The symposium "Quantum Science and form" was held at the Institute of Statistical Mathematics in Tachikawa city, Tokyo on June 2016 by the conference chair: N. Nishigaki (Saitama Univ.) and S. Nakamura (Riken). It was held as the 81st symposium of Society for Science on Form. The purpose of the symposium was to discuss on the form that appears in the quantum theory. Four guest speakers have discussed mainly on the quantum chemistry, and another one from fellow has discussed on the spherical harmonics from algebraic view point.

A special issue on this symposium, which is a collection of papers presented at this symposium as well as a newly contributed paper related to the symposium theme has been published here. There are four contributions, and I will roughly introduce them here.

Profs. Nohira and Nohira have discussed on the correlation diagram of molecular orbitals for the pre- and post-chemical reaction. The essential concepts are to take the chemical reaction as a single step one without unsteady intermediate states and to consider the resultant change of molecular orbitals under the constraint of minimum change of the entire 'shapes'. By using this approach, they succeeded in explaining the conversion reactions of Dewar benzene to benzene and prismane to benzene. The results thus obtained are obeying to the Woodward-Hoffmann rule and Fukui's theory.

Prof. Tokita has shown the graphics of absolute square of the angular wave functions (spherical harmonics) with each quantum numbers, and showed the relation between quantum numbers and the numbers of spherical-, planar-, and conical-nodes, and he also introduced the glass object of spherical harmonics by which we can easily imagine the form of spherical harmonics in three dimension.

On the other hand, I (Ogawa) have discussed the algebraic method to obtain the angular wave function without solving the Schrödinger equation. By introducing the SU(2) operators and mirror operators in matrix representation, the form of spherical harmonics is constructed from the restriction by symmetry.

Prof. Kaino studied magnetic properties of rare-earth metals by using the exchange interaction between conduction-electrons and the f-electrons of all atoms. The angular momentum of f-electron at the n-th atom is replaced by the mean value 〈S_n〉 which is assumed to have a conical structure. He considered the quantum theory of the fluctuation of f-spin system by using the spin-wave theory.

I hope that the readers enjoy reading this issue, and find new viewpoints or ideas for Science on Form.

A three dimensional representation of the probability density distribution of a hydrogen atomic orbital in a glass block was developed. The density of the dots sculptured in the glass block shows the probability density of finding an electron, so the nodal plane is well described as spherical shell(s), planar or conical node(s) where the dots cannot be found. Node types and their numbers are summarized in Table 2. Classification according to the node type leads to an observation of systematic regularity in hydrogen atomic orbitals. Relationship between a square of a complex atomic orbital and a real atomic orbital is explained. Advantage of the probability density sculpture in a spherical glass block is also discussed.

Chemical substances are highly diverse. Here, we discuss the regularities of chemical changes among the substances. As a typical example, the regularity of the change between benzene and its structural isomers, i.e. Dewar benzene and prismane, has been discussed in terms of molecular vibrational theory and orbital correlation diagrams. It has been shown that when a certain molecule changes to another one with a similar structure, the "shapes" of the original electronic states are transferred to the next ones with minimum change. Regarding this, it has been pointed out that Neumann-Wigner's non-crossing rule need not be applied to orbital correlation diagrams. It has also been shown that changes of molecules are "quantized" and cannot be described continuously, i.e. the states which can be accurately described by the Schrödinger equation are limited to "quantum steady states".

The angular wave functions for a hydrogen atom are well known to be spherical harmonics, and are obtained as the solutions of a partial differential equation. However, the differential operator is given by the Casimir operator of the SU(2) algebra and its eigenvalue l(l + 1)ħ² , where l is non-negative integer, is easily obtained by an algebraic method. Therefore the shape of the wave function may also be obtained by extending the algebraic method. In this paper, we describe the method and show that wave functions with different quantum numbers are connected by a rotational group in the cases of l = 0, 1 and 2.

Using a simple band model which produces a helical spin ordering, we show the magnetic phase diagram by minimizing the unperturbed energy which includes the c-f exchange interaction by replacing the f-spin operator S_n with the expectation value 〈 S_n 〉. As the c-f exchange interaction increases, the helix and the cone structures appears without any crystal fields, and then the ferromagnetic structure becomes more stable. Secondly, we obtain formulae of spin-wave dispersions and show their instabilities on the second-order transition boundaries. Near the ferro-helix boundary, the spin-wave constant of the ferromagnetic spin-wave vanishes, while in the helical phase the whole region of the wave-number 0 < q_z < Q shows softening where the helical wave-number Q decreases continuously. Thirdly, by the method of the double-time Green function, we derive the spin-wave dispersion at finite temperatures. Finally, anomalous properties in magnon dispersions at finite temperatures for Gd, Ho and those for diluted Tb-Y alloys are explained by use of numerical calculations.