The Kepler Problem in Two-Dimensional Momentum Space

Shibuya, Tai-ichi; Wulfman, Carl E.

doi:10.1119/1.1971931

Cited by 33 publications

(23 citation statements)

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“…., (0, ±N ), and the degeneracy is d N = 2N + 1. These results are well known [13][14][15][16][17][18]. 2. ν = 0, m 0 = 0.…”

Section: Bound Statessupporting

confidence: 65%

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Lin

1999

Phys. Rev. A

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We propose a simple quantum mechanical equation for n particles in two dimensions, each particle carrying electric charge and magnetic flux. Such particles appear in (2+1)-dimensional Chern-Simons field theories as charged vortex soliton solutions, where the ratio of charge to flux is a constant independent of the specific solution. As an approximation, the charge-flux interaction is described here by the Aharonov-Bohm potential, and the charge-charge interaction by the Coulomb one. The equation for two particles, one with charge and flux (q, Φ/Z) and the other with (−Zq, −Φ) where Z is a pure number is studied in detail. The bound state problem is solved exactly for arbitrary q and Φ when Z > 0. The scattering problem is exactly solved in parabolic coordinates in special cases when qΦ/2πhc takes integers or half integers. In both cases the cross sections obtained are rather different from that for pure Coulomb scattering. PACS number(s): 03.65. Bz, 12.90.+b

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“…., (0, ±N ), and the degeneracy is d N = 2N + 1. These results are well known [13][14][15][16][17][18]. 2. ν = 0, m 0 = 0.…”

Section: Bound Statessupporting

confidence: 65%

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Lin

1999

Phys. Rev. A

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“…In fact, as Monkhorst and Jeriorski have pointed out [4], the many-center one-particle Coulomb problem in reciprocal space reduces essentially to the problem of evaluating these integrals. In an early paper [4], Shibuya and Wulfman developed a method for their evaluation based on the R4 Wigner coefficients. In this paper we shall discuss an alternative method based on a transform defined by the relationship 1 / aS2 eik'Rf(u) =jT(t) (52) 2~ 2 with t = koR.…”

Section: S2/~m(r ) =-[ Do E Ik'r Rn*-llm(u) Yn-llm(u)mentioning

confidence: 97%

“…It has been shown by a number of authors [4][5][6]] that Fock's approach can be generalized in such a way as to yield solutions to the reciprocal-space Schr6dinger equation for a charged particle moving in the many-center potential: …”

Section: Many-center Coulomb Potentialsmentioning

confidence: 99%

Fock transforms in reciprocal-space quantum theory

Avery

1994

J Math Chem

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In Fock's reciprocal-space treatment of the hydrogen atom, k-space is mapped onto the surface of a 4-dimensional hypersphere, and the solutions (apart from an invariant factor) are 4-dimensional hyperspherical harmonics. Fock's method can be generalized to provide solutions for the Schr6dinger equation of a charged particle moving in a many-center Coulomb potential, and in this case the solutions are found by diagonalizing an overlap matrix involving products of hyperspherical harmonics. The present paper discusses a transform which can conveniently be used to evaluate the elements of the overlap matrix. (1) I n t r o d u c t i o nIn eq.(1) we have used a notation which emphasizes the similarity between the hyperspherical harmonics Y~u and the spherical harmonics t3m. However, in the ddimensional case, # is not a single index but stands for a set of indices representing eigenvalues of a complete set of operators which commute with A 2 and with each other. The choice of these operators is not unique, but in physical applications it is convenient to choose the set which commutes with the Hamiltonian of the system. Table 1 shows the first few hyperspherical harmonics for d = 4. (As we shall see below, the 4-dimensional hyperspherical harmonics play an important role in reciprocal-space quantum theory.) The index/z of eq. (1) stands for the two indices l and m in table 1. If the angles X, 0 and q~ are related to the unit vectors in a 4-dimensional space by U 1 = sin X sin 0 cos ~b, © J.C. Baltzer AG, Science Publishers

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“…It is p 2 D and therefore must be the (202) state. The three P states, p 2 P, spP, and psP transform like linear combinations of (111), (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) and (201). The (111) and (1-11) states have zero coupling with the (202) state for the reason mentioned above, and thus we may determine the (111) and (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) states by means of the equations…”

Section: A Mathematicalmentioning

confidence: 99%

The GroupR4in Atomic-Structure Theory: The HydrogenicR4versus the Mathematical
Alper
¹
,
Si̇nanoğlu
²

1969
Phys. Rev.

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The rotation group in four dimensions 2? 4 is applied to the study of the 2s m 2p n configurations of atoms. This Lie group is used in both the mathematical sense, describing the transformation properties of the angular parts of the 2s and 2p electrons, and in the more physical sense of an approximate symmetry group for the first-row atoms. The various states of each configuration are classified by means of i? 4 . By expressing the Coulomb interaction in terms of tensors, transforming according to irreducible representations of i? 4 , Coulomb and exchange integrals are evaluated group theoretically. The method is extended to 3s m Sp n 3d r configurations. The states are classified by means of i? 4 and Coulomb and exchange integrals are approximately evaluated.The theory of groups has long played an important role in the quantum theory of atomic structure. The indistinguishability of electrons and the Pauli exclusion principle lead naturally to the introduction of the permutation group. The use of the three-dimensional rotation group in classifying states of definite angular momentum is of course well known. The group R 3 is not sufficiently large to explain the n 2 -fold degeneracy of the nth level of hydrogen-like atoms, and in 1935 Fock 1 realized that the group R 4 would. He showed that, iif the Fourier-transformed Hamiltonian is stereographically projected from three-dimensional p-space onto a four-dimensional sphere, the Hamiltonian exhibits four-dimensional rotation symmetry. It is this additional symmetry beyond R 3 which is responsible for the degeneracy of orbitals of the same principal quantum number. Each set of orbitals of a given principal quantum number when projected onto the four-dimensional sphere transforms according to one of the irreducible representations of R 4 .In his study of atomic spectra, Racah 2 found that in order to classify states of d? 1 and of f n it is useful to introduce Lie groups larger than the R 3 describing the orbital angular momentum, and smaller than the SU n describing the permutation symmetry and spin. These groups describe in more detail than does R 3 the transformation properties of the angular parts of the orbitals under consideration. Consequently such groups are able to provide the additional quantum numbers that can distinguish identical terms, e. g., the two 2 D states of d 3 are labeled by their seniority, a quantum number arising from R 5 . These groups deal with the mathematical problems of the classification of states and will be called here "mathematical groups" as opposed to the "physical groups" such as the R 4 introduced by Fock.In this paper we consider both descriptions of groups in an investigation of the application of R 4 to the study of the 2s m 2p n configurations of atoms. We discuss the relationship between the two descriptions of groups. For the four 2s and 2p orbitals, the appropriate group for both the mathematical and physical descriptions is R 4 , and the representations used in the analysis of atomic configurations are the same. The four 2...

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The Kepler Problem in Two-Dimensional Momentum Space

Cited by 33 publications

References 0 publications

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Fock transforms in reciprocal-space quantum theory

The GroupR4in Atomic-Structure Theory: The HydrogenicR4versus the Mathematical
Alper
¹
,
Si̇nanoğlu
²

1969
Phys. Rev.

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The Kepler Problem in Two-Dimensional Momentum Space

Cited by 33 publications

References 0 publications

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

Fock transforms in reciprocal-space quantum theory

The GroupR4in Atomic-Structure Theory: The HydrogenicR4versus the MathematicalAlper1, Si̇nanoğlu2 1969Phys. Rev.

Contact Info

Product

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The GroupR4in Atomic-Structure Theory: The HydrogenicR4versus the Mathematical
Alper
¹
,
Si̇nanoğlu
²

1969
Phys. Rev.