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Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures.
      
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The operator-splitting methods for the mathematic model of one kind of oin reactions for the problem of groundwater are considered.
      
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The main difficulty of the one-electron theory of optical activity lies in the calculation of the so-called vicinal actions. This may be overcome, as suggested in the present investiga- tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance...

The main difficulty of the one-electron theory of optical activity lies in the calculation of the so-called vicinal actions. This may be overcome, as suggested in the present investiga- tion by assuming that the vicinal actions are due to the interactions of various chemical bonds in the molecule with the chromophoric electron. For the calculation of these interactions the charge distribution of a single covalent bond is assumed to consist of two nuclei, each of unit positive charge, separated by the bond distance and two electrons with opposite spins and a spherically symmetrical orbital eigenfunction represented by (?) where r is the distance measured from a certain point on the bond axis and near the center of the bond. The location of this point is so determined that the model will give the proper dipole moment of the bond. The exponent α, which determines the extent of diffusivity of the electron cloud, is obtained by an approximate variational treatment. The quadruple moment of this model, calculated for the C-H bond, agrees very well with that estimated by Lassettre and Dean from a study of the potential barrier hindering the internal rotation of the ethane molecule. To test the theory and also to illustrate the procedure which is followed in calculating optical rotations using the new model of vicinal actions, the optical rotations of several methyl derivatives of cyclopentanone have been calculated and the following conclusions have been reached: (1) The proposed model gives rise to optical rotations in agreement with the observed values. (2) The effect of the internal rotation of the methyl group on the molecular rotation [M]_D of 3-methylcyclopentanone is large, [M]_D being +44° and -130° for the cis- trans-conformations, respectively. (3) The observed [M]_D may be regarded as composed of two parts, namely, the contribution from the multipole terms and that from the orbital overlapping. The latter corresponds to the "incomplete screening of atomic nuclei", and has been regarded as the most important vicinal action in earlier calculations. In the present work, it is shown, however, that this effect contributes only a few percent of the total [M]_D, unless the perturbing group is very near to the chromophoric group as it is in the case of 2-methyl- cyclopentanone. The failure of the previous calculations is, therefore, due mainly to the omission of the multipole (especially the quadrupole) terms. (4) The absolute configuration of 3-methylcyclopentanone, indicated by the present calculation, is the opposite of the one suggested by Eyring, since he based on the overlapping contribution alone, which bears a sign opposite to that of the total [M]_D (5) That the observed [M]_D of 2,4,5,5-tetramethylcyclopentanone is considerably smaller than that of 2,4-dimethylcyclopentanone may be due to one or both of the following possibilities: (a) that the conformation of the 2- and 4-methyl groups slightly deviates from the trans-position in the tetra-methyl compound, due to the presence of the two additional methyl groups; (b) that the two additional methyl groups in 5,5-positions are not exactly symmetrical with respect to the ring so that they make contributions to [M]_D.

本文在量子力學的單電子旋光理論的鄰近作用問題上,作了如下的貢獻: 1.指出旋光度應由分子中各化學鍵,而不是分子中各原子(如像前人所假定的)對於生色團電子的微擾作用來計算,兩者的主要不同點在於是否考慮鍵的多極矩。 2.建議在旋光度的計算中,共價單鍵可以看作是由兩個處於鍵端的正電荷和一個以單中心狀態函數,表示出來的電子雲所組成。根據這個假定計算了環戊酮的甲基衍生物的旋光度,其結果與實驗值甚爲一致(詳見結果討論)。 3.計算結果證明甲基的內旋轉對於旋光度的影響很大,例如順式和反式構型的3-甲基環戊酮的旋光度,應分別為+44°和-30°。 4.指定了3-甲基環戊酮的絕對構型,其結果Eyring所指定者相反。

It is shown that the modified Schrodinger equation proposed by Janossy cannot describe the behavior of the electron correctly. However, it is further shown that the equation can be regarded as a model of a possible phenomenalogical theory of nuclei or as a model of a possible theory of elementary particles. The quantization of the equation is discussed.

本文指出杨诺赛波动方程作为电子的个体理论的困难;指出这一类型波动方程可以作为原子核系综理论的数学形式,也可以作为元粒子系综理论的数学形式。本文并讨论了这一类型波动方程的第二级量子化问题。

Uhlenbeck has proposed that the ideal Bose-Einstein condensation is a phase transition of the first order in which the gas condenses to a phase of zero volume. In this paper the difficulty in Uhlenbeck's view is pointed out and discussed. In order to avoid this difficulty a two-gas View-point is proposed, in which the gas is conceived as a mixture of two parts, a super-gas and a normal gas- a view similar to that of the two-fluid model of liquid helium. When the temperature ia decreased beyond the condensation...

Uhlenbeck has proposed that the ideal Bose-Einstein condensation is a phase transition of the first order in which the gas condenses to a phase of zero volume. In this paper the difficulty in Uhlenbeck's view is pointed out and discussed. In order to avoid this difficulty a two-gas View-point is proposed, in which the gas is conceived as a mixture of two parts, a super-gas and a normal gas- a view similar to that of the two-fluid model of liquid helium. When the temperature ia decreased beyond the condensation temperature, the super-gas appears, and under isothermal compression the relative amount of the super-gas increases as the volume is decreased. The chemical potential of the super-gas is always KOTO, and the equilibrium, comli ion of the mixture of super-gas and normal gas explains the fact that the pressure is independent of the volume and satisfies an equation of the Clapeyron type in a phase transition of the first order. In the end of the paper an analogous view for the temperature radiation is discussed.

本文讨论了Bose-Einstein凝结的物理实质问题,主要工作为在指出了Uhlenbeck的凝结模型的困难后,用双气体的观点代替凝结的观点来解释Bose凝结的性质,并讨论了辐射场的热力学性质。

 
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