Classical irreversibility and mapping of the isosceles triangle billiard

Mapping of the two-dimensional isosceles triangle billiard onto the circular one-dimensional motion of two mass points is described. The singular nature of trajectories directly incident on an acute vertex is discussed in the framework of the present mapping. For an obtuse-angled isosceles triangle, dynamical equations in two-particle space applied to an orbit along a hypotenuse incident on the obtuse vertex suggests irreversible behavior at the critical angle =2/3. Thus it is found that the nonsingular motion of a finite smooth-walled disk on this trajectory exhibits irreversibility. A finite spherical smooth-walled particle moving in a uniform right cylinder whose cross section includes this critical vertex angle likewise exhibits irreversibility. Each such example comprises an irreversible orbit for a single-particle Hamiltonian.     

Complex Hyperspherical Equations, Nodal-Partitioning, and First-Excited-State Theorems in ℝ n

Three problems related to the spherical quantum billiard in are considered. In the first, a compact form of the hyperspherical equations leads to their complex contracted representation. Employing these contracted equations, a proof is given of Courant's nodal-symmetry intersection theorem for diagonal eigenstates of spherical-like quantum billiards in . The second topic addresses the first-excited-state theorem for the spherical quantum billiard in . Wavefunctions for this system are given by the product form, ( )Z _q+()Y ⁽ⁿ⁾ , where is dimensionless displacement, is angular-momentum number, qis an integer function of dimension, Z() is either a spherical Bessel function (nodd) or a Bessel function of the first kind (neven) and represents (n– 1) independent angular components. Generalized spherical harmonics are written . It is found that the first excited state (i.e., the second eigenstate of the Laplacian) for the spherical quantum billiard in is n-fold degenerate and a first excited state for this quantum billiard exists which contains a nodal bisecting hypersurface of mirror symmetry. These findings establish the first-excited-state theorem for the spherical quantum billiard in . In a third study, an expression is derived for the dimension of the th irreducible representation (irrep) of the rotation group O(n) in by enumerating independent degenerate product eigenstates of the Laplacian.     

Plasma domains in extrinsic GaAs and InP

Various parameters are introduced relevant to criteria for physical domains in solid-state plasmas. Application is made to extrinsic GaAs and InP at 300 K and varying charge-carrier concentrations. At concentrations less than ~ 10¹⁵ cm^?3, charge-carrier plasmas for both p- and n-type semiconductors, respectively, are classical and weakly coupled. At a concentration of 10¹⁶ cm^?1 the plasmas grow degenerate. At a concentration of 10¹⁷ cm^?3 both p-type materials approach a degenerate state whereas both n-type materials are weakly-coupled degenerate.     

Prime Quasientropy and Quasichaos

We construct a prime symmetry relation for integers that is equivalent to Goldbach's conjecture and show that numerical computations of this prime symmetry property strongly resemble a chaotic sequence. We define and examine the notions of global and local prime quasientropies. Finally, we employ the fact that the prime number sequence satisfies the property of deterministic randomness to consider its utility for the field of quantum computation.     

Quasi-Chaotic Property of the Prime-Number Sequence

The prime-number sequence, viewed as thespectrum of eigenvalues of random matrices, is found tobe quasi-chaotic. Plots of histograms of prime-numbernearest-neighbor spacing Delta p at various values of total number of integers indicate roughagreement with the Wigner distribution and illustratelevel repulsion. A global maximum of these curves isnoted at p = 6. Numerical work further implies that in any maximum integer sampling, no matterhow large, a finite number of nearest neighbor spacingsdo not occur. This quasichaotic property of theprime-number sequence supports the conjecture that a formula for the n-th prime does not exist. Arule for missing spacings is inferred according towhich, as maximum number of integers increases, nearest neighbor vacancies corresponding tosmaller vanish and new, larger value vacancies appear. Inaddition, early values of these histograms illustrate arough oscillatory behavior with periodicity[p] 6. A corollary to the resultsimplies that zeros of the Riemann zeta function likewise comprisea quasi-chaotic sequence. Application of these findingsto the resonant spectra of excited nuclei isnoted.     

Quantum chaos for the radially vibrating spherical billiard

The spherical quantum billiard with a time-varying radius, a(t), is considered. It is proved that only superposition states with components of common rotational symmetry give rise to chaos. Examples of both nonchaotic and chaotic states are described. In both cases, a Hamiltonian is derived in which a and P are canonical coordinate and momentum, respectively. For the chaotic case, working in Bloch variables (x,y,z), equations describing the motion are derived. A potential function is introduced which gives bounded motion of a(t). Poincare maps of (a,P) at x=0 and the Bloch sphere projected onto the (x,y) plane at P=0 both reveal chaotic characteristics. (c) 2000 American Institute of Physics.     

Fusion via metallic deuterium

The possibility of ²H(²H, n)³He fusion occurring in the metallic phase of deuterium at low temperatures is suggested. The argument is based on the finite valuedness of the radial distribution at zero argument for a (charged neutral) charged both fluid. Realization of the scheme further depends on the existence of a liquid phase of metallic deuterium at low temperatures.     

Conjectured superfluidity of deuterium

It is proposed that supercooled deuterium may suffer a phase change to the superfluid state near 3 K. A rough estimate of the possibility of the d(d, n)α³ fusion reaction due to large wavefunction overlap in the bose condensate is found to be infinitesimally small.     

Bohr correspondence principle for large quantum numbers

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI ^?1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.