It is shown that the raising and lowering operators derived from the factorization of the radial Schrdinger equation for the two-dimensional hydrogen atom and the isotropic oscillator are equivalent to the Runge-Lenz vector. This implies that the factorizability of the radial Schr?dinger equation is intimately connected with the closeness of classical orbits and the dynamical symmetry.

The collective spectra of the well-deformed nuclei with small axial asymmetry and the nuclei with large axial asymmetry are investigated separately in the frame of the Bohr Hamiltonian. The collective spectra of well-deformed nuclei manifest the vibrational-rotational band structure and each of these bands can be expressed as a 2-parameter closed expression. The agreement between the calculated results and the observed data is very good. From this expression the radius of convergence for the energy expansion in terms of can be estimated and the empirical fact that the expansion in powers of rotational frequency converges much more rapidly than that in powers of can be understood. Similarly, the ground state quasi-band of the nuclei with large axial asymmetry can be expressed as a 3-parameter closed expression.

Berry pointed out that the Aharonov-Bohm phase could be considered as a Berry phase.In his argument,he gave up the adiabatic approximation,and the nonstationarity of quantum states was not concerned.In this paper we re-investigate Berry’s proof,and find that his final statement is correct,and in his argument,the adiabatic approximation should be used.In the AB effect,a moving electron is described as a moving wave packet (nonstationary),and the appearance of AB phase originates from the requirement that the evolution of the wave packet must satisfy the Schrdinger equation.

Using the factorization method,the angular momentum raising and lowering operators of two-and three-dimensional hydrogen atoms were derived.The other three kinds of raising and lowering operators were also obtained by using the recurrence relations of hypergeometric function.The corresponding selection rules and conserved quantum numbers were discussed.The similarity between two-and three-dimensional hydrogen atoms was found.

Using the factorization method,four kinds of raising and lowering operators of a three-dimensional isotropic harmonic oscillator were derived and the corresponding selection rules and conserved quantum numbers were discussed.