An Introduction to Special Functions with Some Applications to Quantum Mechanics

Abstract

A short review on special functions and solution of the Coulomb problems in quantum mechanics is given. Multiparameter wave functions of linear harmonic oscillator, which cannot be obtained by the standard separation of variables, are discussed. Expectation values in relativistic Coulomb problems are studied by computer algebra methods.

Dedicated to the memory of Professor Arnold F. Nikiforov and Professor Vasilii B. Uvarov

Appendices

Appendix A: Evaluation of an Integral

Let us compute the following integral

$$\displaystyle \begin{aligned} J_{nms}^{\alpha \beta }=\int_{0}^{\infty }e^{-x}x^{\alpha +s}L_{n}^{\alpha }\left( x\right) L_{m}^{\beta }\left( x\right) \ dx, {} \end{aligned} $$

(A.1)

where n ≥ m and α − β = 0, ±1, ±2, … . Similar integrals were evaluated in [28, 49] and [113], see also references therein, but an important relation with the Hahn polynomials seems to be missing.

It is convenient to assume at the beginning that parameter s takes some continuous values such that α + s > −1 for convergence of the integral. Using the Rodrigues formula for the Laguerre polynomials [138, 139, 184]

$$\displaystyle \begin{aligned} L_{n}^{\alpha }\left( x\right) =\frac{1}{n!}e^{x}x^{-\alpha }\left( x^{\alpha +n}e^{-x}\right) ^{\left( n\right) }, {} \end{aligned} $$

(A.2)

see the proof in Sect. 1 of the present chapter, and integrating by parts

$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{nms}^{\alpha \beta } &\displaystyle =&\displaystyle \frac{1}{n!}\int_{0}^{\infty }\left( x^{\alpha +n}e^{-x}\right) ^{\left( n\right) }\left( x^{s}L_{m}^{\beta }\left( x\right) \right) \ dx \\ &\displaystyle =&\displaystyle \frac{1}{n!}\left. \left( \left( x^{\alpha +n}e^{-x}\right) ^{\left( n-1\right) }\left( x^{s}L_{m}^{\beta }\left( x\right) \right) \right) \right\vert {}_{0}^{\infty } \\ &\displaystyle \quad &\displaystyle -\frac{1}{n!}\int_{0}^{\infty }\left( x^{\alpha +n}e^{-x}\right) ^{\left( n-1\right) }\left( x^{s}L_{m}^{\beta }\left( x\right) \right) ^{\prime }\ dx \\ &\displaystyle &\displaystyle \vdots \\ &\displaystyle =&\displaystyle \frac{\left( -1\right) ^{n}}{n!}\int_{0}^{\infty }\left( x^{\alpha +n}e^{-x}\right) \left( x^{s}L_{m}^{\beta }\left( x\right) \right) ^{\left( n\right) }\ dx. \end{array} \end{aligned} $$

However, in view of (B.1),

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left( x^{s}L_{m}^{\beta }\left( x\right) \right) ^{\left( n\right) }= \frac{\Gamma \left( \beta +m+1\right) }{m!\;\Gamma \left( \beta +1\right) } \sum_{k}\frac{\left( -m\right) _{k}}{k!\left( \beta +1\right) _{k}}\left( x^{k+s}\right) ^{\left( n\right) } {} \notag\\ &\displaystyle &\displaystyle \quad =\frac{\Gamma \left( \beta +m+1\right) \Gamma \left( s+1\right) }{ m!\;\Gamma \left( \beta +1\right) \Gamma \left( s-n+1\right) }\sum_{k}\frac{ \left( -m\right) _{k}\left( s+1\right) _{k}}{k!\left( \beta +1\right) _{k}\left( s-n+1\right) _{k}}\ x^{k+s-n} \end{array} \end{aligned} $$

(A.3)

and with the help of Euler’s integral representation for the gamma function [8, 138]

$$\displaystyle \begin{aligned} \int_{0}^{\infty }x^{\alpha +k+s}e^{-x}\ dx=\Gamma \left( \alpha +k+s+1\right) =\left( \alpha +s+1\right) _{k}\Gamma \left( \alpha +s+1\right) , \end{aligned}$$

see also (B.10) below, one gets

$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{nms}^{\alpha \beta } &=&\left( -1\right) ^{n}\frac{\Gamma \left( \alpha +s+1\right) \Gamma \left( \beta +m+1\right) \Gamma \left( s+1\right) }{ n!\;m!\;\Gamma \left( \beta +1\right) \Gamma \left( s-n+1\right) } \notag \\ &&\times ~_{3}F_{2}\left( \begin{array}{c} -m,\ s+1,\ \alpha +s +1 \\ \beta +1,\quad s-n+1 \end{array} \right) . {} \end{array} \end{aligned} $$

(A.4)

See [16] or Eq. (1.12) for the definition of the generalized hypergeometric series ${ }_{3}F_{2}\left ( 1\right ) .$ Thomae’s transformation (B.8), see also [16] or [79], results in [183]

$$\displaystyle \begin{aligned} J_{nms}^{\alpha \beta }& =\int_{0}^{\infty }e^{-x}x^{\alpha +s}\ L_{n}^{\alpha }\left( x\right) L_{m}^{\beta }\left( x\right) \ dx {}\notag\\ & =\left( -1\right) ^{n-m}\frac{\Gamma \left( \alpha +s+1\right) \Gamma \left( \beta +m+1\right) \Gamma \left( s+1\right) }{m!\left( n-m\right) !\;\Gamma \left( \beta +1\right) \Gamma \left( s-n+m+1\right) } \notag \\ & \quad \times \ ~_{3}F_{2}\left( \begin{array}{c} -m,\ s+1,\ \beta -\alpha -s \\ \beta +1,\quad n-m+1 \end{array} \right) ,\quad n\geq m, \end{aligned} $$

(A.5)

where parameter s may take some integer values. This establishes a connection with the Hahn polynomials given by Eq. (B.6) below; one can also rewrite this integral in terms of the dual Hahn polynomials [139].

Letting s = 0 and α = β in (A.5) results in the orthogonality relation for the Laguerre polynomials. Two special cases

$$\displaystyle \begin{aligned} I_{1}=J_{nn1}^{\alpha \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha +1}\left( L_{n}^{\alpha }\left( x\right) \right) ^{2}\ dx=\left( \alpha +2n+1\right) \frac{\Gamma \left( \alpha +n+1\right) }{n!} {} \end{aligned} $$

(A.6)

and

$$\displaystyle \begin{aligned} I_{2}=J_{n,\ n-1,\ 2}^{\alpha -2,\ \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha }L_{n-1}^{\alpha }\left( x\right) L_{n}^{\alpha -2}\left( x\right) \ dx=-2\frac{\Gamma \left( \alpha +n\right) }{\left( n-1\right) !} {} \end{aligned} $$

(A.7)

are convenient for normalization of the wave functions of the discrete spectra in the nonrelativistic and relativistic Coulomb problems [28, 138].

Two other special cases of a particular interest in this chapter are

$$\displaystyle \begin{aligned} \begin{array}{rcl} &&J_{k}=J_{nnk}^{\alpha \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha +k}\ \left( L_{n}^{\alpha }\left( x\right) \right) ^{2}\ dx {} \notag\\ &&\quad =\frac{\Gamma \left( \alpha +k+1\right) \Gamma \left( \alpha +n+1\right) }{n!\;\Gamma \left( \alpha +1\right) }\ ~_{3}F_{2}\left( \begin{array}{c} -k,\ k+1,\ -n \\ 1,\quad \alpha +1 \end{array} \right) \end{array} \end{aligned} $$

(A.8)

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} &&J_{-k-1}=J_{nn,\ -k-1}^{\alpha \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha -k-1}\ \left( L_{n}^{\alpha }\left( x\right) \right) ^{2}\ dx {} \notag\\ &&\quad \quad \ \ =\frac{\Gamma \left( \alpha -k\right) \Gamma \left( \alpha +n+1\right) }{n!\;\Gamma \left( \alpha +1\right) }\ ~_{3}F_{2}\left( \begin{array}{c} -k,\ k+1,\ -n \\ 1,\quad \alpha +1 \end{array} \right) . \end{array} \end{aligned} $$

(A.9)

The Chebyshev polynomials of a discrete variable $t_{k}\left ( x\right )$ are special case of the Hahn polynomials $t_{k}\left ( x,N\right ) =h_{k}^{\left ( 0,\ 0\right ) }\left ( x,N\right )$ [185, 186] and [187]. Thus from (A.8)–(A.9) and (B.6) one finally gets

$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{k} &\displaystyle =&\displaystyle J_{nnk}^{\alpha \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha +k}\ \left( L_{n}^{\alpha }\left( x\right) \right) ^{2}\ dx {} \notag\\ &\displaystyle =&\displaystyle \frac{\Gamma \left( \alpha +n+1\right) }{n!}\ t_{k}\left( n,-\alpha \right) \end{array} \end{aligned} $$

(A.10)

and

$$\displaystyle \begin{aligned} \begin{array}{rcl} J_{-k-1} &\displaystyle =&\displaystyle J_{nn,\ -k-1}^{\alpha \alpha }=\int_{0}^{\infty }e^{-x}x^{\alpha -k-1}\ \left( L_{n}^{\alpha }\left( x\right) \right) ^{2}\ dx {} \notag\\ &\displaystyle =&\displaystyle \frac{\Gamma \left( \alpha -k\right) \Gamma \left( \alpha +n+1\right) }{n!\;\Gamma \left( \alpha +k+1\right) }\ t_{k}\left( n,-\alpha \right) \end{array} \end{aligned} $$

(A.11)

for 0 ≤ k < α. One can see that the positivity of these integrals is related to a nonstandard orthogonality relation for the corresponding Chebyshev polynomials of a discrete variable $t_{k}\left ( x,N\right )$ when the parameter takes negative integer values N = −α. Indeed, according to the method of [139] and [138], these polynomials are orthogonal with the discrete uniform distribution on the interval $\left [ -\alpha ,-1\right ]$ which contains all their zeros and, therefore, they are positive for all nonnegative values of their argument. The explicit representation (B.6) gives also a positive sum for all positive x and negative N.

Appendix B: Hypergeometric Series, Discrete Orthogonal Polynomials, and Useful Relations

This section contains some relations involving the generalized hypergeometric series, the Laguerre and Hahn polynomials, the spherical harmonics and Clebsch–Gordan coefficients, which are used throughout the paper.

The Laguerre polynomials are defined as [8, 138, 139, 184]

$$\displaystyle \begin{aligned} L_{n}^{\alpha }\left( x\right) =\frac{\Gamma \left( \alpha +n+1\right) }{ n!\;\Gamma \left( \alpha +1\right) }\ _{1}F_{1}\left( \begin{array}{c} -n \\ \alpha +1 \end{array} ;\ x\right) . {} \end{aligned} $$

(B.1)

(It is a consequence of Theorem 1.1.) The differentiation formulas [138, 139]

$$\displaystyle \begin{aligned} \frac{d}{dx}L_{n}^{\alpha }\left( x\right) =-L_{n-1}^{\alpha +1}\left( x\right) , {} \end{aligned} $$

(B.2)

$$\displaystyle \begin{aligned} x\frac{d}{dx}L_{n}^{\alpha }\left( x\right) =nL_{n}^{\alpha }\left( x\right) -\left( \alpha +n\right) L_{n-1}^{\alpha }\left( x\right) {} \end{aligned} $$

(B.3)

imply a recurrence relation

$$\displaystyle \begin{aligned} xL_{n-1}^{\alpha +1}\left( x\right) =\left( \alpha +n\right) L_{n-1}^{\alpha }\left( x\right) -nL_{n}^{\alpha }\left( x\right) . {} \end{aligned} $$

(B.4)

The simplest case of the connecting relation (c.f. [8] and [9]) is

$$\displaystyle \begin{aligned} L_{n}^{\alpha }\left( x\right) =L_{n}^{\alpha +1}\left( x\right) -L_{n-1}^{\alpha +1}\left( x\right) . {} \end{aligned} $$

(B.5)

The Hahn polynomials are [138, 139]

$$\displaystyle \begin{aligned} \begin{array}{rcl} h_{n}^{\left( \alpha ,\ \beta \right) }\left( x,N\right) =\left( -1\right) ^{n}\frac{\Gamma \left( N\right) \left( \beta +1\right) _{n}}{n!\;\Gamma \left( N-n\right) }\ _{3}F_{2}\left( \begin{array}{c} -n ,\ \alpha +\beta +n+1,\ -x \\ \beta +1 ,\quad 1-N \end{array} ;\ 1\right). \quad {} \end{array} \end{aligned} $$

(B.6)

(We usually omit the argument of the hypergeometric series ₃F ₂ if it is equal to one.) An asymptotic relation with the Jacobi polynomials is

$$\displaystyle \begin{aligned} \frac{1}{\left. \widetilde{N}\right. ^{n}}\;h_{n}^{\left( \alpha ,\ \beta \right) }\left( \frac{\widetilde{N}}{2}\left( 1+s\right) -\frac{\beta +1}{2} ,\;N\right) =P_{n}^{\left( \alpha ,\ \beta \right) }\left( s\right) +\text{O} \left( \frac{1}{\left. \widetilde{N}\right. ^{2}}\right) , {} \end{aligned} $$

(B.7)

where $\widetilde {N}=N+\left ( \alpha +\beta \right ) /2$ and N →∞; see [139] for more details.

Thomae’s transformation [16, 79] is

$$\displaystyle \begin{aligned} _{3}F_{2}\left( \begin{array}{c} -n ,\ a,\ b \\ c ,~d \end{array} ;\ 1\right) =\frac{\left( d-b\right) _{n}}{\left( d\right) _{n}}\ _{3}F_{2}\left( \begin{array}{c} -n ,\ c-a,\ b \\ c ,~b-d-n+1 \end{array} ;\ 1\right) {} \end{aligned} $$

(B.8)

with n = 0, 1, 2, … .

The summation formula of Gauss [8, 16, 79]

$$\displaystyle \begin{aligned} _{2}F_{1}\left( \begin{array}{c} a ,\ b \\ c \end{array} ;\ 1\right) =\frac{\Gamma \left( c\right) \Gamma \left( c-a-b\right) }{ \Gamma \left( c-a\right) \Gamma \left( c-b\right) },\qquad \text{Re}\left( c-a-b\right) >0. {} \end{aligned} $$

(B.9)

The gamma function is defined as [8, 66, 138]

$$\displaystyle \begin{aligned} \Gamma \left( z\right) =\int_{0}^{\infty }e^{-t}t^{z-1}\;dt,\qquad \text{Re}\ z>0. {} \end{aligned} $$

(B.10)

It can be continued analytically over the whole complex plane except the points z = 0, −1, −2, … at which it has simple poles. Functional equations are

$$\displaystyle \begin{aligned} \Gamma \left( z+1\right) =z\Gamma \left( z\right) , {} \end{aligned} $$

(B.11)

$$\displaystyle \begin{aligned} \Gamma \left( z\right) \Gamma \left( 1-z\right) =\frac{\pi }{\sin \pi z}, {} \end{aligned} $$

(B.12)

$$\displaystyle \begin{aligned} 2^{2z-1}\Gamma \left( z\right) \Gamma \left( z+1/2\right) =\sqrt{\pi }\Gamma \left( 2z\right) . {} \end{aligned} $$

(B.13)

The generating function for the Legendre polynomials and the addition theorem for spherical harmonics give rise to the following expansion formula [138, 189]

$$\displaystyle \begin{aligned} \frac{1}{\left| \boldsymbol{r}_{1}-\boldsymbol{r}_{2}\right| }=\sum_{l=0}^{\infty }\sum_{m=-l}^{l}\frac{4\pi }{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}}\ Y_{lm}\left( \theta _{1},\varphi _{1}\right) Y_{lm}^{\ast }\left( \theta _{2},\varphi _{2}\right) , {} \end{aligned} $$

(B.14)

where $r_{<}=\min \left ( r_{1},r_{2}\right )$ and $r_{>}=\max \left ( r_{1},r_{2}\right ) .$

The Clebsch–Gordan series for the spherical harmonics is [139, 148, 189]

$$\displaystyle \begin{aligned} \begin{array}{rcl} Y_{l_{1}m_{1}}\left( \theta ,\varphi \right) \ Y_{l_{2}m_{2}}\left( \theta ,\varphi \right) &\displaystyle =&\displaystyle \sum_{l=\left| l_{1}-l_{2}\right| }^{l_{1}+l_{2}}\sqrt{ \frac{\left( 2l_{1}+1\right) \left( 2l_{2}+1\right) }{4\pi \left( 2l+1\right) }} {} \notag\\ &\displaystyle &\displaystyle \times C_{l_{1}m_{1}l_{2}m_{2}}^{l,\ m_{1}+m_{2}}\ C_{l_{1}0l_{2}0}^{l,\ 0}\ Y_{l,m_{1}+m_{2}}\left( \theta ,\varphi \right) , \end{array} \end{aligned} $$

(B.15)

where $C_{l_{1}m_{1}l_{2}m_{2}}^{lm}$ are the Clebsch–Gordan coefficients. The special case l ₂ = 1 reads [71],

$$\displaystyle \begin{aligned} \begin{array}{rcl} -\sin \theta e^{i\varphi }\ Y_{l,\ m-1}&\displaystyle =\sqrt{\frac{\left( l+m\right) \left( l+m+1\right) }{\left( 2l+1\right) \left( 2l+3\right) }}\ Y_{l+1,\ m}-\sqrt{\frac{\left( l-m\right) \left( l-m-1\right) }{\left( 2l+1\right) \left( 2l-1\right) }}\ Y_{l-1,\ m}, {} \end{array} \end{aligned} $$

(B.16)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sin \theta e^{-i\varphi }\ Y_{l,\ m+1}&\displaystyle =\sqrt{\frac{\left( l-m\right) \left( l-m+1\right) }{\left( 2l+1\right) \left( 2l+3\right) }}\ Y_{l+1,\ m}-\sqrt{\frac{\left( l+m\right) \left( l+m+1\right) }{\left( 2l+1\right) \left( 2l-1\right) }}\ Y_{l-1,\ m}, {} \end{array} \end{aligned} $$

(B.17)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \cos \theta \ Y_{lm}&\displaystyle =\sqrt{\frac{\left( l+1\right) ^{2}-m^{2}}{\left( 2l+1\right) \left( 2l+3\right) }}\ Y_{l+1,\ m}+\sqrt{\frac{l^{2}-m^{2}}{\left( 2l-1\right) \left( 2l+1\right) }}\ Y_{l-1,\ m}, {} \end{array} \end{aligned} $$

(B.18)

where

$$\displaystyle \begin{aligned} \sqrt{\frac{8\pi }{3}}\ Y_{1,\ \pm 1}=\mp \sin \theta e^{\pm i\varphi },\qquad \sqrt{\frac{4\pi }{3}}\ Y_{10}=\cos \theta . {} \end{aligned} $$

(B.19)

These relations allow to prove (3.34) by a direct calculation.

The required identity (5.12) can be derived from the theory of classical polynomials in the following fashion. Let us start from the difference equation for the Hahn polynomials $y_{m}=h_{m}^{\left ( \alpha ,\ \beta \right ) }\left ( x,N\right )$ [139]:

$$\displaystyle \begin{aligned} \left( \sigma \left( x\right) \nabla +\tau \left( x\right) \right) \Delta y_{m}+\lambda _{m}y_{m}=0, {} \end{aligned} $$

(B.20)

where $\Delta f\left ( x\right ) =\nabla f\left ( x+1\right ) =f\left ( x+1\right ) -f\left ( x\right )$ and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma \left( x\right) &\displaystyle =&\displaystyle x\left( \alpha +N-x\right) , {} \notag\\ \tau \left( x\right) &\displaystyle =&\displaystyle \left( \beta +1\right) \left( N-1\right) -\left( \alpha +\beta +2\right) x, \notag \\ \lambda _{m} &\displaystyle =&\displaystyle m\left( \alpha +\beta +m+1\right) , \end{array} \end{aligned} $$

(B.21)

and use the familiar difference-differentiation formula:

$$\displaystyle \begin{aligned} \Delta h_{m}^{\left( \alpha ,\;\beta \right) }\left( x,N\right) =\left( \alpha +\beta +m+1\right) h_{m-1}^{\left( \alpha +1,\;\beta +1\right) }\left( x,N-1\right) . {} \end{aligned} $$

(B.22)

As a result,

$$\displaystyle \begin{aligned} \left( \sigma \left( x\right) \nabla +\tau \left( x\right) \right) h_{m-1}^{\left( \alpha +1,\;\beta +1\right) }\left( x,N-1\right) +mh_{m}^{\left( \alpha ,\;\beta \right) }\left( x,N\right) =0. {} \end{aligned} $$

(B.23)

Letting α = β and β →−1, one gets

$$\displaystyle \begin{aligned} \begin{array}{rcl} &&x\left( N-x-1\right) \nabla h_{m-1}^{\left( 0,\;0\right) }\left( x,N-1\right) =-m\lim_{\beta \rightarrow -1}h_{m}^{\left( \beta ,\;\beta \right) }\left( x,N\right) {} \notag\\ &&\qquad =\left( -1\right)^{m}m\left( m-1\right) \frac{\Gamma \left( N-1\right) }{\Gamma \left( N-m\right) }x\ _{3}F_{2}\left( \begin{array}{c} 1-m ,\ m,\ 1-x \\ 2 ,\quad 2-N \end{array} \right) \end{array} \end{aligned} $$

(B.24)

by (5.11). The last identity takes the form (5.12), if the Chebyshev polynomials of a discrete variable $h_{m-1}^{\left ( 0,\;0\right ) }\left ( x,N-1\right )$ are replaced by the corresponding generalized hypergeometric functions. (Use of (B.22) in (B.24) gives the special ₃F ₂ transformation.)

Appendix C: Dirac Matrices and Inner Product

We use the standard representations of the Dirac and Pauli matrices (3.3) and (3.4). The inner product of two Dirac (bispinor) wave functions

$$\displaystyle \begin{aligned} \psi =\left( \begin{array}{c} \boldsymbol{u}_{1} \\ \boldsymbol{v}_{1} \end{array} \right) =\left( \begin{array}{c} \psi _{1} \\ \psi _{2} \\ \psi _{3} \\ \psi _{4} \end{array} \right) ,\qquad \phi =\left( \begin{array}{c} \boldsymbol{u}_{2} \\ \boldsymbol{v}_{2} \end{array} \right) =\left( \begin{array}{c} \phi _{1} \\ \phi _{2} \\ \phi _{3} \\ \phi _{4} \end{array} \right) {} \end{aligned} $$

(C.1)

is defined as a scalar quantity

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left\langle \psi ,\ \phi \right\rangle &\displaystyle =&\displaystyle \int_{{\mathbb{R}}^{3}}\psi ^{\dagger }\phi \ dv=\int_{{\mathbb{R}}^{3}}\left( \boldsymbol{u}_{1}^{\dagger }{} \boldsymbol{u}_{2}+\boldsymbol{v}_{1}^{\dagger }{}\boldsymbol{v}_{2}\right) \ dv {} \notag\\ &\displaystyle =&\displaystyle \int_{{\mathbb{R}}^{3}}\left( \psi _{1}^{\ast }\phi _{1}+\psi _{2}^{\ast }\phi _{2}+\psi _{3}^{\ast }\phi _{3}+\psi _{4}^{\ast }\phi _{4}\right) \ dv \end{array} \end{aligned} $$

(C.2)

and the raised asterisk is used to denote the complex conjugate. The corresponding expectation values of a matrix operator A are given by

$$\displaystyle \begin{aligned} \langle A\rangle =\left\langle \psi ,\ A\psi \right\rangle . {} \end{aligned} $$

(C.3)

From this definition one gets

$$\displaystyle \begin{aligned} \langle r^{p}\rangle =A_{p},\qquad \langle \beta r^{p}\rangle =B_{p},\qquad \langle i\boldsymbol{\alpha }\mathbf{n}\beta r^{p}\rangle =-2C_{p}, {} \end{aligned} $$

(C.4)

where the integrals A _p, B _p, and C _p are given by (5.1)–(5.3), respectively.

Indeed, the first relation is derived, for example, in [183] and the second one can be obtained by integrating the identity

$$\displaystyle \begin{aligned} \begin{array}{rcl} r^{p}\psi ^{\dagger }\beta \psi &=&r^{p}\left( \boldsymbol{\varphi }^{\dagger },\ \boldsymbol{\chi }^{\dagger }\right) \left( \begin{array}{cc} \boldsymbol{1} & \boldsymbol{0} \\ \boldsymbol{0} & -\boldsymbol{1} \end{array} \right) \ \left( \begin{array}{c} \boldsymbol{\varphi} \\ \boldsymbol{\chi } \end{array} \right) =r^{p}\left( \boldsymbol{\varphi }^{\dagger },\ \boldsymbol{\chi }^{\dagger }\right) \ \left( \begin{array}{c} \boldsymbol{\varphi } \\ -\boldsymbol{\chi } \end{array} \right) {} \notag\\ &=&r^{p}\left( \boldsymbol{\varphi }^{\dagger }\boldsymbol{\varphi -\chi }^{\dagger } \boldsymbol{\chi }\right) =r^{p}\left( \mathcal{Y}^{\dagger }\mathcal{Y}\right) \left( F^{2}-G^{2}\right) \end{array} \end{aligned} $$

(C.5)

(we leave details to the reader) in a similar fashion.

In the last case, we start from the matrix identity

$$\displaystyle \begin{aligned} \left( \boldsymbol{\alpha} \mathbf{n}\right) \beta \psi =\left( \begin{array}{cc} \boldsymbol{0} & \boldsymbol{\sigma} \mathbf{n} \\ \boldsymbol{\sigma} \mathbf{n} & \boldsymbol{0} \end{array} \right) \left( \begin{array}{c} \boldsymbol{\varphi } \\ -\boldsymbol{\chi } \end{array} \right) =\left( \begin{array}{c} -\left( \boldsymbol{\sigma} \mathbf{n}\right) \boldsymbol{\ \chi} \\ \left( \boldsymbol{\sigma} \mathbf{n}\right) \boldsymbol{\ \varphi } \end{array} \right) {} \end{aligned} $$

(C.6)

and use the Ansatz [183]

$$\displaystyle \begin{aligned} \boldsymbol{\varphi }=\boldsymbol{\varphi }\left( \mathbf{r}\right) =\mathcal{Y} \left( \mathbf{n}\right) \ F\left( r\right) ,\qquad \boldsymbol{\chi }=\boldsymbol{ \chi }\left( \mathbf{r}\right) =-i\left( \left( \boldsymbol{\sigma} \mathbf{n}\right) \mathcal{Y}\left( \mathbf{n}\right) \right) \ G\left( r\right) , {} \end{aligned} $$

(C.7)

where n = r∕r and $\mathcal {Y}=\mathcal {Y}_{jm}^{\pm }\left ( \mathbf {n}\right )$ are the spinor spherical harmonics given by (3.12). As a result,

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} ir^{p}\psi {{}^{\dagger }}\left( \left( \boldsymbol{\alpha} \mathbf{n}\right) \beta \psi \right) &=& ir^{p}\left( \boldsymbol{\varphi }^{\dagger },\ \boldsymbol{\chi } ^{\dagger }\right) \left( \begin{array}{c} -\left( \boldsymbol{\sigma} \mathbf{n}\right) \ \boldsymbol{\chi } \\ \left( \boldsymbol{\sigma} \mathbf{n}\right) \ \boldsymbol{\varphi } \end{array} \right) \notag \\ &=&ir^{p}\left( F\mathcal{Y}^{\dagger },\ iG\mathcal{Y}^{\dagger } \left( \boldsymbol{\sigma} \mathbf{n}\right) \right) \left( \begin{array}{c} i\mathcal{Y}G \\ \left( \boldsymbol{\sigma} \mathbf{n}\right) \mathcal{Y}F \end{array} \right) \notag \\ & =&-r^{p}\left( \mathcal{Y}^{\dagger }\mathcal{Y}\right) FG-r^{p}\left( \mathcal{Y}^{\dagger }\left( \boldsymbol{\sigma} \mathbf{n}\right) ^{2}\mathcal{Y}\right) FG \notag \\ &=&-2r^{p}\left( \mathcal{Y}^{\dagger }\mathcal{Y}\right) FG \end{array} \end{aligned} $$

(C.8)

with the help of the familiar identity $\left ( \boldsymbol {\sigma } \mathbf {n}\right ) ^{2}= {\mathbf {n}}^{2}=\boldsymbol {1}.$ Integration over ${\mathbb {R}}^{3}$ in the spherical coordinates completes the proof.