Coulomb transition matrix at negative energy and integer values of interaction parameter

TL;DR

This paper derives analytical expressions for the Coulomb transition matrix at negative energy for both repulsive and attractive interactions, focusing on integer Coulomb parameters using stereographic projection.

Contribution

It introduces a novel method employing stereographic projection to analytically solve the Coulomb transition matrix at specific energy and parameter values.

Findings

Analytical formulas for repulsive Coulomb transition matrix at positive integer parameters.

Extension of the method to attractive Coulomb interaction at negative integer parameters.

Simplification of the Coulomb transition matrix calculation at negative energy.

Abstract

With the use of the stereographic projection of momentum space into the four-dimensional sphere of unit radius. the possibility of the analytical solution of the three-dimensional two-body Lippmann-Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has been also applied for the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

Equations

$$g^C(E)=g^0(E)+g^0(E)t^C(E)g^0(E),$$

$$<\mathbf{k}|t^C(E)|{\mathbf{k}}^{\prime }>=⟨\mathbf{k}\mid v^C\mid {\mathbf{k}}^{\prime }⟩+\int \frac{d{\mathbf{k}}^{\prime \prime }}{(2\pi {)}^3}⟨\mathbf{k}\mid v^C\mid {\mathbf{k}}^{\prime \prime }⟩\frac{1}{E-\frac{k^{\prime \prime 2}}{2\mu }}<{\mathbf{k}}^{\prime \prime }|t^C(E)|{\mathbf{k}}^{\prime }>,$$

$$<\mathbf{k}|v^C|{\mathbf{k}}^{\prime }>=\frac{4\pi q_1{q}_2}{\mid \mathbf{k}-{\mathbf{k}}^{\prime }{\mid }^2},$$

$$E=-\frac{{\hslash }^2{\kappa }^2}{2\mu },$$

$$<\mathbf{k}|t^C(E)|{\mathbf{k}}^{\prime }>=\frac{2\pi q_1{q}_2\eta }{k{k}^{\prime }}\left[\frac{1}{{\sin }^2\frac{\omega }{2}}-\frac{4\gamma }{\sin \omega }\sum _{n=1}^{\infty }\frac{\sin n\omega }{n+\gamma }\right],$$

$$<\mathbf{k}|t^C(E)|{\mathbf{k}}^{\prime }>=\frac{2\pi q_1{q}_2\eta }{k{k}^{\prime }}\left[\frac{1}{{\sin }^2\frac{\omega }{2}}-4\gamma {\int }_0^{1}d\rho \frac{{\rho }^{\gamma }}{{\rho }^2-2\cos \omega \cdot \rho +1}\right],$$

$$<\mathbf{k}|t^C(E)|{\mathbf{k}}^{\prime }>=\frac{2\pi q_1{q}_2\eta }{k{k}^{\prime }}\{\frac{1}{{\sin }^2\frac{\omega }{2}}-\frac{2}{\sin \omega }[\pi \gamma \cos \gamma \omega +\gamma \sin 2\gamma \omega \cdot \ln \left(\sin \frac{\omega }{2}\right)$$

$$-2\pi \gamma c(\gamma )\cot \gamma \pi \cdot \sin \gamma \omega -\gamma \cos \gamma \omega \cdot x_{\gamma }(\omega )-2{\gamma }^2\sin \gamma \omega \cdot y_{\gamma }(\omega )]\},$$

$$\eta =\frac{2{\kappa }^{2}k{k}^{\prime }}{\left(k^2+{\kappa }^2\right)\left(k^{\prime 2}+{\kappa }^2\right)},$$

$$\gamma =\frac{\mu g_1{q}_2}{{\hslash }^2\kappa }$$

$${\sin }^2\frac{\omega }{2}=\frac{{\kappa }^2\mid \mathbf{k}-{\mathbf{k}}^{\prime }{\mid }^2}{(k^2+{\kappa }^2)(k^{\prime 2}+{\kappa }^2)},0\le \omega <\pi .$$

$$x_{\gamma }(\omega )={\int }_0^{\omega }d\phi \sin \gamma \phi \cot \frac{\phi }{2},y_{\gamma }(\omega )={\int }_{\omega }^{\pi }d\phi \sin \gamma \phi \ln \left(\sin \frac{\phi }{2}\right),$$

$$c(\gamma )=\frac{1}{2}\left[1-\frac{1}{\pi }{x}_{\gamma }(\pi )\right].$$

$$E_{\gamma }=-\frac{{\hslash }^2{\kappa }^2}{2\mu }=-\frac{\mu (q_1{q}_2{)}^2}{2{\hslash }^2{\gamma }^2}.$$

$$E_n=-\frac{\mu (q_1{q}_2{)}^2}{2{\hslash }^2{n}^2},n=1,2,\cdots ,$$

$${\kappa }_n=\frac{\mu \mid q_1{q}_2\mid }{{\hslash }^{2}n}.$$

$${\gamma }_n=\frac{q_1{q}_2}{\mid q_1{q}_2\mid }n$$

$$x_n(\pi )=\pi ,x_{-n}(\pi )=-\pi ,$$

$$c(n)=0,c(-n)=1.$$

$${\left[\frac{2\pi \gamma c(\gamma )}{\tan \gamma \pi }\right]}_{\gamma \to n}=2n{c}^{\prime }(n)\equiv {\rho }_n,$$

$${\rho }_n=(-1{)}^n-2n\ln 2-2n\sum _{m=1}^n\frac{(-1{)}^m}{m}.$$

$$x_1(\omega )=\omega +\sin \omega ,y_1(\omega )=-{\cos }^2\frac{\omega }{2}-{\sin }^2\frac{\omega }{2}\ln {\sin }^2\left(\frac{\omega }{2}\right),$$

$$c(1)=0,{\rho }_1=1-2\ln 2.$$

$$⟨\mathbf{k}\mid t(E)\mid {\mathbf{k}}^{\prime }⟩=\frac{2\pi q_1{q}_2\eta }{k{k}^{\prime }}\left\{\frac{1}{{\sin }^2\frac{\omega }{2}}-2(\pi -\omega )\cot \omega -2\ln {\sin }^2\frac{\omega }{2})-4\ln 2\right\}.$$

$$x_2(\omega )=\omega +2\sin \omega +\frac{1}{2}\sin 2\omega ,y_2(\omega )=-{\cos }^4\frac{\omega }{2}-\frac{1}{2}{\sin }^2\frac{\omega }{2}\ln {\sin }^2\left(\frac{\omega }{2}\right),$$

$$c(2)=0,{\rho }_2=3-4\ln 2$$

$$⟨\mathbf{k}\mid t(E)\mid {\mathbf{k}}^{\prime }⟩=\frac{2\pi q_1{q}_2\eta }{k{k}^{\prime }}\left\{\frac{1}{{\sin }^2\frac{\omega }{2}}-4(\pi -\omega )\frac{\cos 2\omega }{\sin \omega }-8\cos \omega \cdot \ln {\sin }^2\frac{\omega }{2})-16\ln 2\cdot \cos \omega -8\right\}.$$

$$x_3(\omega )=\omega +2\sin \omega +\frac{1}{3}\sin 3\omega ,$$

$$y_3(\omega )=-\left(\frac{16}{3}{\sin }^6\frac{\omega }{2}-8{\sin }^4\frac{\omega }{2}+3{\sin }^2\frac{\omega }{2}\right)\ln {\sin }^2\frac{\omega }{2}$$

$$+\frac{16}{9}{\sin }^6\frac{\omega }{2}-4{\sin }^4\frac{\omega }{2}+3{\sin }^2\frac{\omega }{2}-\frac{7}{9},c(3)=0,{\rho }_3=4-6\ln 2,$$

Full text

Coulomb transition matrix at negative energy and

integer values of interaction parameter

V. F Kharchenko

Bogolyubov Institute for Theoretical Physics, National Academy of Sciences

of Ukraine, UA - 03143, Kyiv, Ukraine

E-mail: vkharchenko@bitp.kiev.ua

Abstract

With the use of the stereographic projection of momentum space into the four-dimensional sphere of unit radius. the possibility of the analytical solution of the three-dimensional two-body Lippmann-Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has been also applied for the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

Keywords: three-dimension Coulomb transition matrix, Lippmann-Schwinger equation, integer values of Coulomb parameter, analytical solution

1. Introduction

It is known that the Coulomb transition matrix for two charged particles, which interact with each other, $t^{C}(E)$, as well as connected with it the Coulomb Green function

[TABLE]

where $g^{0}(E)$ is the free Green’s function ($E$ is the energy of the system), contain all the information about the two-body system. The two-body Coulomb transition matrix off the energy shell is an important quantity in both the atomic and nuclear physics. It immediately appears in the integral kernels of the Faddeev equations [1] describing the three-particle system with two or all three particles of which are charged.

In the literature, a number of the different representations of the Coulomb Green’s function (see the review [2]). The representation, which allows explicitly the known symmetry of the Coulomb system in the four-dimensional Euclidean space, firstly revealed by Fock [3], is particularly perspective. With the help of the stereographic projection of the momentum space onto the four-dimensional sphere of the unit radius, a single-parameter integral representation of the three-dimensional Coulomb Green’s function has been derived in the papers by Bratsev and Trifonov [4] and Schwinger [5]. With the use of the Fock symmetry, the general expressions for the three-dimensional Coulomb transition matrix with explicitly separated singularities in the transfer momentum and the energy have been derived in the papers [6] (in the case of the negative energy) and [7] (in the case of zero and positive energies).

The possibility of the derivation of a simple analytical expression for the partial Coulomb transition matrix, which obey the Lippman-Schwinger equation with the energy of the ground bound state of the two-body complex, has been firstly demonstrated in the papers [8] (in the case of the attractive interaction with the Coulomb parameter $\gamma=-1$) and [9] (in the case of the repulsive interaction with the parameter $\gamma=1$). The analytical expressions for the two-particle partial Coulomb transition matrices at the energy of the first-excited state ($\gamma=2$ and $\gamma=-2$) have been obtained in [10].

In this paper, leaning upon the formalisms developed in [5] and [6], we firstly find the possibility of the analytical solution of the Lippman-Schwinger equation for the three-dimensional Coulomb transition matrices in the case of the repulsive interaction at integer values of the Coulomb parameter $\gamma$.

The worked out method has also been applied for the generalized Coulomb transition matrix in the case of the attractive Coulomb interaction and integer values of the Coulomb parameter.

In the Section 2, the basic formulas for the three-dimensional Coulomb transition matrices obtained with the use of the Fock method of the stereographic projection are presented. The Section 3 is devoted to the derivation of the analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction with positive integer Coulomb parameter. In the Section 4, analytical expressions for the generalized Coulomb transition matrix with negative integer values of the Coulomb parameter. he ection 5 is devoted to the discussion and conclusions.

2. Fock method

In the momentum representation the three-dimensional Coulomb transition matrix $<{\bf k}|t^{C}(E)|{\bf k}^{\prime}>$ satisfies the Lippmann-Schwinger equation

[TABLE]

where $E$, ${\bf k}$ and ${\bf k}{\prime}$ are the energy and the momenta of the relative motion of two particles, $\mu$ is the reduced mass of the particles, $\langle{\bf k}\mid v^{C}\mid{\bf k}^{\prime}\rangle$ is the matrix of the Coulomb interaction of the particles 1 and 2,

[TABLE]

$q_{i}$ is the charge of the particle. In this paper we restrict our consideration of the two-particle system with the negative energy

[TABLE]

$\hbar$ is the reduced Planck constant.

Using the Fock method of the stereographic projection of the momentum space onto the four-dimension sphere with the unit radius [3], the solution of the integral equation for the three-dimensional Coulomb transition matrix at the negative energy ($E<0$)(2) can be written in the form of the sum [4]

[TABLE]

the integral representation [4,5]

[TABLE]

and with explicitly separated singularities in the transfer momentum and the energy [6]

[TABLE]

[TABLE]

where

[TABLE]

$\gamma$ is the known dimensionless Coulomb parameter

[TABLE]

and the variable quantity $\omega$ denotes the angle between two vectors in the four-dimensional space introduced by Fock [3],

[TABLE]

The functions $x_{\gamma}(\omega)$, $y_{\gamma}(\omega)$ and $c(\gamma)$ in (7) are given by the expressions

[TABLE]

[TABLE]

The energy $E_{\gamma}$, which corresponds to the given Coulomb parameter $\gamma$ (9),is equal to

[TABLE]

If the Coulomb potential is attractive (the particles have the charges of opposite signs, $g_{1}q_{2}<0$), the Coulomb parameter $\gamma$ takes the negative value, $\gamma<0$. To the spectrum of the bound states of the two-particle system with the energies

[TABLE]

correspond the negative integer values $\gamma$ ($\gamma=-n,\;n=1,2,3,...$) and, according to (4), the values

[TABLE]

The values of the Coulomb parameter (9) at $\kappa=\kappa_{n}$ is integer

[TABLE]

• positive for the repulsive Coulomb interaction ($q_{1}q_{2}>0$, $\gamma_{n}=n$) and negative for the attractive interaction ($q_{1}q_{2}<0$, $\gamma_{n}=-n$).

In view of that in the expression (11) for $c(\gamma)$ the quantity $x_{\gamma}(\pi)$ with the integer values of $\gamma$ takes the values

[TABLE]

we find that in the expression (7)

[TABLE]

Hence, in the case of the attractive Coulomb interaction ($\gamma<0$), the third term in the square brackets of the expression (7) through the presence of the function $\cot\gamma\pi$ contains the singularities at $\gamma=\gamma_{n}$, which correspond to the energies of bound states $E=E_{n}$.

If the Coulomb interaction is repulsive ($\gamma>0$), the expression for the Coulomb transition matrix (7) does not contain singularities at the energies of bound states. Evaluating in this case in the third term of the expression in (7) the indefinite form of the type $\frac{0}{0}$ at $\gamma=\gamma_{n}$ according to the l’Hospital rule, we find

[TABLE]

where

[TABLE]

3. Three-dimensional Coulomb transition matrix in the case of positive integer value of the Coulomb parameter

Using the formula (7), consider the expression for the three-dimensional Coulomb transition matrix in the case of the particles with the charges of the same sign at the energy $E=E_{n}$ (13). In this case the parameter $\gamma$ takes the positive integer values (15).

When $\gamma=1$ we have in accordance with (11)

[TABLE]

[TABLE]

The three-dimensional transition matrix then takes the simple analytical form

[TABLE]

In an analogous way, using (11) with $\gamma=2$ we find

[TABLE]

[TABLE]

and the expression for the three-dimensional Coulomb transition matrix at $E=E_{2}$ in the form

[TABLE]

At $\gamma=3$ we have

[TABLE]

[TABLE]

[TABLE]

and the corresponding Coulomb t-matrix at $E=E_{3}$ is equal to

[TABLE]

[TABLE]

4. Generalized Coulomb transition matrix in the case of negative integer Coulomb parameter $\gamma=-1$

In the case of the system with the attractive Coulomb interaction, of special interest is known as the generalized Coulomb Green function [4,11] at the energy of the bound ground state $E=E_{1}=-b_{1}$, $\tilde{g}^{C}(E_{1})$. It related with the generalized Coulomb transition matrix $\tilde{t}^{C}(E_{1})$ by the relation

[TABLE]

The expression for $\tilde{t}^{C}(E_{1})$ follows from the expression (5) for $t^{C}(E_{1})$ excluding the singular term at $n=1$, which describes the contribution from ground state,

[TABLE]

or, in view of Eq. (6)

[TABLE]

Writing $(1+\gamma)^{-1}$ inthe form of the definite integral

[TABLE]

we obtain the following expression for the generalized transition matrix

[TABLE]

At the energy of the ground bound state $E=-b_{1}$ we have $\gamma_{1}=-1$ and the integral in (30) is equal to

[TABLE]

As a result, for the generalized Coulomb transition matrix at the energy of the ground bound state (and the corresponding values $\kappa=\kappa_{1},\gamma=\gamma_{1},\omega=\omega_{1}$) we find

[TABLE]

5. Discussion and conclusions.

The study of properties of the two-particle Coulomb transition matrix is the topical problem, which arises in both the atomic and nuclear physics in connection with formulation and solution of the equations describing few-body systems which contain the charged particles. A knowledge of the two-particle Coulomb transition matrix is necessary in particular to determine the electric $2^{\lambda}$-pole polarizabilities of the two-body atomic and nuclear systems [12, 13].

In specific cases, the analytical solution of the Lippmann-Schwinger equation for the partial two-body Coulomb transition matrices has been realizeed by us previously using the symmetry of the Coulomb systems in the Fock four-dimensional space - in [8,9] (the system with unlike charges) and in [10] (the system with like charges).

In the present paper, it has been firstly shown the possibility of the analytical solution of the equationn for the three-dimensional Coulomb transition matrix in the case of the repulsive interaction with the positive integer values of the Coulomb parameter.

The obtained analytical expressions for the three-dimensional Coulomb transition matrix (21), (23) and (25) are in agreement with the corresponding formulae for the partial Coulomb transition matrices

[TABLE]

where $P_{l}(x)$ is the Legendre polynom,

[TABLE]

obtained by us earlier [10] in the case of the repulsive Coulomb interaction.

In the case of the attractive interaction, it has been performed analytical solution of the equation for the generalized Coulomb transition matrix with the negative integer value of the Coulomb parameter $\gamma=-1$.

Note, that that the result (31) obtained by us for the generalized Coulomb transition matrix $<{\bf k}|{\tilde{t}}^{C}(-b_{1})|{\bf k}^{\prime}>$ substantially differs from the result off Bratsev and Trifonov [4] as a consequence of the error when integrating in [4].

Acknowledgment

The present work was partially supported by the National Academy of Sciences of Ukraine (project No. 0117U00237) and by the Program of Fundamental Research of the Department of Physics and Astronomy of NASU (project No. 0117U00240).

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