On the number of discrete levels in the quantum problem of three particles

Andrzej Szymacha, J. Werle

A procedure for finding the discrete quantum numbers that, as previously noted, can replace continuous quantum numbers in a representation for systems of three particles of arbitrary mass and spin values is outlined. (D.C.W.)

M. A. Antonets, G. M. Zhislin, Ilya A. Shereshevskii

A. N. Chaba, Raj K. Pathria

Using techniques developed by Greenspoon and Pathria, and by Chaba and Pathria, a rigorous asymptotic analysis of the onset of Bose-Einstein condensation in a finite three-dimensional system at constant pressure, under Dirichlet boundary conditions, is carried out. Finite size corrections arising from a modification of the density of states of the system, as well as from the discreteness of the single-particle energy levels, play an important role in the critical region. At the critical temperature ${T}_{c}$, the volume of the system changes abruptly from being proportional to the total number of particles $N$ to being proportional to ${N}^{\frac{3}{4}}$, the number of particles in the condensate rises sharply from $\ensuremath{\propto}{N}^{\frac{1}{4}}$ to $\ensuremath{\propto}N$, and the specific heat ${C}_{P}$ turns out to be proportional to ${N}^{\frac{9}{8}}$. At somewhat lower temperatures, the finite size effects arising from the discreteness of the single-particle energy levels become dominant and the volume of the system, together with ${C}_{P}$, becomes $\ensuremath{\propto}{N}^{\frac{3}{5}}$.

L. D. Faddeev

Leo Van Hove, Nicolaas Marinus Hugenholtz, L. P. Howland

John T. Brooker

The theory of discrete extension, introduced here, demonstrates that various re-sults from elementary, non-relativistic quantum mechanics can be obtained without recourse to eigenvalue problems, operators, or state vectors in Hilbert space. Us-ing a mathematical structure similar to that of Newtonian mechanics, the theory generates correct, quasi-discrete values of action increments and energy levels for bound states of quantum-level systems. Quantization of the harmonic oscillator provides a simple illustration. Like Newtonian mechanics the theory provides intuitive conceptualizations for basic entities and kinematics. However, in place of the classical concepts of particle and field, the non-classical, unitary concept of a discretely extended ob-ject emerges from the theory as an alternative characterization of the nature of quantum-level entities. Because such an object is well-defined and unitary, the theory avoids the dilemmas of wave-particle dualism and complementarity. The theory offers insight into the nature of a number of quantum effects such as the zero-point energy of a harmonic oscillator. It also makes a number of testable predictions that distinguish it from standard quantum theory and from Bohmian mechanics.