By Abram I. Fet, Vladimir Slepkov
In this monograph, group-theoretical techniques are used to construct a process of hadrons and qualitatively describe the houses of chemical substances. This serves as a supplement to numerically and nearly resolve the many-electron Schrödinger equation, so as to comprehend the habit of chemical components. in addition to normal conception, particular effects are in comparison with experimentally measured chemical houses.
Symmetries of a quantum system
Observables of a quantum system
Lie teams and Lie algebras
The ideas of particle classification
The symmetry staff of chemical elements
Classification and chemical houses of elements
Appendix A. Fock’s strength spectrum of the hydrogen atom
Appendix B. Representations of a few teams
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Additional info for Group Theory of Chemical Elements: Structure and Properties of Elements and Compounds
From eq. 10), it follows that determinants of sufficiently small matrices X are close to the identity. But det tO = det O , and from tOO = 1, obtain (det O )2 = 1, det O = ± 1; therefore, for small X, we have det O = 1 and О belongs to SO(n). 44 3 Lie groups and Lie algebras Thus, eq. 6) defines a continuous one-to-one correspondence between matrices SO(n) close to identity and antisymmetric matrices close to zero. The latter are much easily parametrized: matrix X can be parametrized by its elements above the diagonal (xij, i < j) which completely determine real antisymmetric matrices.
From the Schrödinger equation, it follows that these two operators coincide for a wavefunction of motion in a real space. 1 Deriving observables from a symmetry group 33 and, since it preserves x0, it coincides with the subgroup of rotations about axis z in a three-dimensional space. 25) which is quite similar to operator Lz (see eq. 16)). The same way obtain operators L23, L31 from one-parameter subgroups of rotations in the planes (x2x3 ) and (x3x1); operators Lik (i, k = 1, 2, 3) play the role of operators of spatial angular momentum in the relativistic theory.
The observables of the group are Lx , Ly , Lz; the maximum system of commuting observables contains only one of them, for example, Lz. 1, where irreducible subspaces are denoted as Rnl. Each of them has the basis ψnlm(x , y , z ) = fn (r )Y (l m)(ϑ, φ) (eq. 28)) composed of eigenfunctions of operator Lz with eigenvalues m = −l, −l + 1, …, l − 1, l. In contrast to the previous example, the quantum 36 2 Observables of a quantum system number of the group SO(3) (single, in this specific case) is insufficient to number the basis of the entire R: it enumerates only the basis for each irreducible subspace.
Group Theory of Chemical Elements: Structure and Properties of Elements and Compounds by Abram I. Fet, Vladimir Slepkov