The virial theorem with boundary conditions applications to the harmonic oscillator and to sine-shaped potentials

New objective formulations of the quantum virial theorem rise from a parallelism drawn with d'Alembert'sprinciple. In particular, boundary conditions correspond to constraints whose virial give a complementary term to the usual formulation of the quantum virial theorem. Two examples are examined. first the non-Coulombic and nonperiodical harmonic oscillator for which all the virials can be calculated and compared; second the periodical sine-shaped potential connected with Mathieu'sfunctions, for which only the complementary term due to the cyclic conditions is investigated, showing that its importance is varying from zero to twice the mean kinetic energy when the amplitude of the interaction is vanishing, in the limiting free-electron case. A remark is also made about normalization and the use of mean energy values over a part of space, when mathematical integration is stoped on a surface (partitioning of a molecule, for example) or on a cell in periodical solid-state problems.