Algebraic evaluation of matrix elements in the Laguerre function basis

McCoy, A. E.; Caprio, M. A.

doi:10.1063/1.4941327

The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfunctions (from the Schrödinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1, 1) × SO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.

I. INTRODUCTION

The Laguerre functions^1–3 constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory,^1,4–6 with application in hadronic^7–12 and nuclear structure^13–15 theory as well. The Laguerre functions share the same functional form as the Coulomb bound-state wave functions (obtained as solutions to the Schrödinger equation central force eigenproblem) or the Coulomb-Sturmian functions (obtained as solutions to a related Sturm-Liouville problem). However, unlike these functions, the Laguerre functions constitute a complete, discrete, orthonormal set of square-integrable functions in three dimensions.

Numerical solution of quantum mechanical problems—for the present discussion, we can take this to mean one-body and many-body Schrödinger equation problems, although the applicability is more general—may be represented in terms of an expansion of the wave function on a set of basis functions. The Schrödinger equation eigenproblem, when expanded in a discrete basis, is recast as a Hamiltonian matrix eigenproblem.¹⁶ To proceed with this approach, it is necessary to be able to evaluate the matrix elements of the operators of the problem, taken with respect to the expansion basis. Relevant operators may include those contributing to the Hamiltonian, transition operators, and operators representing moments or other static observables.

For numerical solution of many-body problems, the many-body basis functions are obtained as products of single-particle basis functions (more precisely, either symmetrized or antisymmetrized products, according to the statistics of the problem¹⁷). When the many-body problem is rotationally invariant, it is natural to consider an expansion in terms of single-particle basis functions which are themselves obtained as solutions to a rotationally invariant central force problem, for instance, the three-dimensional harmonic oscillator functions.¹⁸ The solution sets to central force problems are well known to factorize into radial and angular functions, arising from the factorizability of the Hilbert space of square-integrable functions on ℝ³ as ℒ²(ℝ³) = ℒ²(ℝ⁺) × ℒ²(S²). The angular functions, on the sphere S², are the well-known spherical harmonics.¹⁹ For each choice of angular momentum l, the associated radial functions form a complete set of square-integrable functions on ℝ⁺. Matrix elements of spherical tensor operators (including rotationally invariant scalar operators and vector operators) may be simplified through a corresponding factorization into radial and angular dependences, together with the Wigner-Eckart theorem¹⁹ for the angular dependence.

Most basic among the central force problems are those for the harmonic oscillator potential and the Coulomb (hydrogenic) potential. Both these problems have a rich group theoretical structure (reviewed in Ref. 20), which allows analytic expressions to be obtained for matrix elements of many operators of interest, built from the coordinate vector r and momentum vector p = − iħ∇.

Underlying the convenience of these bases is the presence of an SU(1, 1) × SO(3) dynamical group,²⁰ which reflects the factorization of the problem into radial and angular coordinates—with SU(1, 1) operating on the radial functions and SO(3) on the angular functions. The SU(1, 1) spectrum generating algebra relates radial wave functions with different radial quantum number (node number) n, at the same angular momentum l. This SU(1, 1) algebra has been widely studied for both the oscillator^21–23 and Coulomb^{21,22,24–27} problems. The SO(3) algebra, in turn, is the standard angular momentum algebra, which relates angular wave functions (spherical harmonics) with different angular momentum projection quantum number m, at given l.¹⁹

While the SU(1, 1) × SO(3) algebra thus provides the framework for relating states of the same angular momentum (and, in fact, for calculating matrix elements among these states, at least for certain suitable operators), an additional structure is required to connect basis states of different angular momentum. This connection is provided either by the factorization method (first applied by Schrödinger for the Coulomb problem²⁸ and subsequently generalized by Infeld and Hull²⁹) or, equivalently, through the ideas of supersymmetric quantum mechanics (SUSY QM).³⁰

More concretely, the relations among basis states provided by the algebraic structures are indicated in Fig. 1. To each angular momentum l is associated a set of radial wave functions. These constitute a positive-discrete, infinite-dimensional irreducible representation (irrep) of SU(1, 1). Radial wave functions differing in radial quantum number n by ±1, at fixed l, are related by the SU(1, 1) ladder operators T_±. These change the eigenvalue of the weight operator T₀, i.e., the weight m_t = n + l + 3/2, by ±1. Each SU(1, 1) irrep is labeled by its lowest weight t = l + 3/2. Thus, to each irrep (with label l) of the angular algebra SO(3) there is uniquely associated a dual irrep (with label t = l + 3/2) of the SU(1, 1) radial algebra.³¹ Then, shift operators, A^† and A,²⁶ obtained by the factorization or SUSY QM approaches alluded to above, relate radial wave functions corresponding to different angular momenta. Specifically, the shift operators connect radial wave functions of the same m_t but differing in t by ±1, i.e., radial wave functions associated with angular functions of angular momentum l differing by ±1. The algebraic structure (ladder operators) and shift operators in combination are what enable the evaluation of matrix elements of operators in the full three-dimensional space.²⁰

FIG. 1.

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Algebraic classification of basis functions, with respect to the SU(1, 1) × SO(3) dynamical algebra, and the pattern of laddering operators T_± and shift operators, A^† and A, connecting these functions. The quantum number l labels the SO(3) irrep, and the dual quantum number t labels the SU(1, 1) irrep by its lowest weight, where m_t is the SU(1, 1) weight (see text).

Let us now re-examine the choice of basis functions. Since we shall restrict attention to functions with radial-angular factorization, these would also commonly be termed orbitals. In actual calculations, the basis must be truncated to finite size (either the set of single-particle basis functions may be truncated directly, or some more general limitation may be placed on the set of many-body antisymmetrized product states constructed from this basis³²). Therefore, we must be concerned not only with mathematical completeness of the basis but also with how well matched the basis is to the physical problem at hand. That is, the basis functions should be chosen according to the following criteria (see Ref. 6): (i) The functions should allow for a systematic approach to completeness as the single-particle basis set is expanded. (ii) The functions should provide for rapid convergence of the calculated energies or eigenfunctions to the true results, requiring only a small number of basis functions for an accurate description. (iii) It is also practically desirable that the functions have a simple analytic form which permits convenient evaluation of relevant operator matrix elements. (Ideally, the functions should also be orthogonal and easily normalized, to simplify the formulation of the many-body problem, but this is not essential.)

The harmonic oscillator eigenfunctions are of the form (for simplicity, we omit normalization factors and set the radial length parameter to unity)^3,18

Ω_{nlm} (r) \propto r^{l + 1} L_{n}^{l + 1 / 2} (r^{2}) e^{- r^{2} / 2} Y_{lm} (\hat{r}),

(1)

where $L_{n}^{α} (x)$ is the associated Laguerre polynomial of degree n and order α, i.e., for parameter α in the associated Laguerre equation,³³ and $Y_{lm} (\hat{r})$ is a spherical harmonic. These functions constitute a complete, orthonormal set on ℒ²(ℝ³). However, while the Schrödinger equation solutions for a finite-range potential (or for an infinite-range potential of Coulomb 1/r type) exhibit an exponential asymptotic decay at large r (∝e^−βr), the oscillator functions have Gaussian asymptotics (∝e^−αr²), as an artifact of the infinite (quadratic) binding potential of the harmonic oscillator problem. The oscillator functions are thus poorly matched to the physical solutions in the asymptotic region and may therefore violate criterion (ii) above for a suitable basis, i.e., rapid convergence. In nuclear physics, for instance, while structure involving only deeply bound orbitals might be adequately represented using basis functions with Gaussian asymptotics, weakly bound orbitals with distinctly exponential asymptotics influence many aspects of nuclear structure, especially in light or neutron-rich nuclei (see, e.g., Fig. 4 of Ref. 34 and Fig. 4 of Ref. 35).

The Coulomb wave functions, as solutions to a problem with finite binding, display the appropriate exponential asymptotics. The bound-state (or negative-energy) Coulomb functions^3,6,33 are of the form³⁶

W_{nlm} (r) \propto {(\frac{2 r}{n + l + 1})}^{l} L_{n}^{2 l + 1} (\frac{2 r}{n + l + 1}) e^{- r / (n + l + 1)} Y_{lm} (\hat{r}) .

(2)

The full set of solutions to the Schrödinger equation with the Coulomb potential, including both these bound-state solutions and the continuum (or positive-energy) solutions, constitutes a complete set on ℒ²(ℝ³). However, the bound-state solutions by themselves do not. This is not merely a formal deficiency: attempting to use them as an expansion basis is well-known to result in convergence to erroneous energies and wave functions.¹ It is therefore necessary to supplement the bound-state solutions with the continuum functions. The inconvenience of including continuum functions limits the practical applicability of the Coulomb basis: both in terms of arranging a systematic approach to completeness with increasing basis size [criterion (i)] and due to the complicated nature of the continuum wave functions [criterion (iii)].^3,6

One may instead seek a discrete complete set, retaining the same exponential asymptotics as the Coulomb wave functions, by modifying the Schrödinger equation into a Sturm-Liouville problem associated with the Coulomb potential (as detailed in Sec. II), thereby obtaining the Coulomb-Sturmian functions.^3,37–40 These have been applied as expansion bases in atomic and molecular theory,^{37,38,41–43} nuclear scattering,^44,45 and other few-body problems.⁴⁶ The Coulomb-Sturmian functions are of the form

Ψ_{nlm} (r) \propto {(2 r)}^{l} L_{n}^{2 l + 1} (2 r) e^{- r} Y_{lm} (\hat{r}),

(3)

that is, of the same functional form as Coulomb eigenproblem solutions (2) but independently rescaled in the radial coordinate so that r/(n + l + 1) → r. These radial functions constitute a discrete set, complete on the Sobolev space $W_{2}^{(1)} (R^{3})$ [i.e., the space of all functions in ℒ²(ℝ³), the generalized second derivatives of which are also in ℒ²(ℝ³)],³ which implies completeness on ℒ²(ℝ³).⁴⁷ However, these functions are orthogonal with respect to the Coulomb potential 1/r as weighting function, rather than the usual Euclidean metric.

We therefore finally turn to the Laguerre functions^48,49 which are of the form

Λ_{nlm} (r) \propto {(2 r)}^{l} L_{n}^{2 l + 2} (2 r) e^{- r} Y_{lm} (\hat{r}) .

(4)

They are identical in form to Coulomb-Sturmian functions (3), differing only in the parameter α of the associated Laguerre polynomial: odd α = 2l + 1 for the Coulomb-Sturmian functions vs. even α = 2l + 2 for the Laguerre functions.⁵⁰ The Laguerre functions satisfy all the posited criteria: constituting a discrete set, complete on ℒ²(ℝ³), and orthogonal with respect to the usual Euclidean metric, but with the physically appropriate exponential asymptotics.

We demonstrate that an algebraic framework, analogous to that obtained for the usual central force eigenfunctions, can likewise be constructed for the Laguerre functions. These results permit analytic expressions to be obtained for matrix elements. The Laguerre functions are first introduced (Sec. II). After a brief review of the SO(3) and SU(1, 1) Lie algebras and the structure of their irreps (Sec. III A), we develop the SU(1, 1) algebra of the Laguerre radial functions (Sec. III B), and we use this algebra to obtain analytic expressions for the action of several basic operators (involving powers of the radial coordinate r and radial derivative operator d/dr) on the basis of Laguerre radial functions—or, equivalently, analytic expressions for matrix elements of these operators (Sec. III C). The results thus obtained apply to functions within a single SU(1, 1) irrep, i.e., to the set of radial functions associated with the same angular momentum. We then extend the ideas of factorization (or SUSY QM) to a quantum number dependent formulation (Sec. IV A), to construct shift operators relating Laguerre radial functions in different irreps (Sec. IV B), and thus to obtain analytic expressions for the matrix elements of radial operators between different irreps (Sec. IV C). Finally, returning to the full Laguerre functions in three dimensions, we demonstrate how the results for radial matrix elements and shift operators are combined to yield reduced matrix elements of spherical tensor operators, taking the Laplacian and gradient operators for illustration (Sec. V).

II. LAGUERRE FUNCTIONS

Coulomb functions (2) are the solutions to the Schrödinger equation

(- \nabla^{2} - Z \frac{2}{r}) W (r) = 2 E_{n l} W (r), E_{n l} = - \frac{Z^{2}}{2 {(n + l + 1)}^{2}} .

(5)

As outlined above, these do not form a complete set on ℒ²(ℝ³) without including the continuum,^1,41 making them an impractical choice of basis.

There is, however, a set of functions, closely related to the Coulomb functions, called Coulomb-Sturmian functions (3), which combine the discrete character of the harmonic oscillator functions with the desired large r behavior. These functions are related to the Coulomb functions by making the substitution

\frac{n + l + 1}{Z} \to b,

(6)

which recasts differential equation (5) into the form of a Sturm-Liouville equation

(- \nabla^{2} - \frac{2}{b} \frac{β_{n l}}{r}) Ψ (r) = - \frac{1}{b^{2}} Ψ (r), β_{n l} = n + l + 1,

(7)

and yields the Coulomb-Sturmian solutions

Ψ_{nlm} (r) = {(\frac{{(2 / b)}^{3} n!}{2 (n + l + 1) (n + 2 l + 1)!})}^{\frac{1}{2}} {(\frac{2 r}{b})}^{l} L_{n}^{2 l + 1} (2 r / b) e^{- r / b} Y_{lm} (\hat{r}) .

(8)

Observe that, while in Schrödinger equation eigenproblem (5) one varies the energy E_nl to satisfy the boundary conditions, in Sturm-Liouville eigenproblem (7) one now holds the energy constant and varies the coefficient β_nl, which governs the strength or depth of the potential, to satisfy the boundary conditions.³⁸ From the basic theory of the Sturm-Liouville equation,⁵¹ it follows that the solutions Ψ_nlm are orthogonal with respect to the Coulomb potential 1/r as weighting function.³⁸ These functions Ψ_nlm are also complete in the Sobolev space $W_{2}^{(1)} (R^{3})$ ⁠.³

However, for calculations in physics it is useful to work with functions that are orthonormal with respect to the usual integration metric d³r and complete on ℒ²(ℝ³). To obtain functions with this property, the integration weight and norm are absorbed into the function by multiplying Ψ_nlm by [b(n + l + 1)/r]^1/2, and the shift l → l + 1/2 is introduced.³ The resulting functions are the Laguerre functions,

Λ_{nlm} (r) = {(\frac{2}{b})}^{\frac{3}{2}} {(\frac{n!}{(n + 2 l + 2)!})}^{\frac{1}{2}} {(\frac{2 r}{b})}^{l} L_{n}^{2 l + 2} (2 r / b) e^{- r / b} Y_{lm} (\hat{r}),

(9)

which satisfy the radial differential equation

[- \frac{\partial^{2}}{\partial r^{2}} - \frac{3}{r} \frac{\partial}{\partial r} + \frac{l (l + 2)}{r^{2}} - \frac{2}{b} \frac{α_{n l}}{r}] Λ_{nlm} (r) = - \frac{1}{b^{2}} Λ_{nlm} (r), α_{n l} = n + l + \frac{3}{2} .

(10)

The Laguerre functions Λ_nlm(r) form a complete, discrete basis for L²(ℝ³),⁴⁷ have the correct asymptotic behavior and are orthogonal with respect to the ℝ³ metric. That is,

\int d^{3} r Λ_{n^{'} l^{'} m^{'}} (r) Λ_{nlm} (r) = δ_{n, n^{'}} δ_{l, l^{'}} δ_{m, m^{'}} .

(11)

It is now the matrix elements calculated in this Laguerre basis that are of interest. Certain matrix elements can be calculated via direct integration.^8,10 However, in this paper we use the factorability of the Hilbert space to develop a simpler and more elegant algebraic method for calculating matrix elements.

The Laguerre function Λ_nlm(r) can be factorized into radial and angular functions as

Λ_{nlm} (r) = r^{- 1} S_{n l} (r) Y_{lm} (\hat{r}) .

(12)

The integration weight has been absorbed into the radial function S_nl(r), given by

S_{n l} (r) = {(\frac{2}{b})}^{\frac{1}{2}} {(\frac{n!}{(n + 2 l + 2)!})}^{\frac{1}{2}} {(\frac{2 r}{b})}^{l + 1} L_{n}^{2 l + 2} (2 r / b) e^{- r / b},

(13)

which satisfies the orthogonality condition

\int_{0}^{\infty} d r S_{n^{'} l} (r) S_{n l} (r) = δ_{n, n^{'}} .

(14)

The set of functions {S_nl|n = 0, 1, 2, …} forms a complete and orthogonal set on L²(ℝ⁺), for any l = 0, 1, 2, …, independent of b,³⁹ so we make the simplifying substitution b → 1 in the following discussion. The resulting functions,

S_{n l} (r) = {(\frac{2 n!}{(n + 2 l + 2)!})}^{\frac{1}{2}} {(2 r)}^{l + 1} L_{n}^{2 l + 2} (2 r) e^{- r},

(15)

satisfy the differential equation

[- \frac{d^{2}}{d r^{2}} - \frac{1}{r} \frac{d}{d r} + \frac{{(l + 1)}^{2}}{r^{2}} - \frac{2 α_{n l}}{r}] S_{n l} (r) = - 1 S_{n l} (r),

(16)

with α_nl as defined in (10). Although (16) is not explicitly written in the form of a Sturm-Liouville equation (see Ref. 51), it nonetheless may readily be put in this form by left multiplying by r. For convenience of notation, we will sometimes denote Λ_nlm as |nlm〉, S_nl as |nl〉, and Y_lm as |lm〉.

A. Operator map to the radial basis

Although we want the matrix elements in the Λ_nlm basis, the actual calculations will be carried out more simply using the functions

| n l 〉 | l m 〉,

(17)

since they are orthogonal with respect to the measure dr dΩ rather than the measure r² dr dΩ. We start with the matrix element of a component of some irreducible, rank a tensor operator

O_{a α} = R Θ_{a α},

(18)

where R is a scalar radial operator and Θ_a is a rank a tensor with no radial dependence. Although we are only considering operators that can be factored into radial and angular parts, many operators of interest can be rewritten as sums of such separable operators using various factorization lemmas, and so this method can also be used to calculate matrix elements of these operators. As an example, matrix elements of the gradient operator ∇₁ are calculated in Sec. V C.

The matrix element of the operator $O_{a α}$ separates into radial and angular factors as

〈 n^{'} l^{'} m^{'} | O_{a α} | n l m 〉 = \int d^{3} r Λ_{nlm} (r) O_{a α} Λ_{nlm} (r) = [\int d r r^{2} r^{- 1} S_{n^{'} l^{'}} (r) R (r) r^{- 1} S_{n l} (r)] [\int d Ω Y_{l^{'} m^{'}} (\hat{r}) Θ_{a α} Y_{lm} (\hat{r})] .

(19)

That is,

〈 n^{'} l^{'} m^{'} | O_{a α} | n l m 〉 = 〈 n^{'} l^{'} | γ (R) | n l 〉 〈 l^{'} m^{'} | Θ_{a α} | l m 〉,

(20)

where

γ (R) = r R r^{- 1} .

(21)

If the operator is purely radial (Θ_aα → 1₀₀), then we have that

〈 n^{'} l^{'} m^{'} | R | n l m 〉 = 〈 n^{'} l | γ (R) | n l 〉 .

(22)

The similarity transformation given by (21) will allow us to calculate matrix elements in the Λ_nlm basis by factoring the matrix element into linear combinations of products of a matrix element in the S_nl basis and a matrix element in the well known angular momentum basis, as shown (for the case of positive weights) in Sec. V.

III. ALGEBRAIC STRUCTURE OF LAGUERRE BASIS

Factorization (12) of the Λ_nlm basis for ℒ(ℝ³) has the underlying symmetry SU(1,1) × SO(3), where SU(1,1) is the symmetry of the radial functions and SO(3) is the symmetry of the Y_lm angular functions. It is well known that the SO(3) symmetry can be used to calculate matrix elements of operators on the Y_lm basis. In this section we discuss how SU(1,1) symmetry can be used to calculate the matrix elements on the S_nl basis.

A. A brief review of the SO(3) and SU(1,1) Lie algebras

The Lie algebra SU(1,1) has a very similar structure to that of the Lie algebra SO(3), so we begin by summarizing the properties of the more familiar SO(3) algebra for comparison with the SU(1,1) algebra.

The group SO(3) is the compact Lie group of rotations in three dimensions. Its algebra is generated by the operators L₃ and L_±, which satisfy the commutation relations

[L_{3}, L_{\pm}] = \pm L_{\pm}

(23)

and

[L_{+}, L_{-}] = 2 L_{3},

(24)

where $L_{+}^{†} = L_{-}$ ⁠. The eigenfunctions of L₃ are the spherical harmonics |lm〉, where

L_{3} | l m 〉 = m | l m 〉 .

(25)

The laddering operators L_± have the well known action on the spherical harmonics

L_{\pm} | l m 〉 = \sqrt{(l \mp m) (l \pm m + 1)} | l, m \pm 1 〉 .

(26)

Because SO(3) is a compact group, the weight m is bounded above and below. The bound is determined by the SO(3) Casimir operator

C_{S O (3)} = L_{3}^{2} + \frac{1}{2} (L_{+} L_{-} + L_{-} L_{+}),

(27)

which acts on the eigenfunctions as

C_{S O (3)} | l m 〉 = l (l + 1) | l m 〉 .

(28)

The bounds above and below on m are given by ±l, that is,

m = l, l - 1, \dots, - l,

(29)

as shown in Fig. 2(a).

FIG. 2.

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Weights contained within (a) SO(3) and (b) SU(1, 1) irreps of the types considered in the present work (see text). Weights (m or m_t, respectively, for these two algebras) within an irrep (labeled by extremal weight l or t, respectively) are connected by solid lines. The bounding weights, as functions of l or t, are indicated by dashed lines.

Similarly, the generators of SU(1,1) are T₃ and T_±, which satisfy the commutation relations

[T_{3}, T_{\pm}] = \pm T_{\pm}

(30)

and

[T_{+}, T_{-}] = - 2 T_{3},

(31)

where $T_{+}^{†} = T_{-}$ ⁠. The actions of the generators on the eigenfunctions |tm_t〉 of T₃ are given by

T_{3} | t m_{t} 〉 = m_{t} | t m_{t} 〉

(32)

and

T_{\pm} | t m_{t} 〉 = \sqrt{(t \pm m_{t}) (t \mp m_{t} - 1)} | t, m_{t} \pm 1 〉 .

(33)

The commutator in (31) differs from the commutator in (24) by a negative sign. The group SU(1,1) is a non-compact group, and the weight m_t is either bounded above or bounded below but takes on infinitely many values. As with SO(3), the bound on m_t is determined by the Casimir operator

C_{S U (1, 1)} = T_{3}^{2} - \frac{1}{2} (T_{+} T_{-} + T_{-} T_{+}),

(34)

which acts on the eigenfunction as

C_{S U (1, 1)} | t m_{t} 〉 = t (t - 1) | t m_{t} 〉 .

(35)

If t is an integer or half-integer, then the irrep is discrete. If t > 0 then the irrep is bounded below by t, and if t < 0 then the irrep is bounded above by t. In other words,

m_{t} = {\begin{cases} t, t + 1, \dots & t > 0 \\ - t, - t - 1, \dots & t < 0 \end{cases},

(36)

as shown in Fig. 2(b) for t > 0.

B. Representation of SU(1,1) on the Laguerre radial basis

In this section, we construct a representation of the Lie algebra of the radial group, SU(1,1), using the S_nl(r) radial functions as the basis of the vector space. Recall from (16) that the S_nl basis is generated by varying α_nl. We rearrange (16) to rewrite it as an eigenvalue problem for α_nl,

(- r \frac{d^{2}}{d r^{2}} - \frac{d}{d r} + \frac{{(l + 1)}^{2}}{r} + r) S_{n l} (r) = 2 α_{n l} S_{n l} (r) .

(37)

Following a similar approach to the method used in Refs. 21 and 26 to construct representations of SU(1,1) for the Coulomb functions, we define

T_{3} = - \frac{1}{2} (r \frac{d^{2}}{d r^{2}} + \frac{d}{d r} - \frac{{(l + 1)}^{2}}{r} - r)

(38)

and

T_{\pm} = - \frac{1}{2} (r \frac{d^{2}}{d r^{2}} + \frac{d}{d r} - \frac{{(l + 1)}^{2}}{r} + r \mp 2 r \frac{d}{d r} \mp 1),

(39)

which satisfy the commutation relations in (30) and (31) and the relation $T_{+}^{†} = T_{-}$ ⁠. The Casimir invariant is given by

C_{SU(1,1)} = T_{3}^{2} - \frac{1}{2} (T_{+} T_{-} + T_{-} T_{+}) = (l + \frac{3}{2}) (l + \frac{1}{2}) .

(40)

Thus, the representation is discrete, with t = l + 3/2 and weights m_t = n + l + 3/2 (n = 0, 1, ….). Note that m_t = α_nl, with α_nl as defined in (10). From (32) and (33), the actions of the generators are expressed in terms of the labels n and l as

T_{3} S_{n l} = (n + l + \frac{3}{2}) S_{n l},

(41)

T_{+} S_{n l} = \sqrt{(n + 1) (n + 2 l + 3)} S_{n + 1, l},

(42)

T_{-} S_{n l} = \sqrt{n (n + 2 l + 2)} S_{n - 1, l} .

(43)

C. Deriving the action of radial operators using the SU(1,1) algebra

We can use the SU(1,1) algebra to easily determine the actions of r, 1/r, and d/dr on the functions S_nl(r) following a method similar to that used by Rowe in Ref. 23 to calculate the matrix elements for the generalized harmonic oscillator functions. We begin with the elements of the algebra

r = \frac{1}{2} (2 T_{3} - T_{+} - T_{-}),

(44)

and

r \frac{d}{d r} = \frac{1}{2} (T_{+} - T_{-} - 1) .

(45)

Applying the operators in (44) and (45) to S_nl, we immediately have that

r S_{n l} (r) = (n + l + \frac{3}{2}) S_{n l} (r) - \frac{1}{2} \sqrt{n (n + 2 l + 2)} S_{n - 1, l} (r) - \frac{1}{2} \sqrt{(n + 1) (n + 2 l + 3)} S_{n + 1, 1} (r)

(46)

and

r \frac{d}{d r} S_{n l} (r) = \frac{1}{2} \sqrt{(n + 1) (n + 2 l + 3)} S_{n + 1, l} (r) - \frac{1}{2} \sqrt{n (n + 2 l + 2)} S_{n - 1, l} (r) - \frac{1}{2} S_{n l} (r) .

(47)

We can obtain the action of the operator 1/r, which is not an element of the SU(1,1) algebra, using a recurrence relationship that we deduce using the orthogonality of the S_nl(r) functions. Starting with (46), we divide through by r, multiply through by S_ml(r), and integrate to obtain

δ_{n, m} = (n + l + \frac{3}{2}) f_{n m} - \frac{1}{2} \sqrt{n (n + 2 l + 2)} f_{n - 1, m} - \frac{1}{2} \sqrt{(n + 1) (n + 2 l + 3)} f_{n + 1, m},

(48)

where

f_{n m} = \int_{0}^{\infty} d r S_{m l} (r) \frac{1}{r} S_{n l} (r) .

(49)

Starting with f₀₀ = 1/(l + 1), obtained via direct integration, recurrence relationship (48) can be solved to find that

f_{n m} = \frac{1}{l + 1} \sqrt{\frac{n! (m + 2 l + 2)!}{m! (n + 2 l + 2)!}}, n \geq m .

(50)

A similar result, obtained by simply interchanging m and n, holds for n < m. From these results, we have that

\frac{1}{r} S_{n l} (r) = \frac{1}{l + 1} \sum_{i \geq n + 1} \sqrt{\frac{i! (n + 2 l + 2)!}{n! (i + 2 l + 2)!}} S_{i l} (r) + \frac{1}{l + 1} \sum_{i \leq n} \sqrt{\frac{n! (i + 2 l + 2)!}{i! (n + 2 l + 2)!}} S_{i l} (r) .

(51)

Finally, the action of the radial derivative can be determined by noting that d/dr = r⁻¹(rd/dr). Combining (47) and (51),

\frac{d}{d r} S_{n l} (r) = \sum_{i \geq n + 1} \sqrt{\frac{i! (n + 2 l + 2)!}{n! (i + 2 l + 2)!}} S_{i l} (r) - \sum_{i \leq n - 1} \sqrt{\frac{n! (i + 2 l + 2)!}{i! (n + 2 l + 2)!}} S_{i l} (r) .

(52)

The actions of other operators, built from these basic operators, can be calculated by combining (46), (51), and (52).

Once we have the action of a radial operator R(r) on S_nl, we can extract the matrix element 〈n′l|R(r)|nl〉 by inspection as the coefficient of S_n′l in the sum, using the orthogonality of the radial functions S_nl. For example, if n′ ≥ n + 1, then

\begin{aligned} 〈 n^{'} l | \frac{d}{d r} | n l 〉 & = \sum_{i \geq n + 1} \sqrt{\frac{i! (n + 2 l + 2)!}{n! (i + 2 l + 2)!}} δ_{n^{'} i} - \sum_{i \leq n - 1} \sqrt{\frac{n! (i + 2 l + 2)!}{i! (n + 2 l + 2)!}} δ_{n^{'} i} \\ = \sqrt{\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!}} . \end{aligned}

(53)

Expressions for the matrix elements for r, r², 1/r, 1/r², rd/dr, d/dr, and d²/dr² are given in the Appendix (Table I).

TABLE I.

Matrix elements between functions in the S_nl basis.

〈n′l|r|nl〉

n′ ≥ n + 2

0

n′ = n + 1

- \frac{1}{2} {((n + 1) (n + 2 l + 3))}^{1 / 2}

n′ = n

n + l + \frac{3}{2}

n′ = n − 1

- \frac{1}{2} {(n (n + 2 l + 2))}^{1 / 2}

n′ ≤ n − 2

0

〈n′l|r²|nl〉

n′ ≥ n + 3

0

n′ = n + 2

\frac{1}{4} {((n + 1) (n + 2) (n + 2 l + 3) (n + 2 l + 4))}^{1 / 2}

n′ = n + 1

−(n + l + 2)((n + 1)(n + 2l + 3))^1/2

n′ = n

(n + l + \frac{3}{2}) (n + l + 2) + \frac{1}{2} n (n + 2 l + 2)

n′ = n − 1

−(n + l + 1)(n(n + 2l + 2))^1/2

n′ = n − 2

\frac{1}{4} {(n (n - 1) (n + 2 l + 2) (n + 2 l + 1))}^{1 / 2}

n′ ≤ n − 3

0

〈 n^{'} l | \frac{1}{r} | n l 〉

n′ ≥ n

\frac{1}{l + 1} {(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!})}^{1 / 2}

n′ ≤ n

\frac{1}{l + 1} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 2)!})}^{1 / 2}

〈 n^{'} l | \frac{1}{r^{2}} | n l 〉

n′ ≥ n

(\frac{2 n^{'} (2 l + 3) - 2 n (2 l + 1) + 4 l + 6}{(l + 1) (2 l + 1) (2 l + 3)}) {(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!})}^{1 / 2}

n′ ≤ n

(\frac{2 n (2 l + 3) - 2 n^{'} (2 l + 1) + 4 l + 6}{(l + 1) (2 l + 1) (2 l + 3)}) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 2)!})}^{1 / 2}

〈 n^{'} l | r \frac{d}{d r} | n l 〉

n′ ≥ n + 2

0

n′ = n + 1

\frac{1}{2} {((n + 1) (n + 2 l + 3))}^{1 / 2}

n′ = n

- \frac{1}{2}

n′ = n − 1

- \frac{1}{2} {(n (n + 2 l + 2))}^{1 / 2}

n′ ≤ n − 2

0

〈 n^{'} l | \frac{d}{d r} | n l 〉

n′ ≥ n + 1

{(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!})}^{1 / 2}

n′ = n

0

n′ ≤ n − 1

- {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 2)!})}^{1 / 2}

〈 n^{'} l | \frac{d^{2}}{d r^{2}} | n l 〉

n′ ≥ n + 1

- \frac{(2 l + 4) (2 l + 1) n - 2 l (2 l + 3) n^{'} + (2 l + 3) (2 l + 2)}{(2 l + 3) (2 l + 1)} {(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!})}^{1 / 2}

n′ = n

- \frac{4 n (l + 1) + 2 l + 3}{(2 l + 3) (2 l + 1)}

n′ ≤ n − 1

- \frac{(2 l + 4) (2 l + 1) n^{'} - 2 l (2 l + 3) n + (2 l + 3) (2 l + 2)}{(2 l + 3) (2 l + 1)} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 2)!})}^{1 / 2}

IV. SHIFT OPERATORS

In Sec. III, we derived expressions for matrix elements of operators between Laguerre radial functions within the same SU(1,1) irrep, i.e., matrix elements between the three-dimensional basis functions with the same angular quantum number l. However, many operators of interest in physics are not strictly radial, scalar operators (e.g., r and ∇ are vector operators or, equivalently, spherical tensors of rank 1) and may therefore have nonvanishing matrix elements between states with different l. If we consider a spherical tensor operator $O_{a α}$ which factorizes into radial and angular parts, as discussed in Sec. II A, then recall from (20) that the problem reduces to evaluating the radial matrix element 〈n′l′|γ(R)|nl〉, with l′ ≠ l, that is, connecting Laguerre radial functions lying in different SU(1,1) irreps.

The matrix elements of radial operators between radial functions in different irreps (l′ ≠ l) can be written in terms of the matrix elements we have already calculated, i.e., between members of the same irrep (l′ = l), provided we can obtain analytic relations allowing us to expand the members of one of the irreps (namely, l′) as linear combinations of those in the other irrep (namely, l). This relation may be obtained through shift operators, A^† and A, introduced in Sec. I.

To see how we may obtain such shift operators, recall from (16) that the Laguerre functions are eigenfunctions of a differential operator. In particular, the operator

H_{n l} = - \frac{d^{2}}{d r^{2}} - \frac{1}{r} \frac{d}{d r} + \frac{{(l + 1)}^{2}}{r^{2}} - \frac{2 α_{n l}}{r}

(54)

may be taken as a Hamiltonian operator defining the Laguerre functions (of a single irrep l) as its eigenfunctions, except that this Hamiltonian involves an n-dependent coefficient α_nl and is thus actually a quantum number dependent Hamiltonian (the eigenfunctions of interest are also degenerate, with eigenvalue −1).

The problem of relating Laguerre functions from different irreps (of different l) thus amounts to the problem of relating the eigenfunctions of a sequence of Hamiltonians, which are defined through discrete variation of a parameter l. This is the problem addressed by the factorization method of Infeld and Hull²⁹ or, equivalently, for present purposes, the factorization method used in SUSY QM³⁰ for shape invariant potentials.⁵²

A. Factorization with quantum number dependence

The essential idea of factorization is that if a Hamiltonian can be written in a factorized form H = A^†A, this immediately gives rise to a partner Hamiltonian H′ = AA^†, with (at least some) eigenvalues which are degenerate to those of H, and with eigenfunctions which are related to those of H—in fact, related by the action of the operators A^† and A. Factorization is a particularly powerful technique in problems where we have an extended (perhaps infinite) sequence or “tower” of related Hamiltonians $H_{m} = A_{m}^{†} A_{m}$ (m = 1, 2, …), where m is a discrete parameter in the Hamiltonian. Then we likewise have extended sets of degenerate states ψ_nm (n = 0, 1, … and m = 1, 2, …), related to each other by the action of the operators $A_{m}^{†}$ and A_m. SUSY QM provides a general formalism for finding such a factorization, through the superpotential.³⁰ However, for many familiar exactly solvable Schrödinger equation problems, the factorizations have been cataloged by Infeld and Hull,²⁹ and the problem of finding a factorization reduces to that of transforming the problem into one of their standard forms.

It may be helpful to note both the analogy to the familiar factorization of the one-dimensional harmonic oscillator Hamiltonian, H = a^†a + 1/2, explored by Dirac,⁵³ as well as the differences from this factorization. In the oscillator problem, the operators a^† and a are simply laddering operators, between eigenfunctions of the same Hamiltonian, serving to change the number of nodes. However, the operators A^† and A in the more general factorization are shift operators, between eigenfunctions of different Hamiltonians, though in fact their action also changes the number of nodes.

While factorization is a powerful technique for solving eigenproblems, we are interested here primarily in the shift operation which it provides, towards our goal of relating Laguerre radial functions in different irreps. We therefore summarize the necessary ideas behind the shift operation, then show how the relevant relations can readily be extended to accommodate quantum number dependent Hamiltonians, such as encountered for the Laguerre functions in (54). We refer the reader to the more extensive discussions in Refs. 29, 30, and 54 for a more comprehensive discussion of the families of potentials which lead to factorizable Hamiltonians, the solution process, and the nature of the eigenspectra which result. Our notational conventions for factorization follow those of, e.g., Refs. 23 and 30, in the use of A^† and A for the shift operators.

To start with this simplest case, of two partner Hamiltonians, suppose the Hamiltonian for some one-dimensional eigenproblem can be written as

H_{1} (r) = A_{1}^{†} (r) A_{1} (r) + K_{1},

(55)

where, in addition to the factorized portion of the Hamiltonian, it is convenient to permit a constant term K₁. (Since the primary concern in the present work is with radial wave functions on ℝ⁺, we have chosen to denote the independent variable as r rather than as x, but the discussion here is relevant without modification to one-dimensional problems on ℝ.) Then the eigenspectrum of H₁ is related in a simple way to that of the partner Hamiltonian

H_{2} (r) = A_{1} (r) A_{1}^{†} (r) + K_{1} .

(56)

Specifically, suppose H₁ has eigenfunctions ψ_n1 (n = 0, 1, …), with associated eigenvalues E_n1, and H₂ has eigenfunctions ψ_n2 (n = 0, 1, …), with associated eigenvalues E_n2, that is,

H_{1} ψ_{n 1} = E_{n 1} ψ_{n 1},

(57)

H_{2} ψ_{n 2} = E_{n 2} ψ_{n 2} .

(58)

Then, simply from the form of (55) and (56), one can readily verify (as we shall illustrate below in a more general context) that acting with A₁ on any eigenfunction ψ_n1 of H₁ gives an eigenfunction of H₂, still with eigenvalue E_n1. Similarly, acting with $A_{1}^{†}$ on any eigenfunction ψ_n2 of H₂ gives an eigenfunction of H₁, still with eigenvalue E_n2. Thus, eigenfunctions of H₁ and H₂ must occur as degenerate partners. The exception arises if A₁ (or $A_{1}^{†}$ ⁠) annihilates the wave function on which it acts, that is, if it yields the null function instead of a bona fide eigenfunction of the partner Hamiltonian. We shall restrict our attention to the typical case (see Refs. 29, 30, and 54) in which A₁ψ₀₁ = 0. That is, the ground state eigenfunction of H₁ has no partner. Then the remaining eigenfunctions and eigenvalues follow the correspondence (for n = 1, 2, …)

ψ_{n 1} (r) \propto A_{1}^{†} (r) ψ_{n - 1, 2} (r),

(59)

ψ_{n - 1, 2} (r) \propto A_{1} (r) ψ_{n 1} (r),

(60)

E_{n - 1, 2} = E_{n 1} .

(61)

The factorization process may, however, continue. Suppose H₂, in turn, can be refactored in some way, distinct from (56), as

H_{2} (r) = A_{2}^{†} (r) A_{2} (r) + K_{2} .

(62)

Then the same pattern can be followed, as above, to relate the eigenspectrum of H₂ to that of a partner Hamiltonian

H_{3} (r) = A_{2} (r) A_{2}^{†} (r) + K_{2},

(63)

and relations on the eigenfunctions and eigenvalues analogous to (59)–(61) can be obtained.

More generally, we may consider an infinite “tower” of related Hamiltonians H_m, parametrized by a discrete index m (m = 1, 2, …), where successive Hamiltonians H_m and H_m+1 have related factorizations

H_{m} (r) = A_{m}^{†} (r) A_{m} (r) + K_{m},

(64)

H_{m + 1} (r) = A_{m} (r) A_{m}^{†} (r) + K_{m} .

(65)

We now have a sequence of related eigenproblems

H_{m} (r) ψ_{n m} (r) = E_{n m} ψ_{n m} (r) .

(66)

Their eigenfunctions and eigenvalues are related by

ψ_{n m} (r) \propto A_{m}^{†} (r) ψ_{n - 1, m + 1} (r),

(67)

ψ_{n - 1, m + 1} (r) \propto A_{m} (r) ψ_{n m} (r),

(68)

E_{n - 1, m + 1} = E_{n m},

(69)

etc. The identification of n quantum numbers (n = 0, 1, …) across the different eigenproblems here again rests on the further conventional condition that the ground state for each H_m be annihilated by the A operator, i.e., A_mψ_0m = 0 or, equivalently, as may be verified, E_0m = K_m.

Let us take a moment to interpret the energy and operator relations indicated in (67)–(69), which are illustrated schematically in Fig. 3. From (67), it is seen that the operator $A_{m}^{†}$ acts on the wave function ψ_n−1,m+1 to give the wave function ψ_nm. It thus raises the quantum number n, which counts the number of nodes in the wave function, and is therefore reminiscent of Dirac’s a^† operator. However, the initial and final wave functions are eigenfunctions of different Hamiltonians, as the Hamiltonian index has simultaneously been lowered. Thus, $A_{m}^{†}$ is not merely a raising operator, but rather a shift operator. Similarly, from (68), it is seen that A_m is likewise a shift operator, but with the inverse action on the node number and Hamiltonian indices. Finally, from (69), the eigenvalues form degenerate multiplets across the different Hamiltonians, with E_n0 = E_n−1,1 = E_n−2,2 = ⋯. In fact, by repeated application of (67)–(69), the eigenfunctions and eigenvalues of any two Hamiltonians in the sequence can be related. Comparing the shift relations among eigenspectra arising from factorization (Fig. 3) with those anticipated for shift operators between SU(1, 1) irreps (Fig. 1), we may already see the essential resemblance.

FIG. 3.

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Structure of the relations among eigenfunctions and eigenvalues for a sequence of factorized Hamiltonians H_m, defined in (67)–(69).

We may now generalize these relations to the case in which the Hamiltonians are n-dependent, that is, to eigenproblems of the form

H_{n m} (r) ψ_{n m} (r) = E_{n m} ψ_{n m} (r) .

(70)

Factorizations (64) and (65) then generalize to

H_{n m} (r) = A_{n m}^{†} (r) A_{n m} (r) + K_{n m},

(71)

H_{n - 1, m + 1} (r) = A_{n m} (r) A_{n m}^{†} (r) + K_{n m},

(72)

again with the assumption that the ground state eigenfunctions are annihilated, i.e., A_0m(r) ψ_0m(r) = 0. We label the latter Hamiltonian with the index n − 1, rather than n, in recognition of the anticipated correspondence of eigenfunctions across the Hamiltonians.

We may now obtain n-dependent shift operator relations

ψ_{n m} (r) \propto A_{n m}^{†} (r) ψ_{n - 1, m + 1} (r),

(73)

ψ_{n - 1, m + 1} (r) \propto A_{n m} (r) ψ_{n m} (r),

(74)

while the eigenvalue relation E_n−1,m+1 = E_nm from (69) still holds. These relations follow by first establishing that $A_{n m}^{†}$ acts on the eigenfunction ψ_n−1,m+1 of H_n−1,m+1 to give an eigenfunction of H_nm, with the same eigenvalue E_n−1,m+1,

\begin{aligned} H_{n m} [A_{n m}^{†} (r) ψ_{n - 1, m + 1} (r)] & = [A_{n m}^{†} (r) A_{n m} (r) + K_{n m}] A_{n m}^{†} (r) ψ_{n - 1, m + 1} (r) \\ = A_{n m}^{†} (r) [A_{n m} (r) A_{n m}^{†} (r) + K_{n m}] ψ_{n - 1, m + 1} (r) \\ = A_{n m}^{†} (r) H_{n - 1, m + 1} ψ_{n - 1, m + 1} (r) \\ = E_{n - 1, m + 1} [A_{n m}^{†} (r) ψ_{n - 1, m + 1} (r)], \end{aligned}

(75)

where we have successively applied the definition of H_nm from (71), reassociated, recognized the definition of H_n−1,m+1 from (72), and applied the eigenproblem definition from (70). It then may be verified, by a similar argument, that A_nm acts on the eigenfunction ψ_nm of H_nm to give an eigenfunction of H_n−1,m+1, with the same eigenvalue E_nm, i.e.,

H_{n - 1, m + 1} [A_{n m} (r) ψ_{n m} (r)] = E_{n m} [A_{n m} (r) ψ_{n m} (r)] .

(76)

To apply the shift operators in practice, it is desirable not only to know that a proportionality exists, as in (73) and (74), but also to know the constant of proportionality, so that we may construct normalized shift operators $A_{n m}^{†}$ and $A_{n m}$ ⁠. We require that these operators, when acting on normalized eigenfunctions, then yield normalized eigenfunctions, so that proportionalities (73) and (74) become equalities

ψ_{n m} (r) = A_{n m}^{†} (r) ψ_{n - 1, m + 1} (r),

(77)

ψ_{n - 1, m + 1} (r) = A_{n m} (r) ψ_{n m} (r) .

(78)

The constants of proportionality in (73) and (74) follow uniquely, within phase, from the assumed adjoint relationship between the operators $A_{n m}^{†}$ and A_nm and from the requirement that the wave functions ψ_nm be normalized. In particular, the norm of $A_{n m}^{†} ψ_{n - 1, m + 1}$ is readily evaluated, since (we switch to bracket notation to facilitate representing the inner product)

\begin{aligned} | A_{n m}^{†} ψ_{n - 1, m + 1} |^{2} & = 〈 ψ_{n - 1, m + 1} | A_{n m} A_{n m}^{†} | ψ_{n - 1, m + 1} 〉 \\ = 〈 ψ_{n - 1, m + 1} | (H_{n - 1, m + 1} - K_{n m}) | ψ_{n - 1, m + 1} 〉 \\ = E_{n - 1, m + 1} - K_{n m}, \end{aligned}

(79)

where we have recognized the relationship between $A_{n m} A_{n m}^{†}$ and H_n−1,m+1 by (72) and then applied the eigenproblem definition from (70). The result may simply be rewritten as $| A_{n m}^{†} ψ_{n - 1, m + 1} |^{2} = E_{n m} - K_{n m}$ by eigenvalue relation (69). Similarly, considering the norm of A_nmψ_nm gives an identical value |A_nmψ_nm|² = E_nm − K_nm.

Specifically, to construct the normalized shift operators, let

A_{n m}^{†} = σ_{n m} c_{n m} A_{n m}^{†},

(80)

A_{n m} = σ_{n m}^{'} c_{n m}^{'} A_{n m},

(81)

where σ_nm and $σ_{n m}^{'}$ represent the phases, and c_nm and $c_{n m}^{'}$ represent the normalization factors. Thus, insisting that both $ψ_{n m} = A_{n m}^{†} ψ_{n - 1, m + 1}$ and $ψ_{n - 1, m + 1} = A_{n m} ψ_{n m}$ be normalized gives $c_{n m} = c_{n m}^{'} = {(E_{n m} - K_{n m})}^{- 1 / 2}$ ⁠.

The phases (signs) of the solutions to an eigenproblem (66) or (70) are not determined by the eigenproblem itself, but rather are a matter of convention. For instance, it may be chosen that the wave functions are positive as r → 0 or positive as r → ∞. Thus, the phases σ_nm and $σ_{n m}^{'}$ appearing in the normalized shift operators cannot be deduced purely from the factorization formalism but must rather be chosen so that the shift operators yield wave functions with phases consistent with the adopted convention. The self-consistency requirement that $ψ_{n m} = A_{n m}^{†} ψ_{n - 1, m + 1} = A_{n m}^{†} A_{n m} ψ_{n m}$ imposes the constraint that $σ_{n m} = σ_{n m}^{'}$ ⁠.

If the normalized shift operators are known for all m, then we have a way of expressing eigenfunctions of any Hamiltonian H_m+k in terms of those of H_m, via (77) or (78). In particular, if k is a positive integer, then we repeatedly apply $A$ to obtain

ψ_{n, m + k} = A_{n + 1, m + k - 1} A_{n + 2, m + k - 2} \dots A_{n + k, m} ψ_{n + k, m} .

(82)

B. Shift operators for Laguerre functions

We now deduce shift operators on the l quantum number of the Laguerre radial functions S_nl(r). We deduce these shift operators by recasting differential equation (16) into the form of a factorizable n-dependent Hamiltonian eigenproblem, so that we may use the formalism laid out in Sec. IV A. Specifically, (16) can be rewritten in terms of Hamiltonian operator (54) as

H_{n l} S_{n l} (r) = - 1 S_{n l} (r),

(83)

where E_nl = − 1 for all n and l. However, to put this Hamiltonian into one of the standard factorizable forms cataloged by Infeld and Hull,²⁹ we must first eliminate the first-derivative term, through a similarity transformation r^1/2H_nlr^−1/2. Hamiltonian equation (83) becomes

[- \frac{d^{2}}{d r^{2}} + \frac{(l + \frac{1}{2}) (l + \frac{3}{2})}{r^{2}} - \frac{2 α_{n l}}{r}] R_{n l} (r) = - 1 R_{n l} (r),

(84)

where

R_{n l} (r) = r^{1 / 2} S_{n l} (r) .

(85)

Comparing (84) with the standard form for Coulomb-type problems, given by (8.0.1) of Ref. 29, yields the factorization⁵⁵

{\tilde{A}}_{n l}^{†} (r) = (- \frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}}), {\tilde{A}}_{n l} (r) = (\frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}})

(86)

and constant term

K_{n l} = - \frac{α_{n l}^{2}}{{(l + \frac{3}{2})}^{2}} .

(87)

The tildes on ${\tilde{A}}^{†}$ and $\tilde{A}$ indicate that these are the shift operators corresponding to similarity transformed Hamiltonian operator (84). We also note that K_0l = − 1 = E_0l, thus satisfying the condition to obtain relationships (67)–(69) between partner Hamiltonian eigenfunctions and eigenvalues, equivalent to ${\tilde{A}}_{0 l} R_{0 l} = 0$ and thus A_0lS_0l = 0 (see Sec. IV A).

From (86) and (87), we can write (84) in the n-dependent factorized form of (71), as

[(- \frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}}) (\frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}}) - \frac{α_{n l}^{2}}{{(l + \frac{3}{2})}^{2}}] R_{n l} (r) = - 1 R_{n l} (r) .

(88)

Normalized shift operators (80) and (81) for the functions R_nl(r) are given by

{\tilde{A}}_{n l}^{†} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{n (n + 2 l + 3)}} (- \frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}}),

(89)

{\tilde{A}}_{n l} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{n (n + 2 l + 3)}} (\frac{d}{d r} - \frac{l + \frac{3}{2}}{r} + \frac{α_{n l}}{l + \frac{3}{2}}) .

(90)

Here we adopt phases σ_nl = − 1, to enforce the convention that R_nl(r) [and thus S_nl(r) below] is always positive at the origin.

To obtain the shift operators for the functions S_nl(r), we must apply the inverse similarity transformation. Thus, $A_{n l}^{†} (r) = r^{- 1 / 2} {\tilde{A}}_{n l}^{†} (r) r^{1 / 2}$ gives

A_{n l}^{†} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{n (n + 2 l + 3)}} (- \frac{d}{d r} - \frac{l + 2}{r} + \frac{α_{n l}}{l + \frac{3}{2}}),

(91)

and, similarly, $A_{n l} (r) = r^{- 1 / 2} {\tilde{A}}_{n l} (r) r^{1 / 2}$ gives

A_{n l} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{n (n + 2 l + 3)}} (\frac{d}{d r} - \frac{l + 1}{r} + \frac{α_{n l}}{l + \frac{3}{2}}) .

(92)

C. Deriving the action of radial operators across irreps using the shift operators

Now that we have shift operators relating S_nl(r) functions with different l,

S_{n, l + 1} (r) = A_{n + 1, l} (r) S_{n + 1, l} (r), S_{n l} (r) = A_{n l}^{†} (r) S_{n - 1, l + 1} (r),

(93)

we can derive expressions for the matrix elements between functions S_nl and S_n′l′ for l′ = l ± k, for k = 1, 2, …, by writing S_n′,l+k in terms of S_nl using relationships (93). Note that the matrix elements for l′ = l − k can be obtained from the matrix elements of l′ = l + k by making the appropriate substitutions ( Appendix), and thus it is sufficient to consider only l′ = l + k.

For l′ = l ± 1, we simply apply the raising shift operator to S_n+1,l,

\begin{aligned} S_{n, l + 1} (r) & = A_{n + 1, l} (r) S_{n + 1, l} (r) \\ = \frac{- (l + \frac{3}{2})}{\sqrt{(n + 1) (n + 2 l + 4)}} (\frac{d}{d r} - \frac{l + 1}{r} + \frac{α_{n l}}{α_{0 l}}) S_{n + 1, l} (r) . \end{aligned}

(94)

Applying (51) and (52) to (94),

S_{n, l + 1} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{(n + 1) (n + 2 l + 4)}} [\frac{n + 1}{l + \frac{3}{2}} S_{n + 1, l} (r) - 2 \sum_{i = 0}^{n} {(\frac{(n + 1)! (i + 2 l + 2)!}{i! (n + 2 l + 3)!})}^{1 / 2} S_{i l} (r)] .

(95)

For l′ = l + 2, the raising shift operator must be applied twice,

S_{n, l + 2} (r) = A_{n + 1, l + 1} (r) S_{n + 1, l + 1} (r) = A_{n + 1, l + 1} (r) A_{n + 2, l} (r) S_{n + 2, l} (r) .

(96)

From this, we have that

S_{n, l + 2} (r) = \sqrt{\frac{(n + 1) (n + 2)}{(n + 2 l + 6) (n + 2 l + 5)}} S_{n + 2, l} (r) + \sum_{i = 0}^{n + 1} c_{i} \sqrt{\frac{n! (i + 2 l + 2)!}{i! (n + 2 l + 6)!}} S_{i l} (r),

where

c_{i} = (2 l + 4) (2 l + 3) n - (2 l + 5) (2 l + 4) i + (2 l + 4) (2 l + 3) .

(97)

Similarly, we could obtain the explicit relationship between S_n′,l+k and S_nl for any k by applying the shift operator to S_n−k,l k times. However, we only explicitly calculate examples of matrix elements for l′ = l ± 1 and l′ = l ± 2 in this paper.

Once we have an expression for S_n,l+k(r) in terms of S_nl(r), we can then use the matrix elements already calculated for l = l′ to derive the matrix elements for l′≠l. For example,

r S_{n, l + 1} (r) = \frac{- (l + \frac{3}{2})}{\sqrt{(n + 1) (n + 2 l + 4)}} [\frac{n + 1}{l + \frac{3}{2}} r S_{n + 1, l} (r)) - (2 \sum_{i = 0}^{n} {(\frac{(n + 1)! (i + 2 l + 2)!}{i! (n + 2 l + 3)!})}^{1 / 2} r S_{i l} (r)] .

(98)

From (46), we then deduce that

r S_{n, l + 1} (r) = - \frac{1}{2} \sqrt{(n + 1) (n + 2)} S_{n + 2, l} + \sqrt{(n + 1) (n + 2 l + 4)} S_{n + 1, l} (r) - \frac{1}{2} \sqrt{(n + 2 l + 3) (n + 2 l + 4)} S_{n l} (r) .

(99)

Using the orthonormality of the S_nl(r), we can read off the matrix elements 〈n′l|r|n, l + 1〉 from this expression, i.e.,

〈 n + 2, l | r | n, l + 1 〉 = - \frac{1}{2} \sqrt{(n + 1) (n + 2)},

(100)

〈 n + 1, l | r | n, l + 1 〉 = \sqrt{(n + 1) (n + 2 l + 4)},

(101)

〈 n l | r | n, l + 1 〉 = - \frac{1}{2} \sqrt{(n + 2 l + 3) (n + 2 l + 4)},

(102)

and all other matrix elements are zero. Matrix elements of select operators between functions with l′ = l ± 1 or l′ = l ± 2 are given in the Appendix (Tables II and III).

TABLE II.

Matrix elements between functions in the S_n,l+1 and S_nl bases.

〈n′l|n, l + 1〉

n′ ≥ n + 2

0

n′ = n + 1

- {(\frac{n + 1}{n + 2 l + 4})}^{1 / 2}

n′ ≤ n

(2 l + 3) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 4)!})}^{1 / 2}

〈n′l|r|n, l + 1〉

n′ ≥ n + 3

0

n′ = n + 2

\frac{1}{2} {((n + 1) (n + 2))}^{1 / 2}

n′ = n + 1

−((n + 1)(n + 2l + 4))^1/2

n′ = n

\frac{1}{2} {((n + 2 l + 4) (n + 2 l + 3))}^{1 / 2}

n′ ≤ n − 1

0

〈 n^{'} l | \frac{1}{r} | n, l + 1 〉

n′ ≥ n + 1

0

n′ ≤ n

(2 (n + 1) - 2 n^{'}) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 4)!})}^{1 / 2}

〈 n^{'} l | \frac{d}{d r} | n, l + 1 〉

n′ ≥ n + 2

0

n′ = n + 1

{(\frac{n + 1}{n + 2 l + 4})}^{1 / 2}

n′ ≤ n

(n^{'} (2 l + 4) - n (2 l + 2) + 1) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 4)!})}^{1 / 2}

TABLE III.

Matrix elements between functions in the S_n,l+2 and S_nl bases.

〈n′l|n, l + 2〉^a

n′ ≥ n + 3

0

n′ = n + 2

{(\frac{(n + 1) (n + 2)}{(n + 2 l + 6) (n + 2 l + 5)})}^{1 / 2}

n′ ≤ n + 1

c_{n n^{'}} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 6)!})}^{1 / 2}

〈n′l|r|n, l + 2〉

n′ ≥ n + 4

0

n′ = n + 3

- \frac{1}{2} {(\frac{(n + 3)!}{n! (n + 2 l + 6)})}^{1 / 2}

n′ = n + 2

(n + 3 l + \frac{15}{2}) {(\frac{(n + 2)! (n + 2 l + 4)!}{n! (n + 2 l + 6)!})}^{1 / 2}

n′ = n + 1

- \frac{1}{2} (n (n + 14) + 6 l (n + 2 l + 9) + 60) {(\frac{(n + 1) (n + 2 l + 3)!}{(n + 2 l + 6)!})}^{1 / 2}

n′ ≤ n

(l + 2) (2 l + 3) (2 l + 5) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 6)!})}^{1 / 2}

〈n′l|r²|n, l + 2〉

n′ ≥ n + 5

0

n′ = n + 4

\frac{1}{4} {(\frac{(n + 4)!}{n!})}^{1 / 2}

n′ = n + 3

- {(\frac{(n + 2 l + 6) (n + 3)!}{n!})}^{1 / 2}

n′ = n + 2

\frac{3}{2} {(\frac{(n + 2 l + 6)! (n + 2)!}{n! (n + 2 l + 4)!})}^{1 / 2}

n′ = n + 1

- {(\frac{(n + 2 l + 6)! (n + 1)}{(n + 2 l + 3)!})}^{1 / 2}

n′ = n

\frac{1}{4} {(\frac{(n + 2 l + 6)!}{(n + 2 l + 2)!})}^{1 / 2}

n′ ≤ n − 1

0

〈 n^{'} l | \frac{1}{r^{2}} | n, l + 2 〉

n′ ≥ n + 1

0

n′ ≤ n

\frac{2}{3} (n - n^{'} + 1) (n - n^{'} + 2) (n - n^{'} + 3) {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 6)!})}^{1 / 2}

〈 n^{'} l | \frac{d}{d r} | n, l + 2 〉

^b

n′ ≥ n + 3

0

n′ = n + 2

- {(\frac{(n + 2)! (n + 2 l + 4)!}{n! (n + 2 l + 6)!})}^{1 / 2}

n′ ≤ n + 1

d_{n n^{'}} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 6)!})}^{1 / 2}

〈 n^{'} l | \frac{d^{2}}{d r^{2}} | n, l + 2 〉

^c

n′ ≥ n + 3

0

n′ = n + 2

{(\frac{(n + 2)! (n + 2 l + 4)!}{n! (n + 2 l + 6)!})}^{1 / 2}

n′ ≤ n + 2

f_{n n^{'}} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 6)!})}^{1 / 2}

^a

c_nn′ = (2l + 4)(2l + 3) n − (2l + 5)(2l + 4) n′ + (2l + 4)(2l + 3).

^b

d_nn′ = n′[4n(l + 2)² + l(2l + 9) + 11 − n′(2l + 5)(l + 3)] − (n + 1)(2l + 3)(n(l + 1) − 2).

^c

$f_{n n^{'}} = - \frac{2}{3} {n^{'} [(l + 3) n^{'} ((l + 4) n^{'} - 3 ((l + 2) n + 1)) + (l + 2) n (3 n (l + 1) - 6) - l (l + 7) - 9] - n [(l + 1) n (n l - 9) - l (l + 13) - 9] + 3 l}$ ⁠.

V. MATRIX ELEMENTS IN THE LAGUERRE FUNCTION BASIS

Now that we have a way to calculate matrix elements of radial operators with respect to the S_nl basis on ℝ⁺, we can calculate the matrix elements for irreducible tensor operators in the Λ_nlm basis on ℝ³. In Sec. II A we saw that the matrix element for $O_{a α} = R Θ_{a α}$ can be written (20) as

〈 n^{'} l^{'} m^{'} | O_{a α} | n l m 〉 = 〈 n^{'} l^{'} | γ (R) | n l 〉 〈 l^{'} m^{'} | Θ_{a α} | l m 〉,

(103)

where γ(R) is given by (21). From standard angular momentum theory for spherical tensor operators,¹⁹ matrix elements involving different angular momentum projection quantum numbers m and m′ are all related via the Wigner-Eckart theorem, so we need only consider the reduced matrix elements. The reduced matrix element of $O_{a α}$ is given by the product of the SO(3) reduced matrix element of the angular operator and the matrix element of the radial part in the S_nl basis,

〈 n^{'} l^{'} ‖ O_{a} ‖ n l 〉 = 〈 n^{'} l^{'} | γ (R) | n l 〉 〈 l^{'} ‖ Θ_{a} ‖ l 〉 .

(104)

Note that we follow the normalization and phase convention of, e.g., Refs. 19 and 56 for the reduced matrix elements in the Wigner-Eckart theorem, i.e.,

〈 l^{'} m^{'} | Θ_{a α} | l m 〉 = {\hat{l}}^{- 1} (l m; a α | l^{'} m^{'}) 〈 l^{'} ‖ Θ_{a} ‖ l 〉,

(105)

where the quantity delimited by parentheses is a Clebsch-Gordan coefficient, and $\hat{l} = \sqrt{2 l + 1}$ ⁠.

To illustrate this method of calculating the matrix elements in the Λ_nlm(r) basis, we will consider several examples, specifically, r^k for k ∈ ℤ, ∇² and ∇₁.

A. Matrix elements of the purely radial operator r^k

Matrix elements of polynomial operators that depend only on r are particularly simple to calculate. In spherical tensor form, r^k ≡ r^k1₀₀. Applying the factorization in (104), we can write

〈 n^{'} l^{'} ‖ r^{k} 1_{00} ‖ n l 〉 = 〈 n^{'} l^{'} | γ (r^{k}) | n l 〉 〈 l^{'} ‖ 1_{00} ‖ l 〉 .

(106)

The reduced matrix element of the identity operator is $〈 l^{'} ‖ 1_{00} ‖ l 〉 = \hat{l} δ_{l^{'} l}$ ⁠, and, under the transformation (21), γ(r^k) = r^k for any k. Therefore, (106) reduces to

〈 n^{'} l^{'} ‖ r^{k} 1_{00} ‖ n l 〉 = 〈 n^{'} l | r^{k} | n l 〉 \hat{l} .

(107)

The radial matrix elements are already known from Sec. III C and can be obtained from Table I for select values of k.

B. Matrix elements of the scalar operator ∇²

The Laplacian ∇² has the well known form¹⁹

\nabla^{2} = \frac{1}{r^{2}} \frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r} - \frac{L^{2}}{r^{2}} .

(108)

We factor each of the terms contributing to the matrix element of this operator into radial and angular parts, according to (104), and recall L²|lm〉 = l(l + 1)|lm〉, yielding

〈 n^{'} l^{'} ‖ \nabla^{2} ‖ n l 〉

= 〈 n^{'} l | γ (r^{- 2} \frac{d}{d r} r^{2} \frac{d}{d r}) | n l 〉 \hat{l} δ_{l l^{'}} - 〈 n^{'} l^{'} | γ (r^{- 2}) | n l 〉 〈 l^{'} ‖ L^{2} ‖ l 〉

= 〈 n^{'} l | \frac{d^{2}}{d r^{2}} | n l 〉 \hat{l} - 〈 n^{'} l | r^{- 2} | n l 〉 l (l + 1) \hat{l} .

(109)

The matrix elements of d²/dr² and r⁻² can be read from Table I, and we find that

〈 n^{'} l ‖ \nabla^{2} ‖ n l 〉 = {\begin{cases} - \frac{4 n + 2 l + 3}{2 l + 3} \hat{l} & n^{'} = n \\ \frac{4 n^{'} + 4 l + 6}{2 l + 3} \hat{l} {(\frac{n! (n^{'} + 2 l + 2)!}{n^{'}! (n + 2 l + 2)!})}^{1 / 2} & n^{'} \leq n - 1 \\ \frac{4 n + 4 l + 6}{2 l + 3} \hat{l} {(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 2)!})}^{1 / 2} & n^{'} \geq n + 1 \end{cases} .

(110)

C. Matrix elements of the tensor operator ∇₁

We now want to calculate the reduced matrix elements of the gradient operator ∇, which can be expressed as a rank 1 tensor ∇₁ with components¹⁹

\nabla_{1 μ} = {(\frac{4 π}{3})}^{\frac{1}{2}} \{Y_{1 μ} \frac{\partial}{\partial r} - \frac{\sqrt{2}}{r} {[Y_{1} \otimes L_{1}]}_{1 μ}\},

(111)

where Y₁ is the irreducible tensor whose components are the spherical harmonics $Y_{1 μ} (\hat{r})$ ⁠, and L₁ is the orbital angular momentum operator, likewise considered as a spherical tensor. The reduced matrix element of ∇₁ is thus given by

〈 n^{'} l^{'} ‖ \nabla_{1} ‖ n l 〉 = {(\frac{4 π}{3})}^{\frac{1}{2}} 〈 n^{'} l^{'} ‖ [Y_{1} \frac{\partial}{\partial r} - \frac{\sqrt{2}}{r} {[Y_{1} \otimes L_{1}]}_{1}] ‖ n l 〉 .

(112)

d by applying (104). The angular part of the second term can be factored as

〈 n^{'} l^{'} ‖ {[Y_{1} \otimes L_{1}]}_{1} ‖ n l 〉 = {(- 1)}^{l + l^{'} - 1} \sqrt{\frac{3}{2 l^{'} + 1}} \sum_{\underset{1}{l}} \{\begin{array}{ccc} 1 & 1 & 1 \\ l & l^{'} & l_{1} \end{array}\} \times 〈 n^{'} l^{'} ‖ Y_{1} ‖ n_{1} l_{1} 〉 〈 n_{1} l_{1} ‖ L_{1} ‖ n l 〉

(113)

by the standard reduction formula for reduced matrix elements of product operators on a single space.¹⁹ Using the reduced matrix elements of Y₁ and L₁ given in Ref. 19, the expression for the reduced matrix element of ∇, becomes

〈 n^{'} l^{'} ‖ \nabla_{1} ‖ n l 〉 = \sqrt{\frac{l + 1}{2 l + 3}} 〈 n^{'} l^{'} | (\frac{d}{d r} - \frac{l + 1}{r}) | n l 〉 δ_{l^{'}, l + 1}

- \sqrt{\frac{l}{2 l - 1}} 〈 n^{'} l^{'} | (\frac{d}{d r} + \frac{l}{r}) | n l 〉 δ_{l^{'}, l - 1} .

(114)

The terms contributing to the radial matrix elements can then be read from Table II, and we have

〈 n^{'}, l + 1 ‖ \nabla_{1} ‖ n l 〉 = {\begin{cases} - 2 (n^{'} + l + 3 / 2) {(\frac{l + 1}{2 l + 3})}^{\frac{1}{2}} {(\frac{n^{'}! (n + 2 l + 2)!}{n! (n^{'} + 2 l + 4)!})}^{\frac{1}{2}} & n^{'} \geq n \\ {(\frac{n (l + 1)}{(2 l + 3) (n + 2 l + 3)})}^{1 / 2} & n^{'} = n - 1 \\ 0 & n^{'} \leq n - 2 \end{cases},

(115)

and

〈 n^{'}, l - 1 ‖ \nabla_{1} ‖ n l 〉 = {\begin{cases} 0 & n^{'} \geq n + 2 \\ - {(\frac{(n + 1) l}{(2 l - 1) (n + 2 l + 2)})}^{1 / 2} & n^{'} = n + 1 \\ - 2 (n^{'} + l + 1 / 2) {(\frac{l}{2 l - 1})}^{\frac{1}{2}} {(\frac{n! (n^{'} + 2 l)!}{n^{'}! (n + 2 l + 2)!})}^{\frac{1}{2}} & n^{'} \leq n \end{cases} .

(116)

VI. CONCLUSION

It has long been recognized that an SU(1, 1) × SO(3) algebraic structure, together with shift operators between SU(1, 1) irreps, forms a successful combination in the treatment of the classic sets of central force eigenfunctions—those of the harmonic oscillator and Coulomb problems. We have now demonstrated that a similar framework can be developed and effectively applied to the Laguerre functions, a basis set of practical value in quantum mechanical one-body, few-body, and many-body problems.

The essential results thus obtained provide for the evaluation of matrix elements of spherical tensor operators. This is the immediate application of interest in formulating quantum mechanical problems for numerical solution. The evaluation of matrix elements is accomplished by decomposing the full operator, in the three-dimensional coordinate space, into parts which act only radial or angular coordinates. Then, recall the structure outlined in Fig. 1: (1) the SU(1, 1) algebra of the radial functions is used to deduce radial matrix elements, while (2) the shift operators connect the different families of radial functions, or SU(1, 1) irreps, associated with different angular functions, and (3) the SO(3) angular momentum algebra is used to address the action of the operator on the angular functions themselves. The SU(1, 1) algebra follows directly from the eigenvalue equation satisfied by the radial functions. The ladder operators are obtained by factorization (or, equivalently, SUSY QM) methods, reformulated to allow for quantum number dependence of the shift operators. And the SO(3) algebra provides the familiar tools of the Wigner-Eckart theorem and Racah’s reduction formulas for the angular operators.

Although the results, as presented, have been specialized to the Laguerre functions on ℝ³, their applicability is broader. In particular, the results for the radial functions as a basis on ℝ⁺ stand alone or in conjunction with any decomposition of N-dimensional coordinates on ℝ^N into a radial coordinate on ℝ⁺ and angular coordinates on the (N − 1)-sphere S^N−1. For instance, functions of the Laguerre type considered here are applied as hyperradial basis functions in few-body calculations in Jacobi coordinates (e.g., Ref. 57). Alternatively, much as the SU(1, 1) treatment of the harmonic oscillator radial functions may be extended to a broader family of modified harmonic oscillator functions, by transitioning from a discrete SU(1, 1) irrep label t = l + 3/2 to a continuous label²³ (physically corresponding from the transition from a pure oscillator potential to a Davidson potential,⁵⁸ containing an additional 1/r² dependence), a similar generalization may be carried out for the present results, yielding an algebraic description of families of modified Laguerre functions. The general framework may also be carried over to the Coulomb-Sturmian functions, for which the SU(1, 1) radial algebra is considered in Ref. 59.

Finally, it is worth noting that, so far, we have focused exclusively on the SU(1, 1) character of the basis functions, rather than on the SU(1, 1) tensorial properties of operators on the radial space. Through these properties, it is possible to benefit from the Wigner-Eckart theorem, not only of the SO(3) angular algebra as we have done so far, but also of the SU(1, 1) radial algebra. Matrix elements involving different members of the same irrep (i.e., different radial functions for the same angular momentum) may then be related to each other through SU(1, 1) Clebsch-Gordan coefficients. The SU(1, 1) Wigner-Eckart theorem, applied in this fashion, has been a powerful tool in the construction of closed-form expressions for matrix elements in the classic harmonic oscillator and Coulomb eigenproblems.^20,60

Acknowledgments

We would like to thank C. Kolda and J. de Blas for valuable discussions, M. Wolf for comments on the manuscript, and A. I. Rakoski for verification of expressions for matrix elements. This work was supported by the Research Corporation for Science Advancement under a Cottrell Scholar Award and by the US Department of Energy under Grant No. DE-FG02-95ER-40934.

APPENDIX: MATRIX ELEMENTS BETWEEN LAGUERRE RADIAL FUNCTIONS

In the following tables, we have summarized the matrix elements of some of the most basic radial operators that can be constructed from r, r⁻¹, and d/dr, taken in the Laguerre function radial basis. Matrix elements within a single SU(1, 1) irrep, i.e., between radial functions S_nl of the same l, are given in Table I. Matrix elements between radial functions belonging to different SU(1, 1) irreps, corresponding to angular momenta differing by Δl = 1 and 2, are given in Tables II and III, respectively.

Note that the matrix elements in Tables I–III are of the form 〈n′l|R|nl〉, 〈n′l|R|n, l + 1〉, or 〈n′l|R|n, l + 2〉, respectively, as this is the form that is most naturally derived from applying the shift operators (see Sec. IV). However, one may express the l-changing matrix elements in the form 〈n′, l − 1|R|nl〉 or 〈n′, l − 2|R|nl〉 by making the appropriate substitution. For instance, to obtain 〈n′, l − 1|r|nl〉 from the matrix elements in Table II, one may simply make the substitution l → l − 1. Alternatively, matrix elements of the form 〈n′, l − 1|R|nl〉 or 〈n′, l − 2|R|nl〉 may be obtained by Hermitian conjugation. For instance, to obtain 〈n′, l − 1|r|nl〉 from the matrix elements in Table II, one may simply note that 〈n′, l + 1|r|nl〉 = 〈nl|r|n′, l + 1〉, since the matrix elements are real-valued, and make the interchange n↔n′. When applying Hermitian conjugation to differential operators appearing in Tables I–III, recall that the operator d/dr is anti-Hermitian (e.g., Ref. 16).

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2016

AIP Publishing LLC

Algebraic evaluation of matrix elements in the Laguerre function basis

I. INTRODUCTION

II. LAGUERRE FUNCTIONS

A. Operator map to the radial basis

III. ALGEBRAIC STRUCTURE OF LAGUERRE BASIS

A. A brief review of the SO(3) and SU(1,1) Lie algebras

B. Representation of SU(1,1) on the Laguerre radial basis

C. Deriving the action of radial operators using the SU(1,1) algebra

IV. SHIFT OPERATORS

A. Factorization with quantum number dependence

B. Shift operators for Laguerre functions

C. Deriving the action of radial operators across irreps using the shift operators

V. MATRIX ELEMENTS IN THE LAGUERRE FUNCTION BASIS

A. Matrix elements of the purely radial operator r^k

B. Matrix elements of the scalar operator ∇²

C. Matrix elements of the tensor operator ∇₁

VI. CONCLUSION

Acknowledgments

APPENDIX: MATRIX ELEMENTS BETWEEN LAGUERRE RADIAL FUNCTIONS

REFERENCES

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Algebraic evaluation of matrix elements in the Laguerre function basis

I. INTRODUCTION

II. LAGUERRE FUNCTIONS

A. Operator map to the radial basis

III. ALGEBRAIC STRUCTURE OF LAGUERRE BASIS

A. A brief review of the SO(3) and SU(1,1) Lie algebras

B. Representation of SU(1,1) on the Laguerre radial basis

C. Deriving the action of radial operators using the SU(1,1) algebra

IV. SHIFT OPERATORS

A. Factorization with quantum number dependence

B. Shift operators for Laguerre functions

C. Deriving the action of radial operators across irreps using the shift operators

V. MATRIX ELEMENTS IN THE LAGUERRE FUNCTION BASIS

A. Matrix elements of the purely radial operator rk

B. Matrix elements of the scalar operator ∇2

C. Matrix elements of the tensor operator ∇1

VI. CONCLUSION

Acknowledgments

APPENDIX: MATRIX ELEMENTS BETWEEN LAGUERRE RADIAL FUNCTIONS

REFERENCES

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A. Matrix elements of the purely radial operator r^k

B. Matrix elements of the scalar operator ∇²

C. Matrix elements of the tensor operator ∇₁