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 a finite sum.


Introduction
The computation of the expectation r k 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 where R n (r) is the radial solution of the Schrödinger equation for the hydrogen atom. Conditions 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]: In particular, these expectations are given in terms of the Chebyshev polynomials of discrete variables [18,24]  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 where the normalization constant is is the associated Laguerre function; see [14, formula 8.972.1]. The expectation value of a power of the radial coordinate is given by with n ∈ N, 0 ≤ ≤ n − 1, and k ∈ Z.
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] 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 Here (a) n = a(a + 1) · · · (a + n − 1) is the Pochhammer symbol.

A direct evaluation
This section presents a direct evaluation of the integral 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] To obtain an expression for G ,k,s (µ), the representation L a n (x) = Γ(a + n + 1) Γ(n + 1)Γ(1 + a) 1  Proof . The hypergeometric representation (2.5) shows that L 2(2 +1) 2s (4µr) = Γ(4 + 2s + 3) Γ(2s + 1)Γ(4 + 3) Expanding the hypergeometric function gives 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 → 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 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 2 F 1 in the answer is a negative integer, therefore the series reduces to a finite sum.
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 successfully 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 hydrogenic 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], represented 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 transformations. 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].

The method of brackets
The evaluation of the integral giving the mean value r k (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 is the bracket associated to the (divergent) integral on the right. The symbol is called the indicator associated to the index n. The notation φ i 1 i 2 ···ir , or simply φ 12···r , denotes the product Rules for the production of bracket series.
Here the coefficients a(n) come from an assumed expansion f (x) = n≥0 φ n a(n)x αn+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 P 2 . For α ∈ C, the multinomial power (a 1 +a 2 +· · ·+a r ) α is assigned the r-dimensional bracket series n 1 n 2 · · · nr φ n 1 n 2 ···nr a n 1 1 · · · a nr r −α + n 1 + · · · + n r Γ(−α) .
Rules for the evaluation of a bracket series. Rule E 1 . The one-dimensional bracket series is assigned the value 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 E 2 . Assume the matrix A = (a ij ) is non-singular, then the assignment is n 1 · · · nr φ n 1 ···nr f (n 1 , . . . , n r ) a 11 n 1 + · · · + a 1r n r + c 1 · · · a r1 n 1 + · · · + a rr n r + c 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 E 3 . 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 E 2 . 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.

The evaluation of the expectations. A f irst bracket calculation
This section describes the evaluation of the integral that appeared in (1.4) by the method of brackets. The expectation value of a power of the radial coordinate is then given by r k n = (2µ) 2 A 2 n I n, ,k (µ).
This integral can be scaled to This does not appear in the table [14]. The closest entry is 7.414.10: Note 4.1. In the evaluation of (4.1), it is convenient to write it as 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. φ n 1 a n 1 x n 1 and L α m (x) = Γ(α + 1 + m) .
Proof . The series of the exponential function is elementary. Indeed, To evaluate the series of the Laguerre function, treat m as a real non-integer parameter, and observe that The series for the Laguerre function now follows from the identity valid for n 2 ∈ N and m ∈ N.
This intermediate result is stated next. .
The inner sum is now identified as a hypergeometric function to produce I n, ,k;A,B,C (µ) = Γ 2 (n + + 1)Γ(t) Note 4.4. The same procedure can be used to treat the cases obtained by solving for n 1 or n 2 in the equation (4.3). The corresponding integrals are At this point, the parameters A, B, C are replaced by the value 2µ, in order to continue the evaluation. This gives (t) n 2 (k + 2) n 2 (1 − n + ) n 2 (2 + 2) n 2 (s) n 2 n 2 ! (−1) n 2 2 F 1 t + n 2 , 2 + k + n 2 s + n 2 1 .

The evaluation of the expectations. A second approach
The moment r k n has been expressed in (2.3) as a finite sum values of the integral where the index s is an integer varying from 0 to n − − 1. Corollary 2.3 provides an expression for r k n as a finite sum of values of the hypergeometric function 2 F 1 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.
Evaluating the last integral in terms of the gamma function and simplifying produces a proof of the next result.

A couple of examples
The method of brackets has been used here to produce analytic expressions for the mean radius where r k n is given as a finite sum of hypergeometric terms and r k n = Γ(2 + k + 3)Γ(n − − k − 2) 2n(2µ) k Γ(n − )Γ(2 + 2)Γ(−1 − k) 3 F 2 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.