Surveying the solar system by measuring angles and times: from the solar density to the gravitational constant

A surprisingly large amount of information on our solar system can be gained from simple measurements of the apparent angular diameters of the sun and the moon. This information includes the average density of the sun, the distance between earth and moon, the radius of the moon, and the gravitational constant. In this note it is described how these and other quantities can be obtained by simple earthbound measurements of angles and times only, without using any explicit information on distances between celestial bodies. The pedagogical and historical aspects of these results are also discussed briefly.Comment: 12 pges, one figur

Tunneling out of a time-dependent well

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: One class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like i.e the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this article we discuss and develop analytic methods for solving time-dependent tunneling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunneling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunneling phenomena in Bose-Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.Comment: 11 pages, 3 figure

The rigged Hilbert space approach to the Lippmann-Schwinger equation. Part II: The analytic continuation of the Lippmann-Schwinger bras and kets

The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the rigged Hilbert space rather than just the Hilbert space. It is also argued that this analytic continuation entails the imposition of a time asymmetric boundary condition upon the group time evolution, resulting into a semigroup time evolution. Physically, the semigroup time evolution is simply a (retarded or advanced) propagator.Comment: 32 pages, 3 figure

Rigged Hilbert Space Approach to the Schrodinger Equation

It is shown that the natural framework for the solutions of any Schrodinger equation whose spectrum has a continuous part is the Rigged Hilbert Space rather than just the Hilbert space. The difficulties of using only the Hilbert space to handle unbounded Schrodinger Hamiltonians whose spectrum has a continuous part are disclosed. Those difficulties are overcome by using an appropriate Rigged Hilbert Space (RHS). The RHS is able to associate an eigenket to each energy in the spectrum of the Hamiltonian, regardless of whether the energy belongs to the discrete or to the continuous part of the spectrum. The collection of eigenkets corresponding to both discrete and continuous spectra forms a basis system that can be used to expand any physical wave function. Thus the RHS treats discrete energies (discrete spectrum) and scattering energies (continuous spectrum) on the same footing.Comment: 27 RevTex page

Relativistic and Radiative Energy Shifts for Rydberg States

We investigate relativistic and quantum electrodynamic effects for highly-excited bound states in hydrogenlike systems (Rydberg states). In particular, hydrogenic one-loop Bethe logarithms are calculated for all circular states (l = n-1) in the range 20 <= n <= 60 and successfully compared to an existing asymptotic expansion for large principal quantum number n. We provide accurate expansions of the Bethe logarithm for large values of n, for S, P and circular Rydberg states. These three expansions are expected to give any Bethe logarithms for principal quantum number n > 20 to an accuracy of five to seven decimal digits, within the specified manifolds of atomic states. Within the numerical accuracy, the results constitute unified, general formulas for quantum electrodynamic corrections whose validity is not restricted to a single atomic state. The results are relevant for accurate predictions of radiative shifts of Rydberg states and for the description of the recently investigated laser-dressed Lamb shift, which is observable in a strong coherent-wave light field.Comment: 8 pages; RevTeX

On the inconsistency of the Bohm-Gadella theory with quantum mechanics

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.Comment: 16 page

Singular Short Range Potentials in the J-Matrix Approach

We use the tools of the J-matrix method to evaluate the S-matrix and then deduce the bound and resonance states energies for singular screened Coulomb potentials, both analytic and piecewise differentiable. The J-matrix approach allows us to absorb the 1/r singularity of the potential in the reference Hamiltonian, which is then handled analytically. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian. The remaining part of the potential, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. To exhibit the power of our approach we have considered the most delicate region close to the bound-unbound transition and compared our results favorably with available numerical data.Comment: 14 pages, 5 tables, 2 figure

Explicit solution for a Gaussian wave packet impinging on a square barrier

The collision of a quantum Gaussian wave packet with a square barrier is solved explicitly in terms of known functions. The obtained formula is suitable for performing fast calculations or asymptotic analysis. It also provides physical insight since the description of different regimes and collision phenomena typically requires only some of the terms.Comment: To be published in J. Phys.