Two-dimensional hydrogen-like atom in a weak magnetic field

We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schr\"odinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

Recently, we have come across a need to know exact analytical representations for low-order perturbation-theory corrections to an arbitrary energy level of a two-dimensional analogue of a hydrogen-like atom placed in a weak and uniform magnetic field perpendicular to the atomic plane. The first-order correction may be obtained trivially for any atomic state. Exact values of the second-order corrections for states with the principal quantum numbers 1 n 4 may be derived from a table provided in Ref. [4]. The third-order correction may be shown to vanish identically for any state (in fact, the same happens for all odd-order corrections other than the first-order one), while in Refs. [25,31] the fourth-order correction has been given, but for the ground level only. Approximate expressions for several higher even-order corrections to states with zero radial quantum numbers and with principal quantum numbers not exceeding six are contained in Ref. [35]. However, neither of the publications invoked above, nor any other related one we have had in hands in the course of browsing the literature, contains the general formulas we have been seeking for. This is a bit astonishing in view of the fact that for a similar problem of the planar oneelectron atom placed in a weak, uniform, in-plane electric field, closed-form analytical expressions for Stark-Lo Surdo corrections to energies of discrete parabolic eigenstates are known up to the sixth order in the perturbing field [37,38]. Under the circumstances, we have derived expressions for the second-and fourth-order magnetic-field-induced corrections to an arbitrary energy level of the planar hydrogenic atom. The results of that study are presented in this work. We believe they may be of some interest, in particular because the result for the fourth-order correction to the ground state given in Ref. [25], and then repeated in Ref. [31], has been found to be incorrect.

Preliminaries
We consider a one-electron atom with a point-like and spinless nucleus at rest. The electron is constrained to move in a plane through the nucleus. A potential of interaction between the nucleus (of charge +Ze) and electron (of charge −e and mass m) is taken to be the attractive Coulomb one-over-distance one. The system is subjected to the action of a static uniform magnetic field of induction B, which is perpendicular to the atomic plane. With the electron radius vector r being referred to the nucleus, the two-dimensional time-independent Schrödinger equation for the electron is where r = |r| and A(r) is a vector potential of the magnetic field. Equation (2.1) is to be solved, with the electron energy E chosen as an eigenvalue, subject to the constraint that the wave function Ψ(r) is single-valued and bounded for all r ∈ R 2 , including the point r = 0 and the point at infinity. Throughout this paper, we shall be working in the symmetric gauge, in which the vector potential A(r) is Then, the Schrödinger equation (2.1) may be rewritten as is a (dimensionless) orbital angular momentum operator for the electron. The form of the Hamiltonian operator in the Schrödinger equation (2.3) suggests one introduces the polar coordinates r and ϕ, with 0 r < ∞ and 0 ϕ < 2π; Eq. (2.3) is then transformed into the following one: The benefit from the use of the polar coordinates is that Eq. (2.5) is separable, in the sense that it possesses particular solutions of the form Plugging Eq. (2.6) into Eq. (2.5) and exploiting Eq. (2.7) yields the radial Schrödinger equation which is to be solved subject to the boundary conditions P nlm l (r)/ √ r bounded for r → 0 and for r → ∞.
It is easy to deduce from the standard asymptotic analysis that for B = 0 the constraints displayed in Eq. (2.8b) may be replaced by the following ones: The symbol n that has appeared the first time as a subscript in Eq. (2.6) is the principal quantum number defined as n = n r + l + 1, (2.9) where n r ∈ N 0 is the radial quantum number which counts the number of nodes (zeroes) in the radial wave function.
Since the term linear in B which appears in the differential operator in Eq. (2.8a) is independent of the variable r, it is clear that the energy eigenvalue E nlm l may be written as It is also evident that the radial function P nlm l (r) does depend on m l through l = |m l | only: Consequently, the starting point for further considerations will be the radial eigenvalue problem 3 Perturbation-theory analysis 3

.1 Basics and the zeroth-order problem
Closed-form analytical solutions to the eigenproblem (2.13) are not known. Therefore, below we shall attempt to find its approximate solutions, under the assumption that the magnetic field is weak, with the use of the Rayleigh-Schrödinger perturbation theory. To this end, we write the radial differential operator from Eq. (2.13a) as where and We shall treat the diamagnetic term (3.3) as a small perturbation of the radial Coulomb Hamiltonian (3.2). Since H (2) (r) is of the second order in the perturbing magnetic field, we seek solutions to the eigensystem (2.13) in the form of the perturbation series and (subscripts have been omitted intentionally), which correspond to the discrete part of its spectrum, consisting of the eigenvalues and with being the Bohr radius. Eigenfunctions associated with the eigenvalues (3.7), orthonormal in the sense of For integration purposes, it is frequently convenient to have these functions rewritten as (3.12)

The second-order corrections to Coulomb energies
For the present problem, the second-order correction to energy, E nl , is given by (3.14) Plugging Eq. (3.12) into the integrand and exploiting the integration formula is the atomic unit of magnetic induction. For states with l = n − 1 (i.e., those with n r = 0), the expression in Eq. (3.17) simplifies to

The fourth-order corrections to Coulomb energies
Proceeding along the standard route, one finds that for the present problem the fourth-order correction to energy, E nl , is given by where the second-order correction to the radial wave function, P nl (r), is a solution to the inhomogeneous boundary-value problem subject to the further orthogonality restraint where G (0) nl (r, r ′ ) is a generalized (or reduced) radial Coulomb Green function associated with the Coulomb energy level E (0) n . The latter function is defined as that particular solution to the inhomogeneous boundary-value problem where δ(r − r ′ ) is the Dirac delta function, which obeys the additional orthogonality constraint Since the zeroth-order eigenproblem (3.6) is self-adjoint, the function G   (3.29) or, still more explicitly, in the form A representation of the generalized radial Coulomb Green function G (0) nl (r, r ′ ) which is perhaps the most suitable for the use in Eq. (3.30) is the one in the form of a series expansion in the discrete radial Coulomb Sturmian basis. We shall construct it below.
The discrete radial Coulomb Sturmian functions are defined as solutions to the spectral problem with E < 0 fixed and with the parameter µ (0) nr l (E) chosen as an eigenvalue. The spectrum of this problem is purely discrete, and eigenvalues are given by nrl (E, r) = (4πǫ 0 )n r ! Ze 2 (n r + 2l)! (2kr) l+1/2 e −kr L (2l) nr (2kr). (3.35) In contrary to the discrete Coulomb eigenfunctions (3.11), the Sturmians (3.35) form a complete set, the corresponding closure relation being If the parameter E coincides with the Coulomb energy eigenvalue E l (E, r, r ′ ), is defined to be a solution to the inhomogeneous equation subject to the boundary constraints Since the Sturmian functions (3.35) form a complete set, the Green function G l (E, r, r ′ ) may be sought in the form of the series nr l (E, r). (3.40) To determine the expansion coefficients C  (E, r), then integrate with respect to r over the interval [0, ∞), and apply the orthogonality relation (3.35). Upon the replacement of n ′ r with n r , this yields hence, we obtain the following symmetric Sturmian expansion of G l (E, r, r ′ ): (3.42) It follows from Eqs. l (E, r, r ′ ) through the limit procedure (3.43) By virtue of the de l'Hospital rule, the latter equation is equivalent to the following one: which is particularly suitable for the construction of the Sturmian expansion of G (0) nl (r, r ′ ). Inserting the series representation (3.42) into the right-hand side of Eq. (3.44) and then making use of the relationships ∂S which may be easily derived from the defining Eqs. (3.32) and (3.35), one eventually arrives at the sought Sturmian expansion of the generalized radial Coulomb Green function, which is Once the Sturmian expansion of G (0) nl (r, r ′ ) has been found, we are ready to complete the task to find the fourth-order energy correction E The integrals in Eq. (3.51) may be taken after one exploits Eqs. (3.12) and (3.35), with the use of the integration formula which generalizes the one in Eq. (3.15) and, similarly to the latter, may be derived from the general expression (3.16). Since only terms with n ′ r constrained by 1 |n ′ r − n r | 3 are seen to contribute non-vanishingly to the sum in Eq. (3.51), we eventually obtain The latter one is thus found to be incorrect.

Summary and concluding remarks
On the preceding pages, we have shown that energy levels of the planar hydrogen-like atom placed in a weak, static, uniform magnetic field of induction B perpendicular to the atomic plane may be expressed in the form where In Eq. (4.2), Z is an electric charge of the atomic nucleus in units of the elementary charge e, a 0 is the Bohr radius, is the atomic unit of magnetic induction, while the dimensionless and Z-independent coefficients ε (k) ... are given by ε (0) n = −  with n ∈ N + , m l ∈ Z and 0 l = |m l | n − 1. Numerical values of the coefficients ε (2) nl and ε (4) nl for states with 1 n 4 are displayed in Table I. It has to be emphasized that the formula in Eq. (4.1) is valid only if the electron spin is ignored. If this cannot be done, the Schrödinger equation (2.1) should be replaced with the planar Pauli equation {σ · [−i ∇ + eA(r)} 2 2m − Ze 2 (4πǫ 0 )r Ψ(r) = EΨ(r) (r ∈ R 2 ), (4.8) where σ = (σ x , σ y ) is the two-dimensional Pauli matrix vector, and Ψ(r) is a two-component Pauli spinor. Equation where σ z is the third Pauli matrix. It is then evident that Eq. (4.9), supplemented with the regularity constraints on Ψ(r) analogous to those introduced under Eq. (2.1), possesses separated eigenfunctions of the form Ψ nlm l ms (r, ϕ) = 1 √ r P nl (r) e im l ϕ √ 2π χ ms , (4.11) where P nl (r) is the same radial function which has appeared in the preceding sections, while χ ms is the spin one-half eigenfunction obeying Σ z χ ms = m s χ ms m s = ± 1 2 , (4.12) and that the energy spectrum is of the form E nlm l ms = E (0) n + E (1) m l ms + E (2) nl + E nl and E (4) nl being identical to those derived before, and with (4.14)