1 INTRODUCTION

The fundamental quantum Coulomb problem, which allows one to calculate the spectrum of a system of two opposite charges, is still relevant in the qu-antum theory [1–4]. Such founders of twentieth century physics as N. Bohr, A. Sommerfeld, W. Pauli, E. Schrödinger, and V. Fock contribute to it. The introduction to the theory of atomic spectra begins with it, and it is thoroughly studied using the theory of special functions. Due to its simplicity and the underlying SO(4) group of rotations about a fixed point in four-dimensional (4D) Euclidean space, this problem is an extremely useful and fine tool of theoretical physics for constructing various concepts [5–7].

The transition from the coordinate space to the momentum space is exclusively efficient in theoretical physics particularly in quantum electrodynamics, as far as local differential operators are transformed to polynomials and considered transformations are reduced to algebraic. The Coulomb problem is in this case of a particular significance. The Schrödinger equation in the momentum space becomes an integral equation. Fock applied the stereographic projection to the three-dimensional (3D) momentum space map** it to the 3D sphere embedded in the 4D momentum space [8–10]. The integral Fock equation is transformed to the equation for spherical functions on the 3D sphere in the 4D space, which can be interpreted conditionally as the free motion of a particle on the 3D sphere.

Let us recall the background proceeding Fock’s achievement. Two classical vector integrals, the angular momentum and the Laplace–Runge–Lenz vector, in quantum mechanics correspond to vector operators that commute with the energy operator, i.e., with the Hamiltonian. An analysis of their commutators in [11] shows that they generate a Lie algebra (a linear space with a commutation operation) coinciding with the Lie algebra of infinitesimal rotations in the 4D space [1, 3].

For physicists, this correspondence means that some transformation of variables and operators transforms the original quantum Coulomb problem to the state of a particle on the 3D sphere embedded in the 4D space. The energy operator is then invariant under rotations of the 3D sphere, thus a vector creation operator arises naturally, which was developed for the 2D sphere in [13].

In [14], the Schrödinger equation was transformed so that the radii of all orbits are reduced to unity. This means that the problem is reduced to the quantization of the charge \(Z = n\). The Schrödinger equation is squared; the resulting operator is no longer Hermitian but holds the necessary physical properties. After that, the transition to the momentum space with the locality property can be performed. As a result, a differential equation for eigenfunctions appears in the momentum space. In this work, the physical meaning of this differential equation is revealed, its solutions are determined, and the simple differential Runge–Lenz operator in the momentum space is obtained. The application of the Runge–Lenz operator is thus simplified considerably. The relation between this operator and the infinitesimal rotation operator of the 3D Fock sphere is found as well.

Consequently, the quantum Coulomb problem in the momentum space is adequate and is very useful for theories including the Coulomb interaction and perturbation theory together.

2 FOCK THEORY

In atomic units, where \(\frac{{{{Z}^{2}}m{{e}^{2}}}}{{{{\hbar }^{2}}}}\) is the energy unit and the Bohr radius \({{a}_{B}} = \frac{{{{\hbar }^{2}}}}{{Zm{{e}^{2}}}}\) is the length unit, the Schrödinger equation for eigenfunctions has the form

$$\left( { - \frac{1}{2}\Delta - \frac{1}{2}} \right){{\Psi }_{{nl}}} = - \frac{1}{{2{{n}^{2}}}}{{\Psi }_{{nl}}}.$$
(1)

Further, it is convenient to convert each orbit radius naB to single radius [1], i.e., to substitute the position vector \({\mathbf{x}}{\kern 1pt} ' = \frac{{\mathbf{x}}}{n}\) for each eigenfunction. Thus, Eq. (1) takes the simple form

$$( - \Delta + 1){{\Psi }_{{nl}}} = \frac{{2n}}{r}{{\Psi }_{{nl}}},$$
(2)

where x and r are again the position vector and its magnitude, respectively. Then, eigenfunctions in the momentum representation have the scaled argument \({\mathbf{p}}{\kern 1pt} ' = n{\mathbf{p}}\).

For the transition to the momentum space, eigenfunctions of Eq. (2) are represented as a convolution in the momentum (\(\hbar = 1\))

$${{\Psi }_{{nl}}}({\mathbf{x}}) = \frac{1}{{{{{(2\pi )}}^{3}}}}\int {{a}_{{nl}}}({\mathbf{p}}){{e}^{{i({\mathbf{px}})}}}{{d}^{3}}{\mathbf{p}}.$$
(3)

Since the potential \(\frac{1}{r}\) is transformed in the momentum space to \(\frac{{4\pi }}{{{{p}^{2}}}}\), the Schrödinger equation becomes nonlocal and has the form

$$({{p}^{2}} + 1){{a}_{{nl}}}({\mathbf{p}}) - \frac{{2n}}{{2{{\pi }^{2}}}}\int \frac{{{{a}_{{nl}}}({\mathbf{p}}'){{d}^{3}}{\mathbf{p}}'}}{{{\text{|}}{\mathbf{p}} - {\mathbf{p}}'{\kern 1pt} {{{\text{|}}}^{2}}}} = 0.$$
(4)

Fock applied the stereographic projection [11], which maps the 3D plane to a 3D sphere embedded in the 4D momentum space, to this equation. The relations of the coordinates of the 4D space on the 3D sphere to the momentum and the constraint on these coordinates have the formFootnote 1

$$\boldsymbol{\xi} = \frac{{2{\mathbf{p}}}}{{(1 + {{{\mathbf{p}}}^{2}})}},\quad {{\xi }_{0}} = \frac{{({{{\mathbf{p}}}^{2}} - 1)}}{{({{{\mathbf{p}}}^{2}} + 1)}},\quad {{\boldsymbol{\xi} }^{2}} + \xi _{0}^{2} = 1.$$
(5)

In the new variables, taking into account the factor chosen by Fock for the function \({{a}_{{nl}}}({\mathbf{p}})\), the eigenfunction becomes

$${{b}_{{nl}}}(\boldsymbol{\xi} ,{{\xi }_{0}}) = ({{{\mathbf{p}}}^{2}} + {{1)}^{2}}{{a}_{{nl}}}({\mathbf{p}}).$$
(6)

It is essential here that the stereographic projection is a conformal map. In this case, angles between intersecting curves are conserved. Just for this reason, the stereographic projection was invented for nautical charts. The metric on the sphere is expressed in terms of the coordinates on the 3D p plane as

$$\frac{4}{{{{{({{{\mathbf{p}}}^{2}} + 1)}}^{2}}}}(d{{{\mathbf{p}}}^{2}}).$$
(7)

Hence, the contraction coefficient for elements of the momentum space is \((1 + {{{\mathbf{p}}}^{2}}){\text{/}}2\). The volume element in Eq. (4) is expressed in terms of the 3D surface el-ement:

$${{d}^{3}}{\mathbf{p}} = \frac{1}{8}{{(1 + {{{\mathbf{p}}}^{2}})}^{3}}d{{S}_{3}}.$$
(8)

The kernel of the integral can be (very fortunately but not obviously) transformed as

$$\frac{1}{{{\text{|}}{\mathbf{p}} - {\mathbf{p}}'{\kern 1pt} {{{\text{|}}}^{2}}}} = \frac{2}{{({{{\mathbf{p}}}^{2}} + 1)}}\frac{1}{{[(\boldsymbol{\xi} - \boldsymbol{\xi} {{{')}}^{2}} + {{{({{\xi }_{0}} - \xi _{0}^{'})}}^{2}}]}}\frac{2}{{({\mathbf{p}}{{'}^{2}} + 1)}},$$
(9)

which does not follow from the conformal property.

Now, substituting Eqs. (6), (8), and (9) into integral equation (4), we obtain

$${{b}_{{nl}}}(\boldsymbol{\xi} ,{{\xi }_{0}}) - \frac{n}{{2{{\pi }^{2}}}}\int \frac{{{{b}_{{nl}}}(\boldsymbol{\xi} ',\xi _{0}^{'})dS_{3}^{'}}}{{[(\boldsymbol{\xi} - \boldsymbol{\xi} {{{')}}^{2}} + {{{({{\xi }_{0}} - \xi _{0}^{'})}}^{2}}]}} = 0.$$
(10)

As Fock noted, it is the equation for spherical functions on the 3D sphere [15].

Solutions needed in physics are proportional to (ordinary) 2D spherical functions:

$${{Y}_{{n,l}}}(\theta ,\phi )P_{{n - 1 - l}}^{{l + 1}}({{\zeta }_{0}}),$$
(11)

where the second factor is the Gegenbauer polynomial [15] whose arguments are the coordinates on the Fock sphere related to the momenta by Eq. (5). Therefore, Fock found for the first time the general formula for eigenfunctions in the momentum space.

3 DIFFERENTIAL FORM OF THE SCHRÖDINGER EQUATION IN THE MOMENTUM SPACE

3.1 Derivation

Following [13], we multiply Eq. (2), where the radius of the orbit (when multiplied by n) is reduced to unity, by r and square both sides:

$$r( - \Delta + 1)r( - \Delta + 1){{\Psi }_{{nl}}} = 4{{n}^{2}}{{\Psi }_{{nl}}}.$$
(12)

Hence,

$$[{{r}^{2}}{{(\Delta - 1)}^{2}} + 2({{\hat {l}}_{{\mathbf{r}}}} + 1)(\Delta - 1)]{{\Psi }_{{nl}}} = 4{{n}^{2}}{{\Psi }_{{nl}}},$$
(13)

where \({{\hat {l}}_{{\mathbf{r}}}} = (\mathbf{r}\nabla )\) is the “degree operator.” According to the Euler theorem, which is valid for each homogeneous function (e.g., for \(\frac{1}{r}\) and its degrees) rather than only for polynomials, the operator multiplies a homogeneous polynomial by its degree.

We pass to the new function

$${{\Phi }_{{nl}}} = (\Delta - {{1)}^{2}}{{\Psi }_{{nl}}},$$

which corresponds to multiplying the spectrum by \({{({{p}^{2}} + 1)}^{2}}\) (see the Fock method in Eq. (6)). For this, we apply the operator \({{(\Delta - 1)}^{2}}\) to Eq. (13) from the left and swap the operators \({{\hat {l}}_{{\mathbf{x}}}}\) and Δ. As a result, we obtain the following equation for the function \({{\Phi }_{{nl}}}(\mathbf{x})\):

$$[(\Delta - {{1)}^{2}}{{r}^{2}} + 2(\Delta - 1)({{\hat {l}}_{{\mathbf{r}}}} + 3)]{{\Phi }_{{nl}}} = 4({{n}^{2}} - 1){{\Phi }_{{nl}}}.$$

We can now pass to the spectra \({{a}_{{nl}}}({\mathbf{p}})\) and \({{b}_{{nl}}}({\mathbf{p}})\) by substituting \({{\nabla }_{{\mathbf{r}}}}\) for \(i\mathbf{p}\) and x for \(i{{\nabla }_{{\mathbf{p}}}}\),

$$\left[ { - \frac{{{{{({{p}^{2}} + 1)}}^{2}}}}{4}{{\Delta }_{{\mathbf{p}}}} + \frac{{({{p}^{2}} + 1)}}{2}{{{\hat {l}}}_{{\mathbf{p}}}}} \right]{{b}_{{nl}}} = ({{n}^{2}} - 1){{b}_{{nl}}},$$
(14)

where \({{\hat {l}}_{{\mathbf{p}}}} = ({\mathbf{p}}{{\nabla }_{{\mathbf{p}}}})\) is the operator multiplying any polynomial by its degree and bnl(p) = (p2 + 1)2anl(p). Instead of integral Eq. (10), we obtain the equation that is no more complicated than the Schrödinger equation in the coordinate space because it is similar to the Schrödinger equation with the sum of two potentials. It is now necessary to reveal its symmetry properties.

3.2 Solution

We seek a solution of the equation in the product form

$${{b}_{{nl}}}({\mathbf{p}}) = {{Y}_{l}}({\mathbf{p}})\frac{1}{{{{{({{p}^{2}} + 1)}}^{l}}}}{{P}_{k}}\left( {\frac{1}{{({{p}^{2}} + 1)}}} \right),$$
(15)

where \({{Y}_{l}}({\mathbf{p}})\) is a solid spherical harmonic (which is a homogeneous polynomial) and \({{P}_{k}}(u)\) is a degree-k polynomial. We use the following properties valid for solid spherical harmonics:

$$\begin{gathered} \Delta {{Y}_{l}}({\mathbf{p}}) = 0,\quad {{Y}_{l}}(c{\mathbf{p}}) = {{c}^{l}}{{Y}_{l}}({\mathbf{p}}), \\ ({\mathbf{p}}\nabla ){{Y}_{l}}({\mathbf{p}}) = l{{Y}_{l}}({\mathbf{p}}). \\ \end{gathered} $$
(16)

The solution of Eq. (16) is the polynomial \({{P}_{k}}(u)\) coinciding with the Gauss functionFootnote 2

$$F(\alpha ,\beta ,\gamma ,u) = 1 + \frac{{\alpha \beta }}{\gamma }\frac{u}{{1!}} + \frac{{\alpha (\alpha + 1)\beta (\beta + 1)}}{{\gamma (\gamma + 1)}}\frac{u}{{2!}} + ...,$$

where

$$\alpha = - k,\quad \beta = 2l + k + 2,\quad \gamma = l + \frac{3}{2},$$
$$k = n - l - 1,\quad u = \frac{1}{{({{p}^{2}} + 1)}}.$$
(17)

We recall that the transition to the real spectrum requires multiplying the argument p by n.

The found system of solutions (15) exactly corresponds to the Fourier transforms of solutions of the Schrödinger equation. This follows from the coincidence of their angular dependences (the orbital quantum number l, as well as the angular momentum operator, is the same in the coordinate and momentum spaces). In addition, according to the theory of special functions [15], the coordinate substitution (5) transforms the Gauss function to the Gegenbauer polynomial, i.e., to the Fock solution (11).

Examples.

(a) \(n = 2\), \(l = 0\), and \(k = 1\) (isotropic state). We double the momentum, returning to a radius of 2:

$${{a}_{{20}}}(\mathbf{p}) = ({\text{const}})\frac{1}{{{{{(1 + 4{{p}^{2}})}}^{2}}}}\left( {1 - \frac{2}{{(1 + 4{{p}^{2}})}}} \right).$$

The derivation of this formula using the Hankel transforms is rather cumbersome.

(b) \(n = l + 1\) and \(k = 0\) (l is maximal):

$${{\Psi }_{{n(n - 1)}}}(\mathbf{x}) = {{Y}_{{(n - 1)}}}(\mathbf{x}){{e}^{{ - r}}},$$
$${{a}_{{nl}}}(\mathbf{p}) = \frac{{{{Y}_{l}}(\mathbf{p})}}{{{{{(1 + {{p}^{2}})}}^{{(2 + l)}}}}}.$$

Returning to the physical argument \(n{\mathbf{p}}\) and using the homogeneity of the polynomial \({{Y}_{l}}({\mathbf{p}})\), we obtain

$${{a}_{{nl}}}({\mathbf{p}}) = ({\text{const}})\frac{{{{Y}_{{(n - 1)}}}({\mathbf{p}})}}{{{{{(1 + {{n}^{2}}{{p}^{2}})}}^{{(n + 1)}}}}}.$$

4 RUNGE–LENZ OPERATOR IN THE MOMENTUM SPACE

The Runge–Lenz operator \({{\hat {R}}_{{\mathbf{r}}}}\) in the dimensionless coordinates is equal to [1]

$${{\hat {R}}_{{\mathbf{r}}}} = \frac{{\mathbf{r}}}{r} - \frac{1}{2}([{\mathbf{\hat {p}\hat {L}}}] - [{\mathbf{\hat {L}\hat {p}}}]).$$
(18)

Passing to unit radius, we have to multiply the operator given by Eq. (18) by n for the commutation rules to correspond to the SO(4) group [1, 11]. Calculating the vector products in Eq. (18), we obtain the normalized operator

$${{\hat {A}}_{{\mathbf{r}}}} = \frac{{n{\mathbf{r}}}}{r} - \frac{1}{2}([{\mathbf{\hat {p}\hat {L}}}] - [{\mathbf{\hat {L}\hat {p}}}]) = \frac{{n{\mathbf{r}}}}{r} + {\mathbf{r}}\Delta - ({{\hat {l}}_{{\mathbf{r}}}} + 1)\nabla ,$$
(19)

where \({{\hat {l}}_{{\mathbf{r}}}} = ({\mathbf{r}}\nabla )\).

The transition to the momentum space gives an integral operator because it involves the potential \(\frac{1}{r}\). This is why the Runge–Lenz operator historically was not introduced [1, 3].

We use the method that allowed me in [13] to obtain a differential equation in the momentum space. The substitution of \(\frac{{n{\mathbf{r}}}}{r}\) expressed from Eq. (2) into Eq. (19) gives

$${{\hat {A}}_{{\mathbf{r}}}} = {\mathbf{r}}\frac{{(\Delta + 1)}}{2} - ({{\hat {l}}_{{\mathbf{r}}}} + 1)\nabla .$$
(20)

The Fourier transform of Eq. (20) with the substitutions

$${\mathbf{r}} \to i{{\nabla }_{{\mathbf{p}}}},\quad {{\nabla }_{{\mathbf{r}}}} \to i{\mathbf{p}},\quad {{\hat {l}}_{{\mathbf{r}}}} \to - ({{\hat {l}}_{{\mathbf{p}}}} + 3),$$
(21)

yields the modified the Runge–Lenz operator in the momentum space,

$${{\hat {A}}_{{\mathbf{p}}}} = i({{\hat {l}}_{{\mathbf{p}}}} + 1){\mathbf{p}} - \frac{{({{{\mathbf{p}}}^{2}} - 1)}}{2}i{{\nabla }_{{\mathbf{p}}}},$$
(22)

acting on functions \({{a}_{{nl}}}({\mathbf{p}})\). To compare with the Fock theory, we have to pass to the space of functions:

$$\begin{gathered} {{{\hat {A}}}_{{\mathbf{p}}}} = ({{p}^{2}} + {{1)}^{2}}i\left[ {({{{\hat {l}}}_{{\mathbf{p}}}} + 1){\mathbf{p}} - \frac{{({{{\mathbf{p}}}^{2}} - 1)}}{2}{{\nabla }_{{\mathbf{p}}}}} \right]\frac{1}{{{{{({{p}^{2}} + 1)}}^{2}}}} \\ = i{\mathbf{p}}{{{\hat {l}}}_{{\mathbf{p}}}} - \frac{{({{p}^{2}} - 1)}}{2}i{{\nabla }_{{\mathbf{p}}}}. \\ \end{gathered} $$
(23)

5 CHARACTERISTICS OF THE RUNGE–LENZ OPERATOR

The Runge–Lenz operator in the momentum space is simpler than that in the coordinate space:

$${{\hat {A}}_{{\mathbf{p}}}} = i{\mathbf{p}}{{\hat {l}}_{{\mathbf{p}}}} - \frac{{({{p}^{2}} - 1)}}{2}i{{\nabla }_{{\mathbf{p}}}}.$$
(24)

Its characteristics in momentum space are conserved. It has the commutation properties

$$\{ {{L}_{i}},{{\hat {A}}_{{{\mathbf{p}}k}}}\} = i{{e}_{{ikl}}}{{\hat {A}}_{{{\mathbf{p}}l}}},\quad \{ {{\hat {A}}_{{{\mathbf{p}}i}}},{{\hat {A}}_{{{\mathbf{p}}k}}}\} = i{{e}_{{ikl}}}{{L}_{l}}.$$
(25)

The orthogonality property

$${{\hat {A}}_{{\mathbf{p}}}}\hat {L} = \hat {L}{{\hat {A}}_{{\mathbf{p}}}} = 0$$

and the following property of the sum of squared operators, which is the most important for the SO(4) group:

$${{\hat {L}}^{2}} + \hat {A}_{{\mathbf{p}}}^{2} = - \frac{{{{{({{p}^{2}} + 1)}}^{2}}}}{4}{{\Delta }_{{\mathbf{p}}}} + \frac{{({{p}^{2}} + 1)}}{2}{{\hat {l}}_{{\mathbf{p}}}}.$$
(26)

The comparison with the operator in the square brackets in Eq. (14) shows that the eigenvalue of the operator given by Eq. (26) is equal to n2 – 1; i.e., it is the same as in the coordinate space [1].

What is the operator on the Fock sphere corresponding to the found Runge–Lenz operator? According to Eq. (5), an arbitrary function \(f(\boldsymbol{\xi} ,\zeta )\) on the Fock sphere is transformed after the transition to the momentum plane to the function

$$f\left( {\frac{{2{\mathbf{p}}}}{{({{p}^{2}} + 1)}},\frac{{({{p}^{2}} - 1)}}{{({{p}^{2}} + 1)}}} \right),$$
(27)

where the sum of squared arguments is equal to unity. The successive differentiation of these arguments gives

$$\left( {{{{\hat {l}}}_{{\mathbf{p}}}}\frac{{2{\mathbf{p}}}}{{({{p}^{2}} + 1)}}} \right){{\nabla }_{{\boldsymbol{\xi }}}} = \frac{{2{\mathbf{p}}(1 - {{p}^{2}})}}{{{{{({{p}^{2}} + 1)}}^{2}}}}{{\nabla }_{{\boldsymbol{\xi }}}} = ( - \zeta )({\boldsymbol{\xi }}{{\nabla }_{{\boldsymbol{\xi }}}}),$$
$$\left( {{{{\hat {l}}}_{{\mathbf{p}}}}\frac{{({{p}^{2}} - 1)}}{{({{p}^{2}} + 1)}}} \right)\frac{\partial }{{\partial \zeta }} = \frac{{4{{p}^{2}}}}{{{{{({{p}^{2}} + 1)}}^{2}}}}\frac{\partial }{{\partial \zeta }} = \frac{2}{{({{p}^{2}} + 1)}}(1 + \zeta )\frac{\partial }{{\partial \zeta }},$$
$${{\nabla }_{{\mathbf{p}}}}\left( {\frac{{2{\mathbf{p}}}}{{({{p}^{2}} + 1)}}\frac{\partial }{{\partial {\boldsymbol{\xi }}}}} \right) = \frac{2}{{({{p}^{2}} + 1)}}\frac{\partial }{{\partial {\boldsymbol{\xi }}}} - \frac{{4{\mathbf{p}}}}{{{{{({{p}^{2}} + 1)}}^{2}}}}\left( {{\mathbf{p}}\frac{\partial }{{\partial {\boldsymbol{\xi }}}}} \right),$$
$$\left( {{{\nabla }_{{\mathbf{p}}}}\frac{{({{p}^{2}} - 1)}}{{({{p}^{2}} + 1)}}} \right)\frac{\partial }{{\partial \zeta }} = \frac{{4{\mathbf{p}}}}{{{{{({{p}^{2}} + 1)}}^{2}}}}\frac{\partial }{{\partial \zeta }}.$$

We multiply the first two relations by \(i{\mathbf{p}}\) and these next two relations by \( - \frac{{i({{p}^{2}} - 1)}}{2}\). As a result, the application of the Runge–Lenz operator (24) to the function (27) yields

$${{\hat {A}}_{{\mathbf{p}}}}f(\boldsymbol{\xi } ,\zeta ) = i\left( {{\boldsymbol{\xi }}\frac{\partial }{{\partial \zeta }} - \zeta \frac{\partial }{{\partial {\boldsymbol{\xi }}}}} \right)f({\boldsymbol{\xi }},\zeta ).$$
(28)

It means that the vector Runge–Lenz operator in the momentum space is transformed to three infinitesimal rotation operators of the Fock sphere, which were previously predicted theoretically. We notice that the commutation properties on the Fock sphere are evident. The sum of squared operators (26) is transformed to the angular part of the four-dimensional Laplacian, which has an eigenvalue of n2 – 1.

6 CONCLUSIONS

The differential equation with SO(4) symmetry and the Runge–Lenz operator have been obtained in the momentum space. They return the quantum Coulomb problem to the area of methods acting in the momentum space. The Runge–Lenz operator is also useful for interactions involving SO(4) symmetry. The Fock method is also simplified because an integral equation is no longer necessary and the consideration can be performed in the momentum space. Eigenfunctions are as simple as those in the coordinate space and are well integrated for simple perturbations.