One-dimensional hydrogen atom with minimal length uncertainty and maximal momentum

We present exact energy eigenvalues and eigenfunctions of the one-dimensional hydrogen atom in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP). This form of GUP is consistent with various theories of quantum gravity such as string theory, loop quantum gravity, black-hole physics, and doubly special relativity and implies a minimal length uncertainty and a maximal momentum. We show that the quantized energy spectrum exactly agrees with the semiclassical results.Comment: 10 pages, 1 figur

One dimensional Coulomb-like problem in deformed space with minimal length

Spectrum and eigenfunctions in the momentum representation for 1D Coulomb potential with deformed Heisenberg algebra leading to minimal length are found exactly. It is shown that correction due to the deformation is proportional to square root of the deformation parameter. We obtain the same spectrum using Bohr-Sommerfeld quantization condition.Comment: 11 pages, typos corrected, references adde

Synchronous motion of two vertically excited planar elastic pendula

The dynamics of two planar elastic pendula mounted on the horizontally excited platform have been studied. We give evidence that the pendula can exhibit synchronous oscillatory and rotation motion and show that stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown. We argue that our results are robust as they exist in the wide range of the system parameters.Comment: Submitte

Relativistic quantum mechanics of a Dirac oscillator

The Dirac oscillator is an exactly soluble model recently introduced in the context of many particle models in relativistic quantum mechanics. The model has been also considered as an interaction term for modelling quark confinement in quantum chromodynamics. These considerations should be enough for demonstrating that the Dirac oscillator can be an excellent example in relativistic quantum mechanics. In this paper we offer a solution to the problem and discuss some of its properties. We also discuss a physical picture for the Dirac oscillator's non-standard interaction, showing how it arises on describing the behaviour of a neutral particle carrying an anomalous magnetic moment and moving inside an uniformly charged sphere.Comment: 19 pages, 1 figur

Cancer Biomarker Discovery: The Entropic Hallmark

Background: It is a commonly accepted belief that cancer cells modify their transcriptional state during the progression of the disease. We propose that the progression of cancer cells towards malignant phenotypes can be efficiently tracked using high-throughput technologies that follow the gradual changes observed in the gene expression profiles by employing Shannon's mathematical theory of communication. Methods based on Information Theory can then quantify the divergence of cancer cells' transcriptional profiles from those of normally appearing cells of the originating tissues. The relevance of the proposed methods can be evaluated using microarray datasets available in the public domain but the method is in principle applicable to other high-throughput methods. Methodology/Principal Findings: Using melanoma and prostate cancer datasets we illustrate how it is possible to employ Shannon Entropy and the Jensen-Shannon divergence to trace the transcriptional changes progression of the disease. We establish how the variations of these two measures correlate with established biomarkers of cancer progression. The Information Theory measures allow us to identify novel biomarkers for both progressive and relatively more sudden transcriptional changes leading to malignant phenotypes. At the same time, the methodology was able to validate a large number of genes and processes that seem to be implicated in the progression of melanoma and prostate cancer. Conclusions/Significance: We thus present a quantitative guiding rule, a new unifying hallmark of cancer: the cancer cell's transcriptome changes lead to measurable observed transitions of Normalized Shannon Entropy values (as measured by high-throughput technologies). At the same time, tumor cells increment their divergence from the normal tissue profile increasing their disorder via creation of states that we might not directly measure. This unifying hallmark allows, via the the Jensen-Shannon divergence, to identify the arrow of time of the processes from the gene expression profiles, and helps to map the phenotypical and molecular hallmarks of specific cancer subtypes. The deep mathematical basis of the approach allows us to suggest that this principle is, hopefully, of general applicability for other diseases

The Coulomb problem and Rutherford scattering using the Hamilton vector

The motion of a particle in a Coulomb field is analyzed with the help of the conserved Hamilton vector. This affords a simple way of obtaining both the orbit in configuration space and the hodograph in velocity space. We show how to obtain the Hamilton vector, then, with its help, we get the equations of both trajectories. We next show that the trajectories of the Coulomb problem in velocity space are all circular. We also exhibit a geometric method for calculating the deflection angle in the case of scattering trajectories and then we derive the Rutherford scattering formula. We also discuss an approximate method which takes advantage of the Hamilton vector for studying scattering in a centrally perturbed Coulomb field. As an example of the use of this approach the case of an inverse cubic perturbation is discussed in some detail

An operator solution for the hydrogen atom using the phase as an additional variable

We discuss an operator solution for the bound states of the non-relativistic hydrogen atom. The method adds the phase of a state and its associated operator to the set of variables of the system. The augmented set of operators is found to form a closed set of commutation relations thus comprising an operator Lie algebra. From these relations, the energy spectrum and bounded radial eigenfunctions are calculated. Our approach is analogous to the one employed to compute the angular momentum spectrum and eigenfunctions but with operators satisfying an su(1,1) Lie algebra instead of su(2). This method, with the same operator algebra and minor modifications, may be used to solve the Dirac relativistic hydrogen atom. (c) 2007 American Association of Physics Teachers

An su(1,1) algebraic method for the hydrogen atom

An algebraic solution for the hydrogen atom analogous to the one recently proposed to solve the relativistic version of the system is presented. We add to the usual radial description of the problem an additional angular variable and an associated operator which can be considered as part of an su(1, 1) Lie algebra. The operators of the algebra define radial ladder operator relating the eigenfunctions of the system in unit steps of the principal quantum number. We conclude that the radial bound states of the hydrogen atom in our extended configuration space can be regarded as spanning the minimal M representation of the su(1, 1) Lie algebra. The method can also be extended to solve the s-wave Morse problem and the three-dimensional harmonic oscillator