Solution of the quantum harmonic oscillator plus a delta-function potential at the origin: The oddness of its even-parity solutions

We derive the energy levels associated with the even-parity wave functions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of solution of the harmonic oscillator problem.Comment: 9 pages, 3 figure

Towards absolute calibration of optical tweezers

Aiming at absolute force calibration of optical tweezers, following a critical review of proposed theoretical models, we present and test the results of MDSA (Mie-Debye-Spherical Aberration) theory, an extension of a previous (MD) model, taking account of spherical aberration at the glass/water interface. This first-principles theory is formulated entirely in terms of experimentally accessible parameters (none adjustable). Careful experimental tests of the MDSA theory, undertaken at two laboratories, with very different setups, are described. A detailed description is given of the procedures employed to measure laser beam waist, local beam power at the transparent microspheres trapped by the tweezers, microsphere radius and the trap transverse stiffness, as a function of radius and height in the (inverted microscope) sample chamber. We find generally very good agreement with MDSA theory predictions, for a wide size range, from the Rayleigh domain to large radii, including the values most often employed in practice, and at different chamber heights, both with objective overfilling and underfilling. The results asymptotically approach geometrical optics in the mean over size intervals, as they should, and this already happens for size parameters not much larger than unity. MDSA predictions for the trapping threshold, position of stiffness peak, stiffness variation with height, multiple equilibrium points and `hopping' effects among them are verified. Remaining discrepancies are ascribed to focus degradation, possibly arising from objective aberrations in the infrared, not yet included in MDSA theory.Comment: 15 pages, 20 figure

Prominent effect of soil network heterogeneity on microbial invasion

Using a network representation for real soil samples and mathematical models for microbial spread, we show that the structural heterogeneity of the soil habitat may have a very significant influence on the size of microbial invasions of the soil pore space. In particular, neglecting the soil structural heterogeneity may lead to a substantial underestimation of microbial invasion. Such effects are explained in terms of a crucial interplay between heterogeneity in microbial spread and heterogeneity in the topology of soil networks. The main influence of network topology on invasion is linked to the existence of long channels in soil networks that may act as bridges for transmission of microorganisms between distant parts of soil

Is There a Detectable Vishniac Effect?

The dominant linear contribution to cosmic microwave background (CMB) fluctuations at small angular scales (less than one arcsec) is a second-order contribution known as the Vishniac or Ostriker-Vishniac effect. This effect is caused by the scattering of CMB photons off free electrons after the universe has been reionized, and is dominated by linear perturbations near the R_V =2 Mpc/(h Gamma/0.2) scale in the Cold Dark Matter cosmogony. As the reionization of the universe requires that nonlinear objects exist on some scale, however, one can compare the scale responsible for reionization to R_V and ask if a linear treatment is even feasible in different scenarios of reionization. For an Omega_0 = 1 cosmology normalized to cluster abundances, only about 65% of the linear integral is valid if reionization is due to quasars in halos of mass 10^9 solar, while 75% of the integral is valid if reionization was caused by stars in 10^6 solar mass halos. In lambda or open cosmologies, both the redshift of reionization and z_V are pushed further back, but still only 75% to 85% of the linear integral is valid, independent of the ionization scenario. We point out that all odd higher-order moments from Vishniac fluctuations are zero while even moments are non-zero, regardless of the gaussianity of the density perturbations. This provides a defining characteristic of the Vishniac effect that differentiates it from other secondary perturbations and may be helpful in separating them.Comment: 21 pages, 5 figures, accepted to the Astrophysical Journa