsk, Poland Received 5 December 000, in final form May 001 Abstract Completeness of the Dirac oscillator eigenfunctions is proved in one and three spatial dimensions. Proofs are based on standard properties of the Hermite and the generalized Laguerre polynomials. PACS numbers: Pm, 0.30.Gp 1. Introduction In recent years various mathematical properties of solutions to the Dirac oscillator eigenproblem have been extensively investigated both analytically and algebraically (e.g., [1 5]). The purpose of the present work is to prove completeness of the Dirac oscillator eigenfunctions. To the best of our knowledge, this problem, very important from the point of view of past [6 8] and planned applications, has not been studied yet.. The Dirac oscillator in one spatial dimension.1. Eigenproblem and its solutions The differential eigenproblem for a one-dimensional Dirac oscillator is ( cα i h d ) dx iβmωx (x) + βmc (x) = E (x) ( <x< ) (.1) (x) bounded for x ± (.) where ω > 0 (oscillator frequency) is a fixed parameter, E is an eigenvalue while α and β are matrices obeying α = β = I αβ + βα = 0 (.3) with I denoting the unit matrix. For the present purposes the most convenient representations of α and β are ( ) 0 i α = i 0 β = ( ) (.4) /01/ $ IOP Publishing Ltd Printed in the UK 4991

2 499 R Szmytkowski and M Gruchowski and expressing the wave function (x) in terms of its components ( ) f(x) (x) = (.5) g(x) we rewrite the eigensystem (.1) and (.) in the explicit form c h df(x) + mcωxf (x) = (mc + E)g(x) dx (.6) c h dg(x) mcωxg(x) = (mc E)f(x) dx (.7) f(x)and g(x) bounded for x ±. (.8) Solving the eigensystem (.6) (.8) one finds [6] that its solutions may be labeled by a quantum number n assuming all integer values. The eigenvalues are E n =±mc 1+ n hω mc (n = 0, ±1, ±,...) (.9) where the upper sign should be chosen for n 0 and the lower one for n<0, while components of the eigenfunctions, orthonormal in the sense of dx [f n (x)f m (x) + g n (x)g m (x)] = δ nm (.10) are λ(e n + mc f n (x) = ) n +1 n! H n (λx)e λ x / (.11) πe n λ(e n mc g n (x) =± ) n ( n 1)! H n 1 (λx)e λ x / (.1) πe n (observe that g 0 (x) 0) where H k (ξ) is the Hermite polynomial [9] and mω λ = h. (.13) It will be profitable to notice the following symmetry properties of the eigensolutions: E n = E n (n 0) (.14) E n mc f n (x) = E n + mc f E n + mc n(x) g n (x) = E n mc g n(x) (n 0). (.15).. Proof of completeness Completeness of the one-dimensional Dirac eigenfunctions will be established if we succeed in proving the closure relation ( ) fn (x) ( fn (x ) g n (x ) ) ( ) 1 0 = δ(x x ) ( <x,x < ) g n (x) 0 1 (.16) where δ(x x ) is the Dirac delta function. To this end, we observe that the matrix relation (.16) is equivalent to two diagonal scalar relations I(x,x ) f n (x)f n (x ) = δ(x x ) (.17)

3 Completeness of the Dirac oscillator eigenfunctions 4993 J(x,x ) and one off-diagonal scalar relation K(x,x ) g n (x)g n (x ) = δ(x x ) (.18) f n (x)g n (x ) = 0 (.19) (the second off-diagonal relation follows immediately from the relation (.19)). To prove the relation (.17), we decompose its left-hand side in the following way: I(x,x ) = f 0 (x)f 0 (x ) + [ fn (x)f n (x ) + f n (x)f n (x ) ]. (.0) On substituting here the explicit forms (.11) of f n and collecting terms containing the Hermite polynomials of the same degree, we arrive at I(x,x ) = λ n n! π e λ (x +x )/ H n (λx)h n (λx ). (.1) Application of the following known closure relation obeyed by the normalized Hermite functions 1 n n! π e (ξ +ξ )/ H n (ξ)h n (ξ ) = δ(ξ ξ ) ( <ξ,ξ < ) (.) and the following identity obeyed by the Dirac delta function δ(λx λx ) = λ 1 δ(x x ) ( <x,x < ) (.3) to the right-hand side of equation (.1) immediately leads to equation (.17). To prove the relation (.18), we make use of the fact that g 0 (x) 0 and decompose the left-hand side of (.18) as follows: J(x,x ) = [ gn+1 (x)g n+1 (x ) + g n 1 (x)g n 1 (x ) ]. (.4) Substituting here the explicit forms (.1) of g n and collecting terms containing the Hermite polynomials of the same degree, we find J(x,x ) = λ n n! π e λ (x +x )/ H n (λx)h n (λx ) (.5) hence (cf. equations (.) and (.3)) equation (.18) follows immediately. Finally, to prove the relation (.19), we again make use of the fact that g 0 (x) 0 and write K(x,x [ ) = fn (x)g n (x ) + f n (x)g n (x ) ]. (.6) Since equation (.15) implies that the product f n (x)g n (x ) is an odd function of n, the summand in equation (.6) vanishes and thus equation (.19) has been proved.

4 4994 R Szmytkowski and M Gruchowski 3. The Dirac oscillator in three spatial dimensions 3.1. Eigenproblem and its solutions Now we turn to the three-dimensional Dirac oscillator. The relevant eigenproblem is cα [ i h iβmωr] (r) + βmc (r) = E (r) (r R 3 ) (3.1) (r) bounded everywhere (3.) where again ω>0 (the frequency of the oscillator) is a fixed parameter and E is an eigenvalue. In equation (3.1) α and β are 4 4 Dirac matrices [10]. The eigensystem (3.1) and (3.) possesses solutions of the form κmj (r) = 1 ( ) Pκ (r) κmj (n r ) (3.3) r iq κ (r) κmj (n r ) where n r = r/r, ±κmj (n r ) are spherical spinors, κ =±1, ±,...and m j = κ + 1, κ + 3,..., κ 1. The radial functions P κ(r) and Q κ (r) are solutions of the eigensystem c h dp κ(r) + c h κ dr r P κ(r) + mcωrp κ (r) = (mc + E κ )Q κ (r) (0 <r< ) (3.4) c h dq κ(r) c h κ dr r Q κ(r) mcωrq κ (r) = (mc E κ )P κ (r) (0 <r< ) (3.5) P κ (0) = Q κ (0) = 0 P κ (r) and Q κ (r) bounded for r. (3.6) The structure of the spectrum of the radial eigensystem (3.4) (3.6) depends on the sign of κ. The eigenvalues are found [1, 3] to be E nκ =±mc 1+4 n hω mc (n = 0, ±1, ±,...) for κ<0 (3.7) with the upper sign chosen for n 0 and the lower one for n<0, and E nκ =±mc 1+4( n + l + 1 ) hω mc (n =±0, ±1, ±,...) for κ>0 (3.8) with the upper sign chosen for n = +0, +1, +,...and the lower one for n = 0, 1,,... (Notice that for κ>0it is necessary to distinguish between the cases n = +0 and n = 0.) Associated radial eigenfunctions, normalized according to are [3] 0 dr [P nκ (r)p mκ (r) + Q nκ (r)q mκ (r)] = δ nm (3.9) λ n!(e nκ + mc ) P nκ (r) = Ɣ( n + l + 3 )E (λr) l+1 e λ r / L (l+1/) n (λ r ) (3.10) nκ λ n Q nκ (r) =±sgn(κ)!(e nκ mc ) Ɣ( n + l + 3 )E (λr) l +1 e λ r / L (l +1/) n (λ r ) (3.11) nκ where L (α) (ρ) is a generalized Laguerre polynomial [9]. (The same sign convention as in k equations (3.7) and (3.8) applies in equation (3.11).) It is to be observed that for κ<0one has Q 0κ (r) 0. The integers n, l and l appearing in equations (3.7), (3.8), (3.10) and (3.11) are defined by n = { n 1 for κ<0 n for κ>0 (3.1)

5 Completeness of the Dirac oscillator eigenfunctions 4995 l = κ = { κ 1 for κ<0 κ for κ>0 { κ = l + 1 for κ<0 l = κ 1 1 = κ 1 = l 1 for κ>0. In the next subsection we shall make use of the following symmetry relations: (3.13) (3.14) E nκ = E nκ (except for n = 0 when κ<0) (3.15) E nκ mc P nκ (r) = E nκ + mc P E nκ + mc nκ(r) Q nκ (r) = E nκ mc Q nκ(r) (except for n = 0 when κ<0) (3.16) stemming from equations (3.7), (3.8), (3.10) and (3.11). 3.. Proof of completeness The supposed closure relation obeyed by radial eigenfunctions is ( ) Pnκ (r) ( Pnκ(r ) Q Q nκ(r ) ) ( ) 1 0 = δ(r r ) (0 <r,r < ). nκ (r) 0 1 (3.17) We shall prove it by deriving the following summation formulae: I κ (r, r ) P nκ (r)p nκ (r ) = δ(r r ) (3.18) J κ (r, r ) K κ (r, r ) Q nκ (r)q nκ (r ) = δ(r r ) (3.19) P nκ (r)q nκ (r ) = 0. (3.0) It is to be remembered that in equations (3.17) (3.0), as well as in the rest of this subsection, the summations over n, extending from to +, for κ<0include a term corresponding to n = 0 while for κ>0include terms corresponding to n = +0 and n = 0. We begin with the relation (3.18). If κ < 0, we decompose the left-hand side of that equation in the following way: I l 1 (r, r ) = P 0, l 1 (r)p 0, l 1 (r [ ) + Pn, l 1 (r)p n, l 1 (r ) + P n, l 1 (r)p n, l 1 (r ) ] while if κ>0, we write I l (r, r ) = (3.1) [ Pnl (r)p nl (r ) + P nl (r)p nl (r ) ]. (3.) n=+0 In either case, collecting terms containing the generalized Laguerre polynomials of the same degree, we arrive at I κ (r, r λn! ) = Ɣ(n + l + 3 (λr ) l+1 e λ (r +r )/ L (l+1/) )(λr)l+1 n (λ r )L (l+1/) n (λ r ). (3.3)

6 4996 R Szmytkowski and M Gruchowski The series in equation (3.3) may be summed on making use of the known closure relation n! Ɣ(n + α +1) (ρρ ) α/ e (ρ+ρ )/ L (α) n (ρ)l(α) n (ρ ) = δ(ρ ρ ) (0 <ρ,ρ < ) (3.4) obeyed by the generalized Laguerre functions. Subsequent application of the relationship δ(λ r λ r ) = δ(r r ) (0 <r,r < ) (3.5) λ (rr ) 1/ following from the standard properties of the Dirac delta function, completes the proof of equation (3.18). Next we consider equation (3.19). When κ<0we utilize the fact that Q 0, l 1 (r) 0 and rewrite the left-hand side of this equation in the form J l 1 (r, r [ ) = Qn+1, l 1 (r)q n+1, l 1 (r ) + Q n 1, l 1 (r)q n 1, l 1 (r ) ] (3.6) while for κ>0we decompose it in the following way: J l (r, r [ ) = Qnl (r)q nl (r ) + Q nl (r)q nl (r ) ]. (3.7) n=+0 On collecting terms at the generalized Laguerre polynomials of the same degree, in both cases we find J κ (r, r λn! ) = )(λr)l +1 Ɣ(n + l + 3 (λr ) l +1 e λ (r +r )/ L (l +1/) n (λ r )L (l +1/) n (λ r ). (3.8) Hence and from equations (3.4) and (3.5) we obtain equation (3.19). To prove equation (3.0), for κ<0we rewrite its left-hand side as K l 1 (r, r [ ) = Pn, l 1 (r)q n, l 1 (r ) + P n, l 1 (r)q n, l 1 (r ) ] (3.9) and for κ>0as K l (r, r ) = [ Pnl (r)q nl (r ) + P nl (r)q nl (r ) ]. (3.30) n=+0 Since equation (3.16) implies that the product P nκ (r)q nκ (r ) is an odd function of n, the summands in equations (3.9) and (3.30) vanish and we arrive at equation (3.0). The closure relation (3.17) and the known closure relation κ= (κ 0) κ 1/ m j = κ +1/ κmj (n r ) κm j (n r ) = I δ () (n r n r ) (3.31) obeyed by the spherical spinors, together with the following well known representation of the three-dimensional Dirac delta function δ (3) (r r ) = δ(r r )δ () (n r n r ) rr (3.3)

7 Completeness of the Dirac oscillator eigenfunctions 4997 immediately imply the closure relation for the four-component eigenfunctions (3.3) κ= (κ 0) κ 1/ m j = κ +1/ nκmj (r) nκm j (r ) = I 4 δ (3) (r r ) (3.33) where I 4 is the unit 4 4 matrix. We leave to the reader as an exercise to prove completeness of the Dirac oscillator eigenfunctions in two spatial dimensions [4]. Acknowledgments The work of RSz was supported in part by the Polish State Committee for Scientific Research under Grant no 8/P03/99/17. We thank Professor Cz Szmytkowski for commenting on the manuscript. References [1] Moshinsky M and Szczepaniak A 1989 J. Phys. A: Math. Gen. L817 [] Benítez J, Martínez y Romero R P, Núñez-Yépez H N and Salas-Brito A L 1990 Phys. Rev. Lett [3] de Lange O L 1991 J. Phys. A: Math. Gen [4] Villalba V M 1994 Phys. Rev. A [5] Martínez-y-Romero R P, Núñez-Yépez H N and Salas-Brito A L 1995 Eur. J. Phys [6] Toyama F M, Nogami Y and Coutinho FAB1997 J. Phys. A: Math. Gen [7] Rozmej P and Arvieu R 1999 J. Phys. A: Math. Gen [8] Turek M, Rozmej P and Arvieu R 000 Acta Phys. Pol. B [9] Magnus W, Oberhettinger F and Soni R P 1966 Formulas and Theorems for the Special Functions of Mathematical Physics (Berlin: Springer) [10] Schiff L I 1968 Quantum Mechanics 3rd edn (New York: McGraw-Hill)