McGuire, James B.

Solutions and partial solutions of some three body, one dimensional,
quantum mechanical problems are presented in this thesis. The particles
are assumed to interact with one another through delta function potentials.
The three body mathematical problems are transformed into an equivalent
problem which can be interpreted as one particle moving in a two dimensional
space. Some special cases are then considered which can be worked or
partially worked.

A 4N-dimensional formalism is developed and a corresponding space is found.
The necessity of coupling the particle's proper times to one parameter is
discussed. The appropriate conditions and constraints which relate this
space to the ordinary 4-space are found. The transformation properties which
are consistent with general 4-space transformations are determined. These
transformation properties are used to determine the form of the 4x4 matrices
making up the 4N-dimensional metric tensor. The form of these matrices
indicates they represent interactions between particles. The diagonal
matrices are shown to represent gravitational interaction and the off-diagonal
matrices to represent other interparticle interactions. A metric theory to
cover all interparticle interactions is then proposed. The equations of
motion for one particle in this 4N-dimensional space are found. These
equations are then related to the motion of N interacting particles in 4- space .
Finally, an approximation procedure is applied to determine the first order
equations of motion.

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.