Quantum Mechanics of One- and Two-Electron Atoms

Bethe, Hans A.; Salpeter, E. E.

doi:10.1007/978-1-4613-4104-8

Cited by 1,788 publications

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“…It differs from the usual form for the Hamiltonian for two particles as it contains a coupling term between the momenta of the particles. This is known as the Hughes-Eckart or mass-polarisation term [15,16,17].…”

Section: A Notationmentioning

confidence: 99%

“…For a review on two-electron atoms and ions, including H − and its role in astrophysics, see [16,58,59].…”

Section: B H +mentioning

confidence: 99%

“…The spectroscopy of the He atom is well known and documented [16], and thus will not be discussed further. The existence of at least one bound state of (M ++ , m − , m ′− ) is rather intuitive: the M ++ nucleus can bind the heaviest of m and m ′ , and there is a long-range Coulomb force left to bind the lightest negative charge.…”

Section: Antiprotonic Heliummentioning

confidence: 99%

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Stability of few-charge systems in quantum mechanics

2005

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We consider non-relativistic systems in quantum mechanics interacting through the Coulomb potential, and discuss the existence of bound states which are stable against spontaneous dissociation into smaller atoms or ions. We review the studies that have been made of specific mass configurations and also the properties of the domain of stability in the space of masses or inverse masses. These rigorous results are supplemented by numerical investigations using accurate variational methods. A section is devoted to systems of three arbitrary charges and another to molecules in a world with two space-dimensions.

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Section: A Notationmentioning

confidence: 99%

“…For a review on two-electron atoms and ions, including H − and its role in astrophysics, see [16,58,59].…”

Section: B H +mentioning

confidence: 99%

Section: Antiprotonic Heliummentioning

confidence: 99%

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Stability of few-charge systems in quantum mechanics

2005

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“…Minimizing the energy functional gives E = −(Z − 5/16) 2 , i.e., E=-2.85 a.u. for the helium ground state [1]. However, for H − this trial wave function is noted to fail in obtaining a bounded ground state.…”

Section: Variational Methodsmentioning

confidence: 97%

“…Very high accuracy can be achieved for eigenvalues such that they may be compared with high precision measurements. Bether and Salpeter [1] have summarized the early works in this area. The more recent calculations of the bound states have extended such routines with new numerical schemes and judicious choices of basis functions.…”

Section: Introductionmentioning

confidence: 99%

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Chi

Fang

Hsiang

et al. 2007

Solid State Communications

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Abstract. We present a review of the quantum three-body problem, with emphasis on the different methodologies, different three-body atomic systems and their historical interest. With the review as the background, a more recently proposed non-variational, kinetic energy operator approach to the solution of quantum three-body problem is presented, based on the utilization of symmetries intrinsic to the kinetic energy operator, i.e., the three-body Laplacian operator with the respective masses. Through a four-step reduction process, the nine dimensional problem is reduced to a one dimensional coupled system of ordinary differential equations, amenable to accurate numerical solution as an infinite-dimensional algebraic eigenvalue problem. A key observation in this reduction process is that in the functional subspace of the kinetic energy operator where all the rotational degrees of freedom have been projected out, there is an intrinsic symmetry which can be made explicit through the introduction of Jacobi-spherical coordinates. A numerical scheme is presented whereby the Coulomb matrix elements are calculated to a high degree of accuracy with minimal effort, and the truncation of the linear equations is carried out through a systematic procedure. The resulting matrix equations are solved through an iteration process. Numerical results are presented for (1) the negative hydrogen ion H − , (2) the helium and heliumlike ions (Z = 3 ∼ 6), (3) the hydrogen molecular ion H + 2 , and (4) the positronium negative ion Ps − . Up to thirteen-significant-figure accuracy is achieved for the ground state eigenvalues when double precision programming is used. Comparison with the variational and other approaches shows our ground state eigenvalues to be comparable, generally with less decimal digits than the variational results, but can yield highly accurate wavefunctions as by-products. Results on low-lying excited states and their wavefunctions are obtained simultaneously with the ground state properties, some at accuracies not achieved by other methods. In particular, for the doubly excited state 3 P e of H − and the 1,3 P e states of helium, some results are obtained for the first time. Also, we have calculated fourteen H + 2 excited states, up to its dissociation level. Analysis of the wavefunction characteristics, especially in relation to the electronelectron correlation effects, are presented. A significant advantage of the kinetic energy operator approach is its general applicability to different three-body systems, with only the charges, masses, and the symmetry of the desired state as the required inputs. Potential applications of the present approach to scattering and other problems are noted.

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Distribution of electron orbits having a definite angular momentum in a static magnetic field

Olszewski

1996

Int J of Quantum Chemistry

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Electron orbits having a definite angular momentum in a static magnetic field are calculated with the aid of the Bohr-Sommerfeld quantization rules. The quantization gives that orbits are arranged along a straight line but the distance between the centers of two neighboring orbits decreases with increase of the absolute value of the angular momentum. With the energy correction equal to the zero-point energy of the harmonic oscillator, the distribution of orbits becomes identical to that obtained recently with the aid of a mixed semiclassical and quantum mechafical theory. A well-known semiclassical approach started by Onsager [ 31 allowed us to quantize the flux of the magnetic field together with the electron orbital motion. More recently, the quantum Hall effect raised a problem of the filling factor of the Landau levels, the origin of which is the degeneracy of these levels. The filling factor of the quantum levels in a constant magnetic field plays an essential role in both the integer and the fractional quantum Hall effects [41.Electron orbits are a very convenient picture of the electron behavior, but they have been traced precisely by the old quantum theory in only few cases 15, 61. Perhaps the best-known example are electron orbits in the hydrogen atom where circular orbits were obtained for the quantum number n = k at the distances n2h2 m,e2 y = -from the nucleus; here, n is a principal quantum number, and k, an azimutal quantum number. For

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Quantum Mechanics of One- and Two-Electron Atoms

Cited by 1,788 publications

References 0 publications

Stability of few-charge systems in quantum mechanics

Stability of few-charge systems in quantum mechanics

Kinetic energy operator approach to the quantum three-body problem with Coulomb interactions

Distribution of electron orbits having a definite angular momentum in a static magnetic field

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