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.
We present a new, completely rigorous way of proving the relation between the second virial coefficient and the
S matrix. The method involves the use of the ${U}_{2,\ensuremath{\infty}}$ function introduced by Lee and Yang and proceeds in the most straightforward manner. Although difficulties are encountered for virial coefficients of higher order, there is hope that this type of approach might shed some light on the question of the connection between the virial series and the scattering matrix.
The quantum theory of the third virial coefficient
C is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of
CBE are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for
He4 and
He3, by means of a comparison of the corresponding expansion for the second virial coefficient
B with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to
B. Then the leading term of
C is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for
C is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
Understanding a composite system through its constituents is a fundamental practice in physics. In the context of quantum reference frames (QRFs), however, combining the usual quantum-theoretic notion of composition with QRF perspectives gives rise to subtle issues, such as the 'paradox of the third particle'. Here we study in depth how to compose subsystems in QRF perspectives, building on a recent formalism for QRFs [E
.Castro-Ruiz and O
.Oreshkov, 2025]. We first show how the frames of external observers can be internalised and treated within the framework. This establishes a consistent hierarchy of QRF perspectives, which we use to define the adding and removing of subsystems. We then explain how, owing to the so-called 'extra-particle' degrees of freedom, the formalism gives a consistent treatment of subsystems, avoiding any paradoxes by construction. Consistency then implies that composing subsystems in a QRF perspective differs from, and generalises, the standard quantum-theoretic case. In particular, not every state can be appended in tensor-product form relative to a QRF, and we characterise the set of compatible states. Finally, we introduce 'classicalisation', a procedure that recovers a classical-like perspective from a general QRF. This procedure circumvents the aforementioned state restrictions but carries a different operational meaning, which we study through a concrete example.
The authors calculate the second virial coefficient of the quantum Lorentz gas with point interaction. It can be evaluated in closed form for arbitrary strength of the interaction. Particular attention is given to the critical coupling which gives rise to a zero-energy resonance. Point interactions satisfy a modified Levinson's theorem; as a consequence they show that there is no cancellation between the bound state and the scattering contributions in the high-temperature expansion of the second virial coefficient, contrary to the regular potentials case. They also derive high- and low-energy sum rules for the point interaction. As an application of the sum rules they rederive a high-temperature expansion for the second virial coefficient.