Renormalizing the kinetic energy operator in elementary quantum mechanics

In this paper, we consider solutions to the three-dimensional Schrodinger equation of the form psi(r) = u(r)/r, where u(0) not equal 0. The expectation value of the kinetic energy operator for such wavefunctions diverges. We show that it is possible to introduce a potential energy with an expectation value that also diverges, exactly cancelling the kinetic energy divergence. This renormalization procedure produces a self-adjoint Hamiltonian. We solve some problems with this new Hamiltonian to illustrate its usefulness.

Influence of irradiant energy on degree of conversion, polymerization rate and shrinkage stress in an experimental resin composite system

Objective. This study evaluated the degree of conversion (DC), maximum rate of cure (R(p)(max)), and polymerization stress (PS) developed by an experimental dental composite subjected to different irradiant energies (3,6,12, 24, or 48J/cm(2)) under constant irradiance (500 mw/cm(2)). Methods. DC and R(p)(max) were monitored for 10 min on the bottom surface of 2-mm thick disks and on 150-mu m thick films (representing the top of the specimen) using ATR-FTIR. PS was monitored for 10 min in 2-mm thick disks bonded to two glass rods (O = 5 mm) attached to a universal testing machine. One-way ANOVA/Tukey tests were used and differences in DC and R(p)(max) between top and bottom surfaces were examined using Student`s t-test. Statistical testing was performed at a pre-set alpha of 0.05. Results. For a given surface, DC showed differences among all groups, except at the top between 24 and 48 J/cm(2). R(p)(max) was similar among all groups at the same surface and statistically higher at the top surface. PS also showed significant differences among all groups. Data for 48 J/cm(2) were not obtained due to specimen failure at the glass/composite interface. Significance. Increases in irradiant exposure led to significant increases in DC and PS, but had no effect on R(p)(max) (c) 2008 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.