We report the first isoelectronic differential force measurements between a Au-coated probe and two Au-coated films, made out of Au and Ge. These measurements, performed at submicron separations using soft microelectromechanical torsional oscillators, eliminate the need for a detailed understanding of the probe-film Casimir interaction. The observed differential signal is directly converted into limits on the parameters α and λ which characterize Yukawa-like deviations from Newtonian gravity. We find α < ∼ 10 12 for λ ∼ 200 nm, an improvement of ∼ 10 over previous limits.

The treatment of the time-independent Schrödinger equation in real space is an indispensable part of introductory quantum mechanics. In contrast, the Schrödinger equation in momentum space is an integral equation that is not readily amenable to an analytical solution, and is rarely taught. We present a numerical approach to the Schrödinger equation in momentum space. After a suitable discretization process, we obtain the Hamiltonian matrix and diagonalize it numerically.By considering a few examples, we show that this approach is ideal for exploring bound states in a localized potential, and complements the traditional (analytical or numerical) treatment of the Schrödinger equation in real space.

We provide a formalism to calculate the effect of the finite size of the sample on hypothetical Yukawa-like corrections to the Newtonian gravitational potential. It is explicitly shown that finite size effect contributions are negligible when the extent of the sample is larger than the range of the Yukawa term. In particular we show that these contributions are small in the configuration of a recent experiment. In the experiment a gold coated sphere was moved across the interface between two materials with different mass densities. In view of these new experimental results, we analyze the situation when the error on the hypothetical correction could result to be significant.

We provide a formalism to calculate the effect of the finite size of the sample on hypothetical Yukawa-like corrections to the Newtonian gravitational potential. It is explicitly shown that finite size effect contributions are negligible when the extent of the sample is larger than the range of the Yukawa term. In particular we show that these contributions are small in the configuration of a recent experiment. In the experiment a gold coated sphere was moved across the interface between two materials with different mass densities. In view of these new experimental results, we analyze the situation when the error on the hypothetical correction could result to be significant.