The Moments of the Hydrogen Atom by the Method of Brackets

Expectation values of powers of the radial coordinate in arbitrary hydrogen states are given, in the quantum case, by an integral involving the associated Laguerre function. The method of brackets is used to evaluate the integral in closed-form and to produce an expression for this average value as...

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Datum:	2017
Hauptverfasser:	Gonzalez, I., Kohl, K.T., Kondrashuk, I., Moll, V.H., Salinas, D.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2017
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1485582019-02-19T01:24:22Z The Moments of the Hydrogen Atom by the Method of Brackets Gonzalez, I. Kohl, K.T. Kondrashuk, I. Moll, V.H. Salinas, D. Expectation values of powers of the radial coordinate in arbitrary hydrogen states are given, in the quantum case, by an integral involving the associated Laguerre function. The method of brackets is used to evaluate the integral in closed-form and to produce an expression for this average value as a finite sum. 2017 Article The Moments of the Hydrogen Atom by the Method of Brackets / I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll, D. Salinas // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C45; 33C20; 81V45 DOI:10.3842/SIGMA.2017.001 http://dspace.nbuv.gov.ua/handle/123456789/148558 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Expectation values of powers of the radial coordinate in arbitrary hydrogen states are given, in the quantum case, by an integral involving the associated Laguerre function. The method of brackets is used to evaluate the integral in closed-form and to produce an expression for this average value as a finite sum.
format	Article
author	Gonzalez, I. Kohl, K.T. Kondrashuk, I. Moll, V.H. Salinas, D.
spellingShingle	Gonzalez, I. Kohl, K.T. Kondrashuk, I. Moll, V.H. Salinas, D. The Moments of the Hydrogen Atom by the Method of Brackets Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Gonzalez, I. Kohl, K.T. Kondrashuk, I. Moll, V.H. Salinas, D.
author_sort	Gonzalez, I.
title	The Moments of the Hydrogen Atom by the Method of Brackets
title_short	The Moments of the Hydrogen Atom by the Method of Brackets
title_full	The Moments of the Hydrogen Atom by the Method of Brackets
title_fullStr	The Moments of the Hydrogen Atom by the Method of Brackets
title_full_unstemmed	The Moments of the Hydrogen Atom by the Method of Brackets
title_sort	moments of the hydrogen atom by the method of brackets
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/148558
citation_txt	The Moments of the Hydrogen Atom by the Method of Brackets / I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll, D. Salinas // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 29 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 001, 13 pages The Moments of the Hydrogen Atom by the Method of Brackets Ivan GONZALEZ † 1 , Karen T. KOHL † 2 , Igor KONDRASHUK †3, Victor H. MOLL † 4 and Daniel SALINAS † 5 †1 Instituto de F́ısica y Astronomia, Universidad de Valparaiso, Avda. Gran Bretaña 1111, Valparaiso, Chile E-mail: ivan.gonzalez@uv.cl †2 Department of Mathematics, University of Southern Mississippi, Long Beach, MS 39560, USA E-mail: karen.kohl@usm.edu †3 Grupo de Matemática Aplicada & Grupo de F́ısica de Altas Enerǵıas, Departmento de Ciencias Básicas, Universidad del Bı́o-Bı́o, Campus Fernando May, Av. Andres Bello 720, Casilla 447, Chillán, Chile E-mail: igor.kondrashuk@gmail.com †4 Department of Mathematics, Tulane University, New Orleans, LA 70118, USA E-mail: vhm@tulane.edu URL: http://129.81.170.14/~vhm/ †5 Departamento de Fisica, Universidad Técnica Federico Santa Maŕıa, Casilla 110-V, Valparaiso, Chile E-mail: salinas.a.daniel@gmail.com Received November 23, 2016, in final form December 30, 2016; Published online January 05, 2017 https://doi.org/10.3842/SIGMA.2017.001 Abstract. Expectation values of powers of the radial coordinate in arbitrary hydrogen states are given, in the quantum case, by an integral involving the associated Laguerre function. The method of brackets is used to evaluate the integral in closed-form and to produce an expression for this average value as a finite sum. Key words: non-relativistic hydrogen atom; method of brackets; hypergeometric function; associated Laguerre functions 2010 Mathematics Subject Classification: 33C45; 33C20; 81V45 1 Introduction The computation of the expectation 〈rk〉 of the electron for atoms with a single electron is a standard problem in quantum mechanics, see [19, 29]. For a given energy state n, the problem is expressed as〈 rk 〉 = ∫ ∞ 0 R2 n`(r)r k+2dr, where Rn`(r) is the radial solution of the Schrödinger equation for the hydrogen atom. Condi- tions on the parameters n, `, k are determined by the convergence of this integral. In the non-relativistic situation, the solution is given in terms of the Hahn polynomials [4]: h(α,β)m (x,N) = (1−N)m(β + 1)m m! 3F2 ( −m, α+ β +m+ 1, −x β + 1, 1−N ∣∣∣∣ 1) . (1.1) mailto:ivan.gonzalez@uv.cl mailto:karen.kohl@usm.edu mailto:igor.kondrashuk@gmail.com mailto:vhm@tulane.edu http://129.81.170.14/~vhm/ mailto:salinas.a.daniel@gmail.com https://doi.org/10.3842/SIGMA.2017.001 2 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas In particular, these expectations are given in terms of the Chebyshev polynomials of discrete variables [18, 24] tm(x,N) = h(0,0)m (x,N) in the form〈 rk 〉 = 1 2n (2µ)−ktk+1(n− `− 1,−2`− 1), when k = −1, 0, 1, 2, . . . , (1.2) and 〈 rk 〉 = 1 2n (2µ)−kt−k−2(n− `− 1,−2`− 1), when k = −2,−3, . . . ,−2`− 2. (1.3) The parameters are µ = Z/na0 with a0 = ~2/me2 the Bohr radius and Z is the nuclear charge. The constants m and e are the mass and charge of the electron, respectively. The identity tk(n− `− 1,−2`− 1) = Γ(2`+ k + 2) Γ(2`+ 2) 3F2 ( −k, k + 1, −n+ `+ 1 1, 2`+ 2 ∣∣∣∣ 1) follows from (1.1). Then (1.2) becomes 〈 rk 〉 n` = 1 2n(2µ)k Γ(2`+ k + 3) Γ(2`+ 2) 3F2 ( −1− k, k + 2, −n+ `+ 1 1, 2`+ 2 ∣∣∣∣ 1) for k = −1, 0, 1, 2, . . . , and (1.3) 〈 rk 〉 n` = 1 2n(2µ)k Γ(2`− k) Γ(2`+ 2) 3F2 ( −2 + k, −k + 3, −n+ `+ 1 1, 2`+ 2 ∣∣∣∣ 1) for k = −2,−3, . . . ,−2` − 2, where the dependence upon the parameters n and ` have been made explicit. In the quantum case, the radial component of the wave function for a hydrogen atom with nuclear charge Z is characterized by two quantum numbers: n the principal quantum number and ` the orbital number. The corresponding normalized radial function is Rn`(r) = An`(2µr) ` exp(−µr)L2`+1 n−`−1(2µr), where the normalization constant is An` = √ (2µ)3 2n (n− `− 1)! (n+ `)! and Lαm(x) = Γ(α+m+ 1) Γ(m+ 1)Γ(1 + α) 1F1 ( −m 1 + α ∣∣∣∣x) is the associated Laguerre function; see [14, formula 8.972.1]. The expectation value of a power of the radial coordinate is given by〈 rk 〉 n` = (2µ)2`A2 n` ∫ ∞ 0 r2+2`+ke−2µr [ L2`+1 n−`−1(2µr) ]2 dr, (1.4) with n ∈ N, 0 ≤ ` ≤ n− 1, and k ∈ Z. The Moments of the Hydrogen Atom by the Method of Brackets 3 The goal of the work is to compute the integral in (1.4) by the method of brackets, to illustrate its flexibility. The reader will find in [3, 6, 9, 10, 11, 12, 16] a collection of examples of definite integrals evaluated by this method. The basic procedure is described in Section 3. The examples presented here are to be considered as the beginning of a series of calculations of integrals related to the hydrogen atom. These include the evaluation of the integral [13] Jαβnms = ∫ ∞ 0 e−xxα+sLαn(x)Lβm(x)dx given by S.K. Suslov and B. Trey [25]. The method of brackets provides an alternative method of proof that only uses the hypergeometric representation of the Laguerre function. The method can also be used to discuss the relativistic situation. Details will appear elsewhere. The reductions of the formulas discussed here uses basic properties of the gamma function, such as Γ(a+ n) = Γ(a)(a)n and (a)−n = (−1)n (1− a)n for a ∈ R, n ∈ N. (1.5) Here (a)n = a(a+ 1) · · · (a+ n− 1) is the Pochhammer symbol. 2 A direct evaluation This section presents a direct evaluation of the integral 〈 rk 〉 n` = (2µ)2`A2 n` ∫ ∞ 0 r2+2`+ke−2µr [ L2`+1 n−`−1(2µr) ]2 dr (2.1) given in (1.4). The proof is based on some identities for the associated Laguerre function appearing in the integrand. The methods presented here are then compared with the evaluation by the method of brackets explained in the next section. The first identity used to modify the integrand appears in [14, formula 8.976.3] [ Lαm(x) ]2 = Γ(α+m+ 1) 22mΓ(m+ 1) m∑ s=0 ( 2m− 2s m− s ) Γ(2s+ 1) Γ(α+ s+ 1)Γ(s+ 1) L2α 2s (2x). (2.2) Therefore〈 rk 〉 n` = (2µ)2`A2 n` Γ(`+ n+ 1) 22(n−`−1)Γ(n− `) × n−`−1∑ s=0 ( 2(n− `− 1− s) n− `− 1− s ) Γ(2s+ 1) Γ(2`+ 2 + s)Γ(s+ 1) G`,k,s(µ), (2.3) where G`,k,s(µ) = ∫ ∞ 0 r2+2`+ke−2µrL 2(2`+1) 2s (4µr)dr. (2.4) To obtain an expression for G`,k,s(µ), the representation Lan(x) = Γ(a+ n+ 1) Γ(n+ 1)Γ(1 + a) 1F1 ( −n 1 + a ∣∣∣∣x) (2.5) for the Laguerre function (see [14, formula 8.972.1]) is used. 4 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas Theorem 2.1. The integral G`,k,s(µ) is given by G`,k,s(µ) = Γ(4`+ 2s+ 3)Γ(2`+ k + 3) Γ(2s+ 1)Γ(4`+ 3)(2µ)2`+k+3 2F1 ( −2s, 2`+ k + 3 4`+ 3 ∣∣∣∣ 2) . Proof. The hypergeometric representation (2.5) shows that L 2(2`+1) 2s (4µr) = Γ(4`+ 2s+ 3) Γ(2s+ 1)Γ(4`+ 3) 1F1 ( −2s 4`+ 3 ∣∣∣∣ 4µr) . Expanding the hypergeometric function gives G`,k,s(µ) = Γ(4`+ 2s+ 3) Γ(2s+ 1)Γ(4`+ 3) ∫ ∞ 0 2s∑ j=0 (−2s)j (4`+ 3)j (4µr)j j! r2`+2+ke−2µrdr = Γ(4`+ 2s+ 3) Γ(2s+ 1)Γ(4`+ 3) 2s∑ j=0 (−2s)j (4`+ 3)jj! (4µ)j ∫ ∞ 0 r2`+2+k+je−2µrdr = Γ(4`+ 2s+ 3) Γ(2s+ 1)Γ(4`+ 3) 2s∑ j=0 (−2s)j(4µ)j (4`+ 3)jj! Γ(2`+ k + j + 3) (2µ)2`+k+j+3 = Γ(4`+ 2s+ 3) Γ(2s+ 1)Γ(4`+ 3)(2µ)2`+k+3 2s∑ j=0 (−2s)j2 j (4`+ 3)jj! Γ(2`+ k + j + 3) = Γ(4`+ 2s+ 3)Γ(2`+ k + 3) Γ(2s+ 1)Γ(4`+ 3)(2µ)2`+k+3 2s∑ j=0 (−2s)j(2`+ k + 3)j (4`+ 3)jj! 2j = Γ(4`+ 2s+ 3)Γ(2`+ k + 3) Γ(2s+ 1)Γ(4`+ 3)(2µ)2`+k+3 ∞∑ j=0 (−2s)j(2`+ k + 3)j (4`+ 3)jj! 2j = Γ(4`+ 2s+ 3)Γ(2`+ k + 3) Γ(2s+ 1)Γ(4`+ 3)(2µ)2`+k+3 2F1 ( −2s, 2`+ k + 3 4`+ 3 ∣∣∣∣ 2) . This is the stated form for G`,k,s(µ). � Note 2.2. Observe that s ∈ N, so the hypergeometric function in the expression for G`,k,s(µ) is actually a polynomial in its last variable. Thus, there are no convergence issues. The expression for G`,k,s(µ) and (2.3) are used to produce the next result (after the change s 7→ n− `− 1− s). Corollary 2.3. For n = 1, 2, . . . , ` = 0, 1, . . . , n − 1 and k ∈ Z with 2` + k + 3 > 0. The moments of the hydrogen atom are given by 〈 rk 〉 n` = Γ(2`+ k + 3)(2n+ 2`)! n22n−2`−1(4`+ 2)!(2µ)k(n+ `)!(n− `− 1)! × n−`−1∑ s=0 ( n+` s )( n−`−1 s )( 2n+2` 2s ) 2F1 ( −2(n− `− 1− s), 2`+ k + 3 4`+ 3 ∣∣∣∣ 2) . Note 2.4. The restriction 2`+k+3 > 0 avoids the singularities of the gamma factor Γ(2`+k+3). Also observe that the first entry in the series 2F1 in the answer is a negative integer, therefore the series reduces to a finite sum. The Moments of the Hydrogen Atom by the Method of Brackets 5 In this article the expectation values of the powers of the radial coordinate of the hydrogen atom in a framework of quantum mechanics, that is, in the non-relativistic case are computed. In the Introduction it was stated that this already has appeared in the literature. In the relativistic case, results for these expectation values of the powers of the radial coordinate appeared in 2009. Indeed, the relativistic Coulomb integrals are contained in [22, 23]. The treatment of the results obtained in [22, 23] by computer algebra methods is described in [17, 20]. In the nonrelativistic case of quantum mechanics, the corresponding questions were success- fully solved by direct calculation. For example, in [21] useful relations between different Laguerre polynomials were found. In [5] the radial expectation values are given for D-dimensional hydro- genic states with D > 1. The same quantities are discussed in a more general setting in [26, 27]. The radial expectation values of hydrogenic states in momentum space appear in [28], repre- sented in terms of Gegenbauer polynomials instead of Laguerre polynomials. All these results were obtained by direct calculations too. The method of brackets may significantly simplify the calculations for these tasks. This will be discussed in a future publication. The method of brackets is not the unique successful method which involves integral transfor- mations. Traditional methods based on Mellin–Barnes transformation may be efficient tools in order to obtain new results in quantum field theory [1, 2, 7, 8, 15]. 3 The method of brackets The evaluation of the integral giving the mean value 〈rk〉 (2.1) presented in the previous section, used the relation (2.2) in a fundamental way. A method to evaluate integrals over the half line [0,∞), based on a small number of rules has been developed in [11, 12]. This method of brackets is described next. The heuristic rules are currently being placed on solid ground [3]. The reader will find in [6, 9, 10] a large collection of evaluations of definite integrals that illustrate the power and flexibility of this method. For a ∈ C, the symbol 〈a〉 = ∫ ∞ 0 xa−1dx is the bracket associated to the (divergent) integral on the right. The symbol φn = (−1)n Γ(n+ 1) is called the indicator associated to the index n. The notation φi1i2···ir , or simply φ12···r, denotes the product φi1φi2 · · ·φir . Rules for the production of bracket series. Rule P1. Assign to the integral ∫∞ 0 f(x) dx a bracket series:∑ n φna(n)〈αn+ β〉. Here the coefficients a(n) come from an assumed expansion f(x) = ∑ n≥0 φna(n)xαn+β−1. The extra ‘−1’ in the exponent is set for convenience. The coefficients are written as a(n) because these will soon be evaluated at complex numbers n, not necessarily positive integers. Now we need to state how to convert the bracket series into a number. Rule P2. For α ∈ C, the multinomial power (a1+a2+· · ·+ar)α is assigned the r-dimensional bracket series∑ n1 ∑ n2 · · · ∑ nr φn1n2···nra n1 1 · · · a nr r 〈−α+ n1 + · · ·+ nr〉 Γ(−α) . 6 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas Rules for the evaluation of a bracket series. Rule E1. The one-dimensional bracket series is assigned the value∑ n φnf(n)〈an+ b〉 = 1 \|a\| f(n∗)Γ(−n∗), where n∗ is obtained from the vanishing of the bracket; that is, n∗ solves an + b = 0. This is precisely the Ramanujan’s master theorem. The next rule provides a value for multi-dimensional bracket series of index 0, that is, the number of sums is equal to the number of brackets. Rule E2. Assume the matrix A = (aij) is non-singular, then the assignment is∑ n1 · · · ∑ nr φn1···nrf(n1, . . . , nr)〈a11n1 + · · ·+ a1rnr + c1〉 · · · 〈ar1n1 + · · ·+ arrnr + cr〉 = 1 \|det(A)\| f(n∗1, . . . , n ∗ r)Γ(−n∗1) · · ·Γ(−n∗r), where {n∗i } is the (unique) solution of the linear system obtained from the vanishing of the brackets. There is no assignment if A is singular. Rule E3. Each representation of an integral by a bracket series has associated an index of the representation via index = number of sums− number of brackets. It is important to observe that the index is attached to a specific representation of the integral and not just to integral itself. The experience obtained by the authors using this method suggests that, among all representations of an integral as a bracket series, the one with minimal index should be chosen. The value of a multi-dimensional bracket series of positive index is obtained by computing all the contributions of maximal rank by Rule E2. These contributions to the integral appear as series in the free parameters. Series converging in a common region are added and divergent series are discarded. Any series producing a non-real contribution is also discarded. There is no assignment to a bracket series of negative index. 4 The evaluation of the expectations. A first bracket calculation This section describes the evaluation of the integral In,`,k(µ) := ∫ ∞ 0 r2+2`+ke−2µr [ L2`+1 n−`−1(2µr) ]2 dr, (4.1) that appeared in (1.4) by the method of brackets. The expectation value of a power of the radial coordinate is then given by〈 rk 〉 n` = (2µ)2`A2 n`In,`,k(µ). This integral can be scaled to In,`,k(µ) = 1 (2µ)3+2`+k ∫ ∞ 0 t2+2`+ke−t [ L2`+1 n−`−1(t) ]2 dt. (4.2) This does not appear in the table [14]. The closest entry is 7.414.10:∫ ∞ 0 e−bxx2a [ Lan(x) ]2 dx = 22aΓ ( a+ 1 2 ) Γ ( n+ 1 2 ) π(n!)2b2a+1 Γ(a+ n+ 1) 2F1 ( −n, a+ 1 2 1 2 − n ∣∣∣∣(1− 2 b )2) . The Moments of the Hydrogen Atom by the Method of Brackets 7 Note 4.1. In the evaluation of (4.1), it is convenient to write it as In,`,k:A,B,C(µ) := ∫ ∞ 0 r2+2`+ke−ArL2`+1 n−`−1(Br)L 2`+1 n−`−1(Cr)dr and then consider the limiting value as A, B, C tend to 2µ. The computation of (4.2) described in this section is obtained without any further identities for the Laguerre function. Next section describes the computation of the function G`,k,s(µ), defined in (2.4). The first step is to compute a series representation for the factors in the integrand. Lemma 4.2. The functions in the integrand of (4.1) have series given by e−ax = ∑ n1 φn1a n1xn1 and Lαm(x) = Γ(α+ 1 +m) ∑ n2 φn2 xn2 Γ(1 +m− n2)Γ(1 + α+ n2) . Proof. The series of the exponential function is elementary. Indeed, e−ax = ∑ n1≥0 (−a)n1 n1! xn1 = ∑ n1≥0 (−1)n1 n1! (ax)n1 = ∑ n1 φn1(ax)n1 . To evaluate the series of the Laguerre function, treat m as a real non-integer parameter, and observe that Lαm(x) = Γ(α+ 1 +m) Γ(α+ 1)Γ(m+ 1) ∞∑ n2=0 (−m)n2 (α+ 1)n2 xn2 n2! = Γ(α+ 1 +m) Γ(m+ 1) ∞∑ n2=0 Γ(n2 −m) Γ(−m)Γ(α+ 1 + n2) xn2 n2! . The series for the Laguerre function now follows from the identity Γ(n2 −m) Γ(−m) = (−1)n2 Γ(1 +m) Γ(1 +m− n2) valid for n2 ∈ N and m 6∈ N. � The series given in Lemma 4.2 are now used directly to evaluate the integral (4.1). This gives In,`,k;A,B,C(µ) = ∫ ∞ 0 r2+2`+k [∑ n1 An1φn1r n1 ] × [∑ n2 Γ(`+ n+ 1) Γ(n− `− n2)Γ(2`+ 2 + n2) φn2B n2rn2 ] × [∑ n3 Γ(`+ n+ 1) Γ(n− `− n3)Γ(2`+ 2 + n3) φn3C n3rn3 ] dr = ∑ n1,n2,n3 ∫ ∞ 0 r2+2`+k+n1+n2+n3drAn1Bn2Cn3φn1,n2,n3 8 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas × Γ2(`+ n+ 1) Γ(n− `− n2)Γ(2`+ 2 + n2)Γ(n− `− n3)Γ(2`+ 2 + n3) = ∑ n1,n2,n3 〈n1 + n2 + n3 + 3 + 2`+ k〉An1Bn2Cn3φn1,n2,n3 × Γ2(`+ n+ 1) Γ(n− `− n2)Γ(2`+ 2 + n2)Γ(n− `− n3)Γ(2`+ 2 + n3) . This intermediate result is stated next. Proposition 4.3. A bracket series for the integral In,`,k;A,B,C(µ) is given by In,`,k;A,B,C(µ) = ∑ n1,n2,n3 〈n1 + n2 + n3 + 3 + 2`+ k〉An1Bn2Cn3φn1,n2,n3 × Γ2(`+ n+ 1) Γ(n− `− n2)Γ(2`+ 2 + n2)Γ(n− `− n3)Γ(2`+ 2 + n3) . The bracket series above contains one bracket and three indices, thus it is expected that the method will produce a double series as an expression for In,`,k;A,B,C(µ). The vanishing of the bracket gives n1 + n2 + n3 = −3− 2`− k, (4.3) showing the two free indices. Solving for n3. Replacing n3 = −n1 − n2 − t, with t = 2`+ k+ 3, in the bracket series yields the expression In,`,k;A,B,C(µ) = Γ2(n+ `+ 1) Ct × ∞∑ n1,n2=0 Γ(n1 + n2 + t) ( −A C )n1 ( −B C )n2 Γ(n− `− n2)Γ(2`+ 2 + n2)Γ(n1 + n2 + s)Γ(−1− k − n1 − n2)n1!n2! with s = n+ `+ 3 + k. Using (1.5) yields In,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) CtΓ(n− `)Γ(2`+ 2)Γ(s)Γ(−1− k) × ∞∑ n1,n2=0 (t)n1+n2(1− n+ `)n2(k + 2)n1+n2 (2`+ 2)n2(s)n1+n2n1!n2! (−1)n2 ( A C )n1 ( B C )n2 . Then use (b)n1+n2 = (b)n2(b+ n2)n1 to produce In,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) CtΓ(n− `)Γ(2`+ 2)Γ(s)Γ(−1− k) × ∞∑ n1,n2=0 (t)n2(t+ n2)n1(1− n+ `)n2(k + 2)n2(k + 2 + n2)n1 (2`+ 2)n2(s)n2(s+ n2)n1n1!n2! (−1)n2 ( A C )n1 ( B C )n2 . The sum corresponding to the index n1, which appears only in 3 places, is chosen as the internal sum. This yields In,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) CtΓ(n− `)Γ(2`+ 2)Γ(s)Γ(−1− k) The Moments of the Hydrogen Atom by the Method of Brackets 9 × ∞∑ n2=0 (t)n2(k + 2)n2(1− n+ `)n2 (2`+ 2)n2(s)n2n2! ( −B C )n2 ∞∑ n1=0 (t+ n2)n1(k + 2 + n2)n1 (s+ n2)n1n1! ( A C )n1 . The inner sum is now identified as a hypergeometric function to produce In,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) CtΓ(n− `)Γ(2`+ 2)Γ(s)Γ(−1− k) × ∞∑ n2=0 (t)n2 , (k + 2)n2(1− n+ `)n2 (2`+ 2)n2(s)n2n2! ( −B C )n2 2F1 ( t+ n2, 2 + k + n2 s+ n2 ∣∣∣∣AC ) . Note 4.4. The same procedure can be used to treat the cases obtained by solving for n1 or n2 in the equation (4.3). The corresponding integrals are I (1) n,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) AtΓ2(n− `)Γ2(2`+ 2) × ∞∑ n2=0 (t)n2(1− n+ `)n2 (2`+ 2)n2n2! ( B A )n2 2F1 ( t+ n2, 1− n+ ` 2`+ 2 ∣∣∣∣CA ) and I (2) n,`,k;A,B,C(µ) = Γ2(n+ `+ 1)Γ(t) AtΓ2(n− `)Γ2(2`+ 2) × ∞∑ n3=0 (t)n3,(1− n+ `)n3 (2`+ 2)n3n3! ( C A )n3 2F1 ( t+ n3, 1− n+ ` 2`+ 2 ∣∣∣∣BA ) . At this point, the parameters A, B, C are replaced by the value 2µ, in order to continue the evaluation. This gives In,`,k(µ) = Γ2(n+ `+ 1)Γ(t) (2µ)tΓ(n− `)Γ(2`+ 2)Γ(s)Γ(−1− k) × ∞∑ n2=0 (t)n2(k + 2)n2(1− n+ `)n2 (2`+ 2)n2(s)n2n2! (−1)n2 2F1 ( t+ n2, 2 + k + n2 s+ n2 ∣∣∣∣ 1) . Observe that 1 − n + ` is a negative integer, so this is actually a finite sum. Using Gauss’ evaluation 2F1 ( a, b c ∣∣∣∣1) = Γ(c)Γ(c− a− b) Γ(c− a)Γ(c− b) for c− a− b > 0, and expressing the resulting gamma factors in terms of Pochhammer symbols to obtain In,`,k(µ) = Γ(n+ `+ 1)Γ(2`+ k + 3)Γ(n− `− k − 2) (2µ)2`+k+3Γ2(n− `)Γ(2`+ 2)Γ(−1− k) × ∞∑ n2=0 (k + 2)n2(1− n+ `)n2(2`+ k + 3)n2 (2`+ 2)n2(`+ k + 3− n)n2n2! . The final step identifies this series as a hypergeometric series to produce: In,`,k(µ) = Γ(n+ `+ 1)Γ(2`+ k + 3)Γ(n− `− k − 2) (2µ)2`+k+3Γ2(n− `)Γ(2`+ 2)Γ(−1− k) × 3F2 ( k + 2, 1 + `− n, 2`+ k + 3 2`+ 2, l + k + 3− n ∣∣∣∣ 1) . The results of this section are summarized in the next statement. 10 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas Theorem 4.5. For n, `, k as above,〈 rk 〉 n` = Γ(2`+ k + 3)Γ(n− `− k − 2) 2n(2µ)kΓ(n− `)Γ(2`+ 2)Γ(−1− k) 3F2 ( k + 2, 1 + `− n, 2`+ k + 3 2`+ 2, l + k + 3− n ∣∣∣∣ 1) . 5 The evaluation of the expectations. A second approach The moment 〈rk〉n` has been expressed in (2.3) as a finite sum values of the integral G`,k,s(µ) = ∫ ∞ 0 r2+2`+ke−2µrL 2(2`+1) 2s (4µr)dr, (5.1) where the index s is an integer varying from 0 to n − ` − 1. Corollary 2.3 provides an expres- sion for 〈rk〉n` as a finite sum of values of the hypergeometric function 2F1 evaluated at the argument 2. The hypergeometric terms appearing in the mentioned representation are actually finite sums, so the convergence of the series is not an issue. An alternative form is derived in this section that extends the range of validity of G`,k,s(µ) to a larger range for the parameter s. The goal is to produce a representation of the series for the Laguerre polynomials, given initially by Lαn(x) = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1) 1F1 ( −n α+ 1 ∣∣∣∣x) . This series is now written in a form suitable for the application of the method of brackets: Lαn(x) = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1) ∞∑ k1=0 (−n)k1 (α+ 1)k1 xk1 k1! = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1) ∞∑ k1=0 (−1)k1(−n)k1(−α)−k1 xk1 k1! = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1) ∑ k1 φ1(−n)k1(−α)k1x k1 = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1)Γ(−n)Γ(−α) ∑ k1 φ1Γ(−n+ k1)Γ(−α− k1)xk1 . To produce a bracket series representation of the last expression, observe that Γ(β) = ∑ ` φ`〈β + `〉 and this leads to Lαn(x) = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1)Γ(−n)Γ(−α) ∑ k1,k2k3 φ123〈−n+ k1 + k2〉〈−α− k1 + k3〉xk1 . The vanishing of the brackets provides two representations for the Laguerre function, denoted by Tj . Case 1. Take k1 as a free index. Then k∗2 = n− k1 and k∗3 = k1 + α yields the expression T1 = Γ(α+ n+ 1) Γ(n+ 1)Γ(α+ 1) ∞∑ k1=0 (−n)k1 (α+ 1)k1 xk1 k1! . This is the original series for Lαn(x). The Moments of the Hydrogen Atom by the Method of Brackets 11 Case 2. Take k2 as a free index. Then k∗1 = n− k2 and k∗3 = α+ n− k2 yields T2 = Γ(α+ n+ 1)xn Γ(n+ 1)Γ(α+ 1)Γ(−n)Γ(−α) ∞∑ k2=0 Γ(−n+ k2)Γ(−α− n+ k2) (−x)−k2 k2! . (5.2) Case 3. Taking k3 as a free index does not produce a representation for Lαn(x). The next step is to use the T2 representation to evaluate the integral G`,k,s(µ). By equa- tion (5.2), the expression for Lαn(x) is now written as Lαn(x) = Γ(α+ n+ 1)xn Γ(n+ 1)Γ(α+ 1)Γ(−n)Γ(−α) ∞∑ j=0 φjΓ(−n+ j)Γ(−α− n+ j)x−j . Using this representation in (5.1) produces G`,k,s(µ) = Γ(4`+ 3 + 2s)(4µ)2s Γ(2s+ 1)Γ(4`+ 3)Γ(−2s)Γ(−4`− 2) × ∞∑ j=0 φjΓ(−2s+ j)Γ(−4`− 2− 2s+ j)(4µ)−j ∫ ∞ 0 r2+2`+k+2s−je−2µrdr. Evaluating the last integral in terms of the gamma function and simplifying produces a proof of the next result. Theorem 5.1. The integral G`,k,s(µ) = ∫ ∞ 0 r2+2`+ke−2µrL 2(2`+1) 2s (4µr)dr is given by G`,k,s(µ) = 4s (2µ)3+2`+k Γ(3 + 2`+ k + 2s) Γ(2s+ 1) 2F1 ( −2s, −2s− 4`− 2 −2− 2`− k − 2s ∣∣∣∣ 1 2 ) . 6 A couple of examples The method of brackets has been used here to produce analytic expressions for the mean radius〈 rk 〉 n` = (2µ)2`A2 n` ∫ ∞ 0 r2+2`+ke−2µr [ L2`+1 n−`−1(2µr) ]2 dr, stated first in (1.4). The physically relevant parameters are n = 0, 1, 2, . . . , 0 ≤ ` ≤ n− 1, k ∈ R. The expressions include〈 rk 〉 n` = Γ(2`+ k + 3)(2n+ 2`)! n22n−2`−1(4`+ 2)!(2µ)k(n+ `)!(n− `− 1)! × n−`−1∑ s=0 ( n+` s )( n−`−1 s )( 2n+2` 2s ) 2F1 ( −2(n− `− 1− s), 2`+ k + 3 4`+ 3 ∣∣∣∣ 2) , (6.1) where 〈rk〉n` is given as a finite sum of hypergeometric terms and〈 rk 〉 n` = Γ(2`+ k + 3)Γ(n− `− k − 2) 2n(2µ)kΓ(n− `)Γ(2`+ 2)Γ(−1− k) 3F2 ( k + 2, 1 + `− n, 2`+ k + 3 2`+ 2, l + k + 3− n ∣∣∣∣ 1) given in Theorem 4.5. This section compares these expressions with the results found in the literature. 12 I. Gonzalez, K.T. Kohl, I. Kondrashuk, V.H. Moll and D. Salinas Example 6.1. Take ` = n− 1. Then the sum (6.1) reduces to 1 since the index s must vanish. Then 〈 rk 〉 n,n−1 = Γ(k + 2n+ 1) (2µ)k(2n)! . In particular, for k ∈ N, this becomes〈 rk 〉 n,n−1 = (2n+ k)! (2µ)k(2n)! . Example 6.2. The case ` = n− 2 reduces the sum (6.1) to two terms. The result is 〈rk〉n,n−2 = (k2 + 3k + 2n)Γ(k + 2n− 1) 2(2µ)k(2n− 2)! . Acknowledgments The work of I.K. was supported in part by Fondecyt (Chile) Grants Nos. 1040368, 1050512 and 1121030, by DIUBB (Chile) Grant Nos. 102609, GI 153209/C and GI 152606/VC. V.H.M. acknowledges the partial support of NSF-DMS 1112656. References [1] Allendes P., Guerrero N., Kondrashuk I., Notte-Cuello E.A., New four-dimensional integrals by Mellin– Barnes transform, J. Math. Phys. 51 (2010), 052304, 18 pages, arXiv:0910.4805. [2] Allendes P., Kniehl B.A., Kondrashuk I., Notte-Cuello E.A., Rojas-Medar M., Solution to Bethe–Salpeter equation via Mellin–Barnes transform, Nuclear Phys. 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A 143 (1934), 679–681. https://doi.org/10.1016/j.nuclphysb.2013.08.002 http://arxiv.org/abs/1304.3004 https://doi.org/10.1007/978-3-642-54479-8_6 https://doi.org/10.1007/978-3-642-54479-8_6 http://arxiv.org/abs/1306.1362 https://doi.org/10.1007/978-3-642-74748-9 https://doi.org/10.1007/978-3-642-74748-9 https://doi.org/10.1073/pnas.23.2.91 https://doi.org/10.1007/978-3-7091-1616-6 http://arxiv.org/abs/1206.2071 https://doi.org/10.1016/S0377-0427(97)00243-4 https://doi.org/10.1016/S0377-0427(97)00243-4 https://doi.org/10.1088/0953-4075/42/18/185003 http://arxiv.org/abs/0906.3338 https://doi.org/10.1103/PhysRevA.81.032110 http://arxiv.org/abs/0911.0111 https://doi.org/10.1088/0953-4075/43/7/074006 http://arxiv.org/abs/0908.3021 https://doi.org/10.1063/1.2830804 http://arxiv.org/abs/0707.1887 https://doi.org/10.1209/0295-5075/113/48003 https://doi.org/10.1209/0295-5075/113/48003 http://arxiv.org/abs/1603.09494 https://doi.org/10.1063/1.4961322 http://arxiv.org/abs/1609.01113 https://doi.org/10.1063/1.1286984 https://doi.org/10.1098/rspa.1934.0027 1 Introduction 2 A direct evaluation 3 The method of brackets 4 The evaluation of the expectations. A first bracket calculation 5 The evaluation of the expectations. A second approach 6 A couple of examples References

The Moments of the Hydrogen Atom by the Method of Brackets

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