Hypervirial analysis of enclosed quantum mechanical systems. II. von Neumann boundary conditions and periodic potentials

Abstract

Previous results on the hypervirial analysis of confined systems are extended. Periodic potentials are discussed and an efficient alternative way to solve the Mathieu equation is proposed.

Hypervirial analysis of enclosed quantum mechanical systems. IV. Further comments

“…Furthermore, certain previous details which were not properly explained with care [2] are pointed out. Our aim is the study of the eigenfunctions and eigenvalues of the Hamiltonian …”

“…Besides, the HPM was successfully used for periodical potentials [2]. In this case it was necessary to choose different hypervirial operators with respect to those formerly utilized [4,51.…”

“…We mean finite or bounded systems at those models whose boundary conditions (BC) are satisfied for finite values of the coordinates. In these communications we discussed the application of diagonal hypervirial theorems (DHT) to finite systems constrained to Dirichlet BC (DBC) [l, 31, von Neumann BC (VNBC) [2,3], and periodic BC (PBC) [2]. Furthermore we considered in a brief manner another BC [2].…”

“…In these communications we discussed the application of diagonal hypervirial theorems (DHT) to finite systems constrained to Dirichlet BC (DBC) [l, 31, von Neumann BC (VNBC) [2,3], and periodic BC (PBC) [2]. Furthermore we considered in a brief manner another BC [2]. The deduced relationships are general enough and they are valid for symmetrical as well as unsymmetrical models.…”

“…Through the application of the HPM we got in an analytical form, the firstand second-order corrections for the harmonic oscillator model symmetrically and unsymmetrically bounded by DBC [ l , 3,6] and symmetrically bounded by VNBC [3]. Besides, the HPM was successfully used for periodical potentials [2]. In this case it was necessary to choose different hypervirial operators with respect to those formerly utilized [4, 51. The main advantage and characteristic value of the HPM consists in that it enables to obtain perturbational corrections of any desired order by way of a simple recurrence relationship which is easily handled.…”