The virial theorem for bound, virtual, and resonance states is derived, based on the virial theorem for scattering states proved by Demkov. The association of the complex virial theorem with Demkov's virial theorem is discussed.
Heretofore, the hypervirial theorems recently introduced by Hirschfelder have only been applied to bound-state systems. In this paper it is shown that, with certain modifications, these theorems can also be applied to free systems, and, in particular, to scattering problems. The new form of the general hypervirial theorem is derived, and the theory is illustrated with the problem of a particle scattered by a central field. The ordinary virial theorem is deduced, together with other results of physical interest. Both classical and quantum-mechanical formalisms are considered, and in some cases the semiclassical approximation links corresponding results.
Expansion formulas for density matrices are derived with the use of the calculus of ordered exponentials. Using the formulas the expressions for virial coefficients are explicitly obtained. The expression of the third virial coefficient is calculated to obtain its expansion formula in terms of ℏ2β/M. The expansion formulas for density matrices can be used to derive a new formula of multiple scattering, which involves Luttinger and Kohn's formula as a special case. The expansion formula for the normalized density matrix is also given. Expansion formulas which are to the applied to irreversible processes and relaxation phenomena are also given.
We present a general virial theorem for quantum particles with arbitrary zero-range or finite-range interactions in an arbitrary external potential. We deduce virial theorems for several situations relevant to trapped cold atoms: zero-range interactions with and without Efimov effect, hard spheres, narrow Feshbach resonances, and finite-range interactions. If the scattering length a is varied adiabatically in a two-component Fermi gas, we find that the trapping potential energy as a function of 1∕a has an inflexion point at unitarity.