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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hydrogen-like atom ： ウィキペディア英語版 Hydrogen-like atom A hydrogen-like ion is any atomic nucleus with one electron and thus is isoelectronic with hydrogen. Except for the hydrogen atom itself (which is neutral), these ions carry the positive charge $e(Z-1)$, where $Z$ is the atomic number of the atom. Examples of hydrogen-like ions are He+, Li2+, Be3+ and B4+. Because hydrogen-like ions are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form, as can the (relativistic) Dirac equation. The solutions are one-electron functions and are referred to as ''hydrogen-like atomic orbitals''.〔In quantum chemistry an orbital is synonymous with "a one-electron function", a square integrable function of $x$, $y$, $z$.〕 Other systems may also be referred to as "hydrogen-like atoms", such as muonium (an electron orbiting a muon), positronium (an electron and a positron), certain exotic atoms (formed with other particles), or Rydberg atoms (in which one electron is in such a high energy state that it sees the rest of the atom practically as a point charge). ==Schrödinger solution== In the solution to the Schrödinger equation, which is non-relativistic, hydrogen-like atomic orbitals are eigenfunctions of the one-electron angular momentum operator ''L'' and its ''z'' component ''L''z. A hydrogen-like atomic orbital is uniquely identified by the values of the principal quantum number ''n'', the angular momentum quantum number ''l'', and the magnetic quantum number ''m''. The energy eigenvalues do not depend on ''l'' or ''m'', but solely on ''n''. To these must be added the two-valued spin quantum number ''ms'' = ±½, setting the stage for the Aufbau principle. This principle restricts the allowed values of the four quantum numbers in electron configurations of more-electron atoms. In hydrogen-like atoms all degenerate orbitals of fixed ''n'' and ''l'', ''m'' and ''s'' varying between certain values (see below) form an atomic shell. The Schrödinger equation of atoms or atomic ions with more than one electron has not been solved analytically, because of the computational difficulty imposed by the Coulomb interaction between the electrons. Numerical methods must be applied in order to obtain (approximate) wavefunctions or other properties from quantum mechanical calculations. Due to the spherical symmetry (of the Hamiltonian), the total angular momentum ''J'' of an atom is a conserved quantity. Many numerical procedures start from products of atomic orbitals that are eigenfunctions of the one-electron operators ''L'' and ''L''z. The radial parts of these atomic orbitals are sometimes numerical tables or are sometimes Slater orbitals. By angular momentum coupling many-electron eigenfunctions of ''J''2 (and possibly ''S''2) are constructed. In quantum chemical calculations hydrogen-like atomic orbitals cannot serve as an expansion basis, because they are not complete. The non-square-integrable continuum (E > 0) states must be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.〔This was observed as early as 1928 by E. A. Hylleraas, ''Z. f. Physik'' vol. 48, p. 469 (1928). English translation in H. Hettema, ''Quantum Chemistry, Classic Scientific Papers'', p. 81, World Scientific, Singapore (2000). Later it was pointed out again by H. Shull and P.-O. Löwdin, ''J. Chem. Phys.'' vol. 23, p. 1362 (1955).〕 In the simplest model, the atomic orbitals of hydrogen-like ions are solutions to the Schrödinger equation in a spherically symmetric potential. In this case, the potential term is the potential given by Coulomb's law: :$V(r)\; =\; -\backslash frac\; \backslash frac$ where * ε0 is the permittivity of the vacuum, * ''Z'' is the atomic number (number of protons in the nucleus), * ''e'' is the elementary charge (charge of an electron), * ''r'' is the distance of the electron from the nucleus. After writing the wave function as a product of functions: :$\backslash psi(r,\; \backslash theta,\; \backslash phi)\; =\; R\_(r)Y\_(\backslash theta,\backslash phi)\backslash ,$ (in spherical coordinates), where $Y\_$ are spherical harmonics, we arrive at the following Schrödinger equation: :$$ \left(- \frac \left(\left(r^2 \right) - \right) + V(r)R(r) \right )= E R(r), where $\backslash mu$ is, approximately, the mass of the electron (more accurately, it is the reduced mass of the system consisting of the electron and the nucleus), and $\backslash hbar$ is the reduced Planck constant. Different values of ''l'' give solutions with different angular momentum, where ''l'' (a non-negative integer) is the quantum number of the orbital angular momentum. The magnetic quantum number ''m'' (satisfying $-l\backslash le\; m\backslash le\; l$) is the (quantized) projection of the orbital angular momentum on the ''z''-axis. See here for the steps leading to the solution of this equation. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hydrogen-like atom」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.