We present a simple method of experimentally studying the elliptic shape of the joined apices of parabolic projectile trajectories in the undergraduate laboratory. The experimental data agrees well with theoretical results, and we find that this experiment provides an interesting twist to the venerable undergraduate experiment on projectile motion.
The one-dimensional (1D) hydrogen atom with potential energy V(q) = −e2/|q|, with e the electron charge and q its position coordinate, has been a source of discussion and controversy for more than 55 years. A number of incorrect claims have been made about its spectrum; for example, that its ground state has infinite binding energy, that bound states associated with a continuum of negative energy values exist, or that anomalous non-Balmer energy levels are present in the system. Given such claims and the ongoing controversy, we have re-analysed the 1D hydrogen atom, first from a classical and then from a quantum perspective both in the coordinate and in the momentum representations. This work exhibits that certain classical properties of the system may serve to clarify the properties of the quantum problem. Using the Dirichlet boundary condition, we show that the singularity of the potential prevents any relation between the right and left sides of the origin. Hence we prove that the attractive potential V(q) acts in that case as an impenetrable barrier splitting the coordinate space into two independent regions. We show that such splitting appears both in the classical and in the quantum descriptions of the system. The analysis presented in this paper may serve as a pedagogical tool for the comparison between classical and quantum problems, as well as an illustrative example of a problem involving a singular potential that can be approached both from its position and momentum representations.
In the referred paper, the authors use a numerical method for solving ordinary differential equations and a softened Coulomb potential -1/√[x(2)+β(2)] to study the one-dimensional Coulomb problem by approaching the parameter β to zero. We note that even though their numerical findings in the soft potential scenario are correct, their conclusions do not extend to the one-dimensional Coulomb problem (β=0). Their claims regarding the possible existence of an even ground state with energy -∞ with a Dirac-δ eigenfunction and of well-defined parity eigenfunctions in the one-dimensional hydrogen atom are questioned.
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