Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle

The topics of confinement, average forces, and the Ehrenfest theorem are examined for a particle in one spatial dimension. Two specific cases are considered: (i) A free particle moving on the entire real line, which is then permanently confined to a line segment or ‘a box’ (this situation is achieved by taking the limit V ₀?→?∞ in a finite well potential). This case is called ‘a particle-in-an-infinite-square-well-potential’. (ii) A free particle that has always been moving inside a box (in this case, an external potential is not necessary to confine the particle, only boundary conditions). This case is called ‘a particle-in-a-box’. After developing some basic results for the problem of a particle in a finite square well potential, the limiting procedure that allows us to obtain the average force of the infinite square well potential from the finite well potential problem is re-examined in detail. A general expression is derived for the mean value of the external classical force operator for a particle-in-an-infinite-square-well-potential, $\hat{F}$ . After calculating similar general expressions for the mean value of the position ( $\hat{X}$ ) and momentum ( $\hat{P}$ ) operators, the Ehrenfest theorem for a particle-in-an-infinite-square-well-potential (i.e., $\mathrm{d}\langle\hat{X}\rangle/\mathrm{d}t=\langle\hat{P}\rangle/M$ and $\mathrm{d}\langle\hat{P}\rangle/\mathrm{d}t=\langle\hat{F}\rangle$ ) is proven. The formal time derivatives of the mean value of the position ( $\hat{x}$ ) and momentum ( $\hat{p}$ ) operators for a particle-in-a-box are re-introduced. It is verified that these derivatives present terms that are evaluated at the ends of the box. Specifically, for the wave functions satisfying the Dirichlet boundary condition, the results, $\mathrm{d}\langle\hat{x}\rangle/\mathrm{d}t=\langle\hat{p}\rangle/M$ and $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\mathrm{b.t.}+\langle\hat{f}\rangle$ , are obtained where b.t. denotes a boundary term and $\hat{f}$ is the external classical force operator for the particle-in-a-box. Thus, it appears that the expected Ehrenfest theorem is not entirely verified. However, by considering a normalized complex general state that is a combination of energy eigenstates to the Hamiltonian describing a particle-in-a-box with v(x)?=?0 ( $\Rightarrow\hat{f}=0$ ), the result that the b.t. is equal to the mean value of the external classical force operator for the particle-in-an-infinite-square-well-potential is obtained, i.e., $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t$ is equal to $\langle\hat{F}\rangle$ . Moreover, the b.t. is written as the mean value of a quantity that is called boundary quantum force, f _B. Thus, the Ehrenfest theorem for a particle-in-a-box can be completed with the formula $\mathrm{d}\langle\hat{p}\rangle/\mathrm{d}t=\langle{{f_\mathrm{B}}}\rangle$ .     

Thermodynamic and transport properties of nitrogen and butane mixtures

A force field has been developed to describe the phase behaviour, interfacial, and transport properties of nitrogen and hydrocarbon mixtures under conditions relevant to those found in the high pressure extraction of oil from underground reservoirs. A Gibbs ensemble Monte Carlo method is used to parametrize intermolecular potentials for the pure components by matching experimental and simulated liquid and vapour coexisting densities. Also the surface tension, diffusion coefficient and shear viscosity of nitrogen and its mixtures with butane have been determined. The latter properties were obtained by canonical molecular dynamics simulations. The diffusion coefficient and shear viscosity were calculated by a Green-Kubo method. Results for pure nitrogen are given for temperatures ranging from 70 K to 110K. For mixtures of nitrogen with butane, results are presented at 339.4 K and 380.2 K. Good agreement is found between the results of simulations and available experimental data.     

The Numerical Solution of Laplace's Equation on a Wedge

Laplace's equation is considered on regions in the plane, withthe boundary having corners; and the double-layer potentialis used to derive a solution. The essential difficulties, boththeoretically and numerically, are reduced to the case in whichthe boundary is a simple open wedge. The theoretical behaviourof the double layer integral equation is studied explicitly,and then piecewise linear and piecewise quadratic collocationmethods are applied to the numerical solution of the equation.The major question of interest is the stability of the inversesof the approximating equations. The behaviour of the numericalmethods is somewhat surprising, and it is much better than pastanalyses would have led one to expect.     

拟桶式空间的强对偶空间值的向量测度

证明了在某种紧性条件下拟桶式空间的强对偶空间值的向量测度的唯一存在性。     

Comparison of isoscalar and isovector (e,e′) M1 form factors at high momentum transfer

Electron scattering M1 form factors have been measured for the ground state and for the 2.313 MeV M1 transition in ¹⁴N. Whereas the ground-state form factor is in good accord with 1p-shell models, the data for the 2.313 MeV transition show an unexplained enhancement at high momentum transfers.     

Scattering of emitted radiation from inhomogeneous and nonisothermal layers

A parametric study is performed for the exiting monochromatic intensities scattered from finite, plane-parallel, inhomogeneous layers that are driven solely by a distribution of thermal sources. Intensities are obtained by invariantly imbedding the standard and thermal scattering functions. The single scattering albedo ω and the Henyey-Greenstein phase-function parameter g are varied independently, and both linear and exponential profiles are considered. Linear temperature profiles are used, including temperature inversions. The resulting intensities I(μ), μ representing the direction cosine of propagation, are discussed from a remote sensing point of view. For an isothermal and homogeneous medium, the gross characteristics of I(μ) represented by its overall slope I(0)/I(1), mean value (magnitude), and an interior maximum value can be related to the total optical depth t₀, ω, and g, respectively. For a homogeneous medium, linearly decreasing (in the line of sight) temperature profiles tend to obscure the g information and decrease the apparent optical depth. On the other hand, linearly increasing temperature profiles tend to retain g information and increase the apparent optical depth. Temperature inversion profiles give intensities very similar to those for purely linear profiles. Linear and exponential variations of both ω and g for constant temperatures give similar intensity fields. Results for a variation in g can be reproduced fairly well with an average g value. This cannot be done, however, for ω profiles.     

The onset of thermal runaway in partially insulated or cooled reactors

The conditions for the onset of thermal runaway in partiallyinsulated or cooled reactors are investigated. The temperaturein the reactor is taken to satisfy a nonlinear elliptic equationand the reaction is modelled by an Arrhenius heat generationterm with finite activation energy. To determine the onset ofthermal runaway, the method of matched asymptotic expansionsis used to derive expressions for the critical Frank-Kamenetskiiparameter _c() for reactors containing either a small coolingrod or having a small cooling patch on their boundary. The theoryused to determine _c() is an extension of the results of Wardand Keller (1991). These previous results of Ward and Kellerare also extended to the case of finite activation energiesby using a numerical scheme to evaluate the coefficients inthe asymptotic results for _c(). In some special cases, the asymptoticexpansions for _c() are compared with numerical results for _c(),and clear agreement is found.     

Cis-UROCANIC ACID DOWN-REGULATES THE INDUCTION OF ADENOSINE 3'', 5''-CYCLIC MONOPHOSPHATE BY EITHER trans-UROCANIC ACID OR HISTAMINE IN HUMAN DERMAL FIBROBLASTS in vitro

It has been demonstrated that UVB radiation (290-320 nm) suppresses mammalian cell-mediated immunity by effecting the trans to cis isomerization of urocanic acid (UCA) in the stratum corneum, the uppermost layer of the skin. Trans-urocanic acid has been shown to be the photoreceptor for UVB-induced immune suppression and the cis-isomer has been demonstrated to be immunosuppressive. Little is known, however, about how the isomerization of UCA may affect the proximal or distal cells of the skin or the immune system. We report here that trans-UCA is biologically active in vitro in human dermal fibroblasts, inducing adenyl cyclase as measured by cAMP (adenosine 3',5'-cyclic monophosphate) formation in a dose-dependent manner similar to the action of histamine. Trans-UCA and histamine stimulate 50% of maximum activity at concentrations of 3.3 microM and 13.8 microM respectively. Cis-UCA does not increase cAMP in these human fibroblasts but actively down regulates the increase of cAMP induced by either histamine or trans-UCA. Cis-UCA down regulated the histamine response by 75% and the trans-UCA response by 60% at a concentration range of 1 mM to 1 nM. The trans-UCA induction of cAMP can also be downregulated with an H2 histamine receptor antagonist cimetidine. These results support the hypothesis that a cellular target for cis-UCA is the dermal fibroblast and the effects reported here may represent the initial biochemical and cellular event for UVB-induced immune suppression i.e. the immediate step following the isomerization of trans to cis-UCA is the down regulation of cAMP by cis-UCA. Regulation of such an important second messenger such as cAMP could then allow cascading signals to occur, leading to immune suppression.     

CHEMIEXCITATION IN THE PEROXIDATIVE METABOLISM OF N- METHYLCARBAZOLE: MECHANISTIC IMPLICATIONS

Abstract –The peroxidative metabolism of TV-methylcarbazole emits light independently of the presence of oxygen. It is likely that two chemiexcited transients are formed by electron transfer to the activated peroxidase, the cation radical by one electron transfer and a cation biradical by two electron transfer consistent with the failure to observe horseradish peroxidase-II in the steady state of the reaction. In the spectral range investigated (390–700 nm) the observed emission (570–700 nm) is ascribed to the biradical, as the latter is equivalent to an excited state of the postulated iminium cation.
While lipoxygenase has no effect upon JV-methylcarbazole, it markedly enhances the emission if peroxidase is present. This effect requires oxygen and is ascribed to an excited product formed by lipoxygenase acting upon an intermediate hydroperoxide of the aerobic process promoted by peroxidase.