Translation-invariant Systems Research Articles

Some exact solutions to the translation-invariant N-body problem

It is shown that Schrodinger's equation for a translation-invariant system consisting of N particles with arbitrary masses interacting via Hooke's law pair potentials with the same coupling constant can be solved exactly; explicit solutions are found for the case N=3. Exact solutions are also found explicitly for the translation-invariant problem in which a particle with mass m0 interacts with N identical particles of mass m1 via a Hooke's law pair potential with coupling constant k02, and the identical particles interact with each other via Hooke's law pair potentials with coupling constant k12. The latter solution provides a basis problem for an energy lower-bound method for translation-invariant atom-like systems.

On the paper by RJM Carr: 'derivation of energy lower bound models for translation-invariant many-fermion systems'

A detailed derivation of the energy lower bound model (SHRIMP model) for a translation-invariant many-fermion system has recently been proposed (Carr 1978). The author asserts that an N-particle shell model retaining antisymmetry in all N particles is obtained. The present author wishes to point out the error in this derivation which makes the SHRIMP model wrong leaving correct the related RIP and HIP models.

Exact solutions of Schrodinger's equation for translation-invariant harmonic matter

Continuation of previous work (ibid., vol.11, no.7, p.1227 (1978)). Earlier exact solutions of Schrodinger's equation for translation-invariant systems of particles interacting by Hooke's law pair potentials are augmented to include systems consisting of an arbitrary number S of groups of identical particles. Exact solutions can always be found whenever the number of distinct masses plus the number of distinct coupling constants does not exceed (2S+1). The case S=2 is solved in detail and is applied to show that harmonic matter is never stable.

Peierls condition and number of ground states

Recently Pirogov and Sinai developped a theory of phase transitions in systems satisfying Peierls condition. We give a criterium for the Peierls condition to hold and apply it to several systems. In particular we prove that ferromagnetic system satisfies the Peierls condition iff its (internal) symmetry group is finite. And using an algebraic argument we show that in two dimensions the symmetry groups of reduced translation invariant systems is finite.

Derivation of energy lower bound models for translation-invariant many-fermion systems

A derivation of the SHRIMP energy lower bound model for translation-invariant many-fermion systems is given. Previous models are obtained as special cases.

Causality of translation-invariant systems

Causality of translation-invariant systems

Infinite Systems of Classical Particles

In this paper, infinite translation-invariant and periodic systems of classical particles are treated directly—not as limits of finite systems. A formalism of classical creation and annihilation operators that create and destroy classical identical particles at points of phase space is employed. By use of this formalism, classical dynamical variables are expressed without reference to the canonical coordinates and momenta of individual particles, or to the number of particles. Different physical situations are described by different representations of the algebra of creation and annihilation operators. The concept of thermal equilibrium is generalized so as to be meaningful in infinite systems. Stationary states and thermal-equilibrium states for an infinite system of noninteracting particles (possibly in a periodic external potential) are exhibited explicitly.