Transquantum Dynamics

Segal proposed transquantum commutation relations with two transquantum constants , besides Planck's quantum constant and with a variable i. The Heisenberg quantum algebra is a contraction—in a more general sense than that of Inönü and Wigner—of the Segal transquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit ,0. One physical consequence is a discrete spectrum for canonical variables and space-time coordinates. Another is an interconversion of time and energy accompanying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second.     

On the pointwise behavior of semi-classical measures

In this paper we concern ourselves with the small asymptotics of the inner products of the eigenfunctions of a Schrödinger-type operator with a coherent state. More precisely, let _j and E _j denote the eigenfunctions and eigenvalues of a Schrödinger-type operator H with discrete spectrum. Let _(x,) be a coherent state centered at the point (x, ) in phase space. We estimate as 0 the averages of the squares of the inner products ( _a ^(x,) , _j ) over an energy interval of size around a fixed energy, E. This follows from asymptotic expansions of the form for certain test function and Schwartz amplitudes a of the coherent state. We compute the leading coefficient in the expansion, which depends on whether the classical trajectory through (x, ) is periodic or not. In the periodic case the iterates of the trajectory contribute to the leading coefficient. We also discuss the case of the Laplacian on a compact Riemannian manifold.Research supported in part by NSF grant DMS-9303778     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

Twisted Lie Group C*-Algebras as Strict Quantization

A nonzero 2-cocycle Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C (g*()). Similarly, a multiplier c Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C (g*()) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Plancks constant assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by 0 (Z\{0})-1, for which A0= C0(g*) and A=C*(G,c) for 0, along with a cross-section of the field satisfying Diracs condition asymptotically relating the commutator in A to the Poisson bracket on C(g*()). Note that the quantization of does not occur for =0.     

Classical limit of Klein-Gordon theory in the coherent-state representation

It is shown that holomorphic solutions of the Klein-Gordon equation concentrate around the classical world-line in the limit 0.     

Short-time propagator on curved spaces

It is shown that the standard WKB-approximation (the approximation of the first order in) to the propagator is not sufficient for the construction of the short-time propagator on curved spaces. The proper short-time propagator can be obtained by means of the second order (in) WKB-approximation and then no subtraction of a quantum correction proportional to ² from the original Lagrangian is necessary.The authors are indebted to J. Tolar for valuable critical comments and advices.     

Electronic structure and optical spectra of YBa2Cu3O7−δ thin films in the infrared and UV region

Optical reflectance of YBa₂Cu₃O_7- thin films grown by laser ablation is measured within photon energies of 0.1 eV to 3.5 eV at room temperature. The spectra can be fit congruently with the anisotropic dielectric constants which take account of the intraband free carrier transition and interband transition. The anisotropic plasma frequencies are simulated to be _pl=2.18 eV and _ch=2.80 eV contributed from free carriers on the plane and in the chain, respectively. The interband transition occurs near 2.5 eV and is pertinent to a rather broad line width.     

Some results on electronic two- and one photon raman scattering in CdS

Two photon Raman scattering (TPRS) via virtually excited biexcitons is observed in CdS over a rather large spectral region in a scattering configuration which favours stimulated emission. We observe either a pure longitudinal exciton or-for the first time—a bound exciton (I ₂) as final state particles. Furthermore, the anomaly in the relation between _exc and _R at _exc= E_blex is observed for the first time in a II–VI compound, indicating an energy of the ₁ biexciton level of 5.098 eV in agreement with two photon absorption measurements. With an applied magnetic fieldB, the corresponding shift of the exciton eigenenergies can be observed. For the longitudinal exciton, the diamagnetic shift is 0.35 meV atB=10T forBc in agreement with theoretical predictions. In this configuration also a stimulated one photon spin flip Raman scattering is observed, yielding the well known electronicg-value of 1.78.     

Classical Limit of Quantum Dissipative Systems

We discuss the Lindblad equation for the density matrix where the dissipation is linear in the position operator. We consider a potential which is a bounded perturbation of the harmonic oscillator. We show that the perturbation of the potential leads to an analytic perturbation of the Wigner distribution. Then the Wigner distribution of the quantum dissipative system tends (uniformly in time) to the classical phase space distribution of the classical dissipative system (if the initial distribution converges when 0).     

Notes on the classical and the nonrelativistic limits in quantum mechanics

A method is presented which can be used to discuss both the classical and also the nonrelativistic limit of quantum mechanics. A one-to-one correspondence may be established between the asymptotic convergence of the resolvent and that of the timedependent solution. In so far as the question of dynamics is concerned we investigate the relation between families of nonrelativistic Hamiltonians and the corresponding Dirac-Hamiltonians when c± or when c±0. The nonrelativistic free theory formally shows the same pattern when ±0 (the classical limit) or when ±. The investigation finally shows how the asymptotic convergence of the relativistic theory can take place under some fairly general conditions of the radiation field.     

The Phase of a Quantum Mechanical Particle in Curved Spacetime

We investigate the quantum mechanical wave equations for free particles of spin 0, 1/2, 1 in the background of an arbitrary static gravitational field in order to explicitly determine if the phase of the wavefunction is S/ = p dx /, as is often quoted in the literature. We work in isotropic coordinates where the wave equations have a simple manageable form and do not make a weak gravitational field approximation. We interpret these wave equations in terms of a quantum mechanical particle moving in medium with a spatially varying effective index of refraction. Due to the first order spatial derivative structure of the Dirac equation in curved spacetime, only the spin 1/2 particle has exactly the quantum mechanical phase as indicated above. The second order spatial derivative structure of the spin 0 and spin 1 wave equations yield the above phase only to lowest order in . We develop a WKB approximation for the solution of the spin 0 and spin 1 wave equations and explore amplitude and phase corrections beyond the lowest order in . For the spin 1/2 particle we calculate the phase appropriate for neutrino flavor oscillations.     

Interaction of excitons in indirect solid solutions of In1−xGaxP at high levels of excitation

The cathode luminescence spectra of GaP and solid solutions of In_1–xGa_xP (with an indirect energy band structure) obtained by liquid-phase epitaxy are investigated at T=77 and 9°K. It is established that the recombination of excitons on neutral donors and the pure exciton recombination with the participation of TA (=13±1 meV), LA (=31±1 meV), and TO (=43±2 meV) phonons. dominate in the low-temperature luminescence spectra of GaP and In_1–xGa_xP with a small concentration of photon impurities. The complex luminescence band that arises for sufficiently high levels of excitation and at T = 9°K is attributed to the luminescence of an electron-hole drop (EHD). The transformation of spectral form of the EHD band with the transition from GaP to In_1–xGa_xP is attributed to a simultaneous reduction in the contribution of the TA and TO replicas and to the appearance of a phononless component in the luminescence of the EHD.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 37–40, September, 1985.     

Small ħ asymptotics for quantum partition functions associated to particles in external Yang-Mills potentials

To a gauge field on a principalG-bundlePM is associated a sequence of quantum mechanical Hamiltonians, as Planck's constant 0 and a sequence of representations _n ofG is taken. This paper studies the associated quantum partition functions, trace exp (–tH _n ), and produces a complete asymptotic expansion, as 0, =1/n, of which the principal term, proportional to the classical partition function, is the familiar classical limit.Research supported in part by Deutsche Forschungsgemeinschaft and NSF grant NoPhy 81-09011A-01On leave of absence from Freie Universität, BerlinResearch supported by NSF grant MCS 820176A01     

Probability interpretation of particle(s) along classical trajectories

The theory of divergent series is used to show that the probability density of a wavemechanical state becomes classical trajectories (for a single particle or many interacting particles) in the limit 0. The probability interpretation is claimed to be valid even if a classical particle idea is inserted into the theory. The idea particle is claimed as a classical entity valid for 0. Finally, the double slit experiment is illustrated as an example and interpreted by the above theory. The interpretation is that the interaction between the incident and interference beams and the double slit is approximately described by the wave theory (interference mechanism).     

Isospectral transformations in relativistic and nonrelativistic mechanics

The modified Korteweg de-Vries hierarchy of partial differential equations generating transformations of the one-dimensional Dirac equation, is shown to reduce in the limitc to the Korteweg de-Vries hierarchy, generating isospectral transformations of the Schrödinger equation. The former hierarchy reduces into relativistic and the latter into nonrelativistic isoperiodic transformation in the limit0.     

Heat equation on phase space and the classical limit of quantum mechanical expectation values

The expectation value of a quantum mechanical operator, taken in coherent states and suitably rescaled, is the solution of an initial value problem for the heat equation on phase space, in which plays the role of time, and the classical observable is the distribution of temperature at =0.     

Spin-orbit coupling in the chlorobenzene molecule

The possible ways of drawing of the intensity of the S₀ T* transition in the chlorobenzene molecule are investigated. The effective spin-orbit coupling of either S* or S* states with the T* state leads to the out-of-plane polarized component of phosphorescence in the chlorobenzene. The in-plane polarized component cannot be explained by the mechanism under consideration — by drawing of intensity of the S₀ S₀* transitions through spin-orbit coupling of first order.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 134–138, August, 1974.     

On the failure of the time-energy uncertainty principle

We establish, for the quantum system made up of a single free particle, the formula E t(v/c) , where E is the precision to whichE can be ascertained in time t. The measurement can be carried out with zero disturbance inE itself.     

Solutions of the braid equation related to a Hopf algebra

Two solutionsT andT of the braid equation acting onA A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=. IfA is commutative, thenT= ( denotes the flip: (a b) =b a for anya,b A).Supported by a grant of the Ministry of Education of Poland.     

Field theoretical construction of an infinite set of quantum commuting operators related with soliton equations

The quantum version of an infinite set of polynomial conserved quantities of a class of soliton equations is discussed from the point of view of naive continuum field theory. By using techniques of two dimensional field theories, we show that an infinite set of quantum commuting operators can be constructed explicitly from the knowledge of its classical counterparts. The quantum operators are so constructed as to coincide with the classical ones in the 0 limit (; Planck's constant divided by 2). It is expected that the explicit forms of these operators would shed some light on the structure of the infinite dimensional Lie algebras which underlie a certain class of quantum integrable systems.