A calculation of the classical analogue for the quantum wave function and local denity of states, in energy representation, is presented for simple Hamiltonian systems. Sucha analogous were proposed by M. V. Berry and A. voros considering the intersection of energy shells of two systems as the only semiclassical object which can give support to eigenfunctions. One of them is the system unser study and the other is the "unperturbed system" used to express the wave functions, even in the case that both systems are not close. For simple systems and as for scalable ones analytical expressions are obtainable. In the present work we offer examples of both.
We propose a simple approach to predict the angular momentum I ground state
(I g.s.) probabilities of many-body systems that does not require the diagonalization of Hamiltonians with random interactions. This method is found to be applicable to all cases that have been discussed: even and odd fermion systems (both in single-
j and many-
j shells), and boson (both
sd and
sdg) systems. A simple relation for the highest angular momentum g.s. probability is found. Furthermore, it is suggested for the first time that the 0 g.s. dominance in boson systems and in even-fermion systems is given by two-body interactions with specific features.
We shall present a new strategy for handling mean field limits of quantum mechanical systems. The new method is simple and effective. It is simple, because it translates the idea behind the mean field description of a many particle quantum system directly into a mathematical algorithm. It is effective because the strategy yields with lesser effort better results than previously achieved. As an instructional example we treat a simple model for the time dependent Hartree equation which we derive under more general conditions than what has been considered so far. Other mean field scalings leading e.g. to the Gross-Pitaevskii equation can also be treated.