Schrödinger and Dirac equations for the hydrogen atom, and Laguerre polynomials

It is usually claimed that the Laguerre polynomials were popularized by Schrödinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were not very popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre.     

Sperm viability in the reproductive tract of females in a population of Sceloporus mucronatus exhibiting asynchronous reproduction

Asynchronous reproduction is a common phenomenon in high-elevation populations of lizards from Central México. Sperm storage in the reproductive tract of females is the mechanism for making oocyte fertilization possible. Our study addresses questions related to functional oviductal sperm storage of females mating on different dates throughout the reproductive season. A population of Sceloporus mucronatus with copulation in the summer and ovulation in the fall was chosen for this experiment. Eleven females that copulated in the field during June and 13 females that copulated in captivity during August were maintained in the laboratory until parturition. The number of pregnant females and the litter sizes produced in each experimental group were indicative of the viability of the stored sperm. Sperm stored in the reproductive tract of females were able to fertilize eggs after 4 months. No significant differences were found in the number of pregnant females between the 2 experimental groups nor in the litter sizes that they produced. We found that the amount of time sperm were held in the female reproductive tract (ca. 3 months) had no effect on the capacity of sperm to fertilize eggs. Histological examination of 8 oviducts collected before the mating season eliminated the possibility of sperm storage from one year to the next. In this system, sperm retention could have evolved as a response mechanism to deal with the asynchrony between sexes in the reproductive cycles. However, we cannot rule out alternative hypotheses.