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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 fine structure ： ウィキペディア英語版 fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines. The scale of the fine structure splitting relative to the gross structure energies is on the order of (''Zα'')2, where ''Z'' is the atomic number and ''α'' is the fine-structure constant, a dimensionless number equal to approximately $1/137$. The fine structure can be separated into three corrective terms: the kinetic energy term, the spin-orbit term, and the Darwinian term. The full Hamiltonian is given by : $H=H\_+H\_\}+H\_\{\backslash mathrm\{Darwinian\}\}.\backslash !$ This can be seen as a non-relativistic approximation of the Dirac equation. ==Kinetic energy relativistic correction== Classically, the kinetic energy term of the Hamiltonian is : $T\; =\; \backslash frac,$ where $p$ is the momentum and $m$ is the mass of the electron. However, when considering a more accurate theory of nature viz. perturbation, we can calculate the first order energy corrections due to relativistic effects. : $E\_n^\; =\; \backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; H\text{'}\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle\; =\; -\backslash frac\backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; p^4\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle\; =\; -\backslash frac\backslash left\backslash langle\backslash psi^0\backslash right\backslash vert\; p^2\; p^2\; \backslash left\backslash vert\backslash psi^0\backslash right\backslash rangle$ where $\backslash psi^$ is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see : $\backslash begin$ H^\left\vert\psi^\right\rangle &= E_\left\vert\psi^\right\rangle \\ \left(\frac + V\right)\left\vert\psi^\right\rangle &= E_\left\vert\psi^\right\rangle \\ p^\left\vert\psi^\right\rangle &= 2m(E_ - V)\left\vert\psi^\right\rangle \end We can use this result to further calculate the relativistic correction: : $\backslash begin$ E_n^ &= -\frac\left\langle\psi^0\right\vert p^2 p^2 \left\vert\psi^\right\rangle \\ E_n^ &= -\frac\left\langle\psi^0\right\vert (2m)^2 (E_n - V)^2\left\vert\psi^0\right\rangle \\ E_n^ &= -\frac\left(E_n^2 - 2E_n\langle V\rangle + \left\langle V^2\right\rangle \right) \end For the hydrogen atom, $V(r)\; =\; \backslash frac$, $\backslash left\backslash langle\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash frac$, and $\backslash left\backslash langle\; \backslash frac\; \backslash right\backslash rangle\; =\; \backslash frac^\}$ where $a\_$ is the Bohr Radius, $n$ is the principal quantum number and $l$ is the azimuthal quantum number. Therefore the first order relativistic correction for the hydrogen atom is : $\backslash begin$ E_^ &= -\frac\left(E_^2 + 2E_n\frac\frac + \frac\frac) n^3 a_0^2}\right) \\ &= -\frac\left(\frac} - 3\right) \end where we have used: :$E\_n\; =\; -\; \backslash frac$ On final calculation, the order of magnitude for the relativistic correction to the ground state is $-9.056\; \backslash times\; 10^\backslash \; \backslash text$. 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「fine structure」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.