We propose the second moment of the Husimi distribution as a measure of complexity of quantum states. The inverse of this quantity represents the effective volume in phase space occupied by the Husimi distribution, and has a good correspondence with chaoticity of classical system. Its properties are similar to the classical entropy proposed by Wehrl, but it is much easier to calculate numerically. We calculate this quantity in the quartic oscillator model, and show that it works well as a measure of chaoticity of quantum states.

We propose a new method to analyze fluctuations in the strength function phenomena in highly excited nuclei. Extending the method of multifractal analysis to the cases where the strength fluctuations do not obey power scaling laws, we introduce a new measure of fluctuation, called the local scaling dimension, which characterizes scaling behavior of the strength fluctuation as a function of energy bin width subdividing the strength function. We discuss properties of the new measure by applying it to a model system which simulates the doorway damping mechanism of giant resonances. It is found that the local scaling dimension characterizes well fluctuations and their energy scales of fine structures in the strength function associated with the damped collective motions.

We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) and the isovector (IV) giant quadrupole resonances (GQR) in 40 Ca, where the strength functions are obtained by the shell model calculation within up to the 2p2h configurations. It is found that at small energy scale, fluctuation of the strength function almost obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. On the other hand, we found a deviation from the GOE limit at the intermediate energy scale about 1.7MeV for the IS and at 0.9MeV for the IV. The results imply that different types of fluctuations coexist at different energy scales. Detailed analysis strongly suggests that GOE fluctuation at small energy scale is due to the complicated nature of 2p2h states and that fluctuation at the intermediate energy scale is associated with the spreading width of the Tamm-Dancoff 1p1h states.

We investigate the transition from integrable to chaotic dynamics in the quantum mechanical wave functions from the point of view of the nodal structure by employing a two dimensional quartic oscillator. We find that the number of nodal domains is drastically reduced as the dynamics of the system changes from integrable to nonintegrable, and then gradually increases as the system becomes chaotic. The number of nodal intersections with the classical boundary as a function of the level number shows a characteristic dependence on the dynamics of the system, too. We also calculate the area distribution of nodal domains and study the emergence of the power law behavior with the Fisher exponent in the chaotic limit.

We performed fluctuation analysis by means of the local scaling dimension for the strength function of the isoscalar (IS) giant quadrupole resonance (GQR) in 208 Pb where the strength function is obtained by the shell model calculation including 1p1h and 2p2h configurations. It is found that at almost all energy scales, fluctuation of the strength function obeys the Gaussian orthogonal ensemble (GOE) random matrix theory limit. This is contrasted with the results for the GQR in 40 Ca, where at the intermediate energy scale of about 1.7 MeV, a deviation from the GOE limit was detected. It is found that the physical origin for this different behavior of the local scaling dimension is ascribed to the difference in the properties of the damping process.