Richard L. Liboff

The polygon quantum‐billiard problem

The present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground‐state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited‐state theorem’’ is established which states that for every N‐sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N‐sided regular polygon is spanned by any two of these N nodal‐diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the corresponde...

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 Plancks 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.

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.

Criteria for physical domains in laboratory and solid‐state plasmas

Physical domains relevant to laboratory and solid‐state plasmas are described in terms of relevant characteristic parameters. Strongly‐ and weakly‐coupled classical plasmas are divided according to the plasma parameter Γ, whereas quantum and classical domains are separated according to the thermal DeBroglie wavelength Λ, nondimensionalized through mean interparticle spacing. These parameters are found to obey the relation Λ2=(π/16)1/3(kBT/R*)Γ4/3, where T is temperature and the Rydberg constant R* includes the dielectric constant of the medium and effective mass of charge carriers. The weakly‐coupled degenerate plasma is described in terms of the quantum compression parameter rs, which represents interparticle spacing measured in Bohr radii. An alternative description of this domain is given in terms of a new quantum parameter (labeled ΓQ) whose definition includes the Thomas–Fermi screening length in place of the Debye length in the classical plasma parameter. A graphical display in terms of appropriatel...

On the potentialx 2N and the correspondence principle

Eigenenergies and frequencies are obtained for a particle oscillating in the potential (1/2)kN×2N, wherek is a constant,x is displacement, andN is a real number. These potentials include the harmonic oscillator (N = 1) and the square well (N = ∞). Thenth eigenenergy has the formANnλ(N), whereλ(N) = 2N/(N + 1), andAN is independent ofn. Application is made to the correspondence principle for the potentialsN > 1 and it is concluded the classical continuum is not obtained in Bohrs limitn → ∞. Complete correspondence is attained in Plancks limith → 0, so that for these configurations the limitsh → 0 andn → ∞ are not equivalent. A classical analysis of these potentials is included which reveals the relation logE(ν/νN) = (N − 1)/2N between frequencyv and energyE, where the constantνN is independent ofE.

The Hexagon Quantum Billiard

A subset of eigenfunctions and eigenvalues for the hexagon quantum billiard are constructed by way of tessellation of the plane and incorporation of symmetries of the hexagon. These eigenfunctions are given as a double Fourier series, obeying C6 symmetry. A table of the lower lying eigen numbers for these states is included. The explicit form for these eigenstates is given in terms of a sum of six exponentials each of which contains a pair of quantum numbers and a symmetry integer. Eigenstates so constructed are found to satisfy periodicity of the hexagon array. Contour read-outs of a lower lying eigenstate reveal in each case hexagonal 6-fold symmetric arrays. Derived solutions satisfy either Dirichlet or Neumann boundary conditions and are irregular in neighborhoods about vertices. This singular property is intrinsic to the hexagon quantum billiard. Dirichlet solutions are valid in the open neighborhood of the hexagon, due to singular boundary conditions. For integer phase factors, Neumann solutions are valid over the domain of the hexagon. These doubly degenerate eigenstates are identified with the basis of a two-dimensional irreducible representation of the C6v group. A description is included on the application of these findings to the hexagonal nitride compounds.

Circular‐sector quantum‐billiard and allied configurations

The circular‐sector quantum‐billiard problem is studied. Numerical evaluation of the zeros of first‐order Bessel functions finds that there is an abrupt change in the nodal‐line structure of the first excited state of the system (equivalently, second eigenstate of the Laplacian) at the critical sector‐angle θc=0.354π. For sector‐angle θ0, in the domain 0<θ0<θc, the nodal curve of the first excited state is a circular‐arc segment. For θc<θ0≤π, the nodal curve of the first excited state is the bisector of the sector. Otherwise nondegenerate first excited states become twofold degenerate at the critical‐angle θc. The ground‐ and first‐excited‐state energies (EG,E1) increase monotonically as θ0 decreases from its maximum value, π. A graph of E1 vs θ0 reveals an inflection point at θ0=θc, which is attributed to the change in Bessel‐function contribution to the development of E1. A proof is given for the existence of a common zero for two Bessel functions whose respective orders differ by a noninteger. Applicat...

On the Radial Momentum Operator

The validity of the radial momentum operator as an observable equivalent is investigated. In addition to the fundamental mathematical prescription, purely physical arguments are presented which demonstrate its inadmissibility. This remains in spite of its correct commutator relationship with the radial coordinate. In the course of presenting these arguments, the condition r1/2Ψ(r)←0, r←0, is derived to be a necessary condition that Ψ be an acceptable wave function.

VIBRATING QUANTUM BILLIARDS ON RIEMANNIAN MANIFOLDS

Quantum billiards provide an excellent forum for the analysis of quantum chaos. Toward this end, we consider quantum billiards with time-varying surfaces, which provide an important example of quantum chaos that does not require the semiclassical (ℏ → 0) or high quantum-number limits. We analyze vibrating quantum billiards using the framework of Riemannian geometry. First, we derive a theorem detailing necessary conditions for the existence of chaos in vibrating quantum billiards on Riemannian manifolds. Numerical observations suggest that these conditions are also sufficient. We prove the aforementioned theorem in full generality for one degree-of-freedom boundary vibrations and briefly discuss a generalization to billiards with two or more degrees-of-vibrations. The requisite conditions are direct consequences of the separability of the Helmholtz equation in a given orthogonal coordinate frame, and they arise from orthogonality relations satisfied by solutions of the Helmholtz equation. We then state and prove a second theorem that provides a general form for the coupled ordinary differential equations that describe quantum billiards with one degree-of-vibration boundaries. This set of equations may be used to illustrate KAM theory and also provides a simple example of semiquantum chaos. Moreover, vibrating quantum billiards may be used as models for quantum-well nanostructures, so this study has both theoretical and practical applications.

Criterion for exciton lasing in pure crystals

A scheme for lasing action involving excitons in a pure crystal is described. The lasing mechanism depends on the nonboson quality of excitons. We find this property to be exhibited at high exciton density. A criterion for lasing under these conditions is obtained.