Quantum theory

A Test Space is a mathematical object which models the process of scientific inquiry. We examine the motivation of defining the Test Spaces and discuss connections between the Test Spaces and Metamathematics. A variant of Test Spaces called orthogonal partitions is introduced and we draw comparisons between the two. The combinatorial problems of counting finite Test Spaces and orthogonal partitions is highlighted. Some issues in manipulating infinite Test Spaces are discussed as well.

The thesis first discusses the history and general
properties of entropy, concentrating on the mathematical
properties. It then introduces the formalism of
stochastic quantum mechanics and the stochastic
Boltzmann-Gibbs-Shannon entropy, emphasizing the
advantages of the stochastic formalism. The relative
entropy for the spin 1/2 system is constructed in the
formalisms of usual quantum mechanics and stochastic
quantum mechanics. The maximum spin entropy states are
then found for the quantum mechanical relative spin
entropy and the stochastic classical relative spin
entropy. The latter is shown to be an acceptable
candidate for a new relative spin entropy for the
spin 1/2 system.

Remarkable similarities are found in the problem of three impenetrable quantum mechanical particles on a ring and the problem of the quantum billiard in a triangle. If the energy contributed by the motion of the center of mass in the ring problem is subtracted from the total energy, the energy eigenvalues of the particles on the ring are proportional to the energy eigenvalues of the wave function in the corresponding triangle. The eigenvalues derived from the ring solution are unique to that triangle and that energy level. A mathematical relationship is derived, which connects the masses of the particles on the ring to the angles of the triangle. There are three quantum billiard triangles that have previously been solved by the method of separation of variables. The three quantum particles on a ring problem, however, has now been solved for many cases. By correlating the three known triangle solutions to the masses on the ring problem we derive and verify a relationship between the two problems. For the three known triangle solutions, the eigenvalues found in the ring problem are proportional to those found in the triangle. The correlation between the masses on the ring and the triangle is then used to find solutions to other triangles, which do not yield to solution by separation of variables.

Ethylene is the simplest alkene. The carbon-carbon double bond is ubiquitous in the field of chemistry. Ethylene serves as the basis for understanding these molecules. Thus, the assignment of the electronic transitions in ethylene is an important endeavor that many scientists have undertaken, but are yet to decipher theoretically or experimentally. Synchrotron Radiation in the vacuum ultraviolet region allows for magnetic circular dichroism (MCD) measurements of ethylene and other simple alkenes. Studies of ethylene and propylene revealed that the páap* (AgáaB1u ethylene notation) transition is not the lowest energy transition. The páa3s(R) (AgáaB3u ethylene notation) is the lowest energy transition. To further this investigation, MCD and absorption measurement were carried out on isobutene. The isobutene spectra clearly showed four electronic transitions in the 156 to 212 nm wavelength region. These four isobutene transitions have been assigned as páa3s, páap*, páa3p(Sv (Band páa3px proceeding from lower energy to higher energy. The present results support the assignments in ethylene and propylene.