Citation
Diestler, Dennis J. (1968) I. Quantum mechanics of onedimensional twoparticle models. II. A quantum mechanical treatment of inelastic collisions. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/70MF3B21. https://resolver.caltech.edu/CaltechTHESIS:12142015085217224
Abstract
Part I
Solutions of Schrödinger’s equation for system of two particles bound in various stationary onedimensional potential wells and repelling each other with a Coulomb force are obtained by the method of finite differences. The general properties of such systems are worked out in detail for the case of two electrons in an infinite square well. For small well widths (110 a.u.) the energy levels lie above those of the noninteresting particle model by as much as a factor of 4, although excitation energies are only half again as great. The analytical form of the solutions is obtained and it is shown that every eigenstate is doubly degenerate due to the “pathological” nature of the onedimensional Coulomb potential. This degeneracy is verified numerically by the finitedifference method. The properties of the squarewell system are compared with those of the freeelectron and hardsphere models; perturbation and variational treatments are also carried out using the hardsphere Hamiltonian as a zerothorder approximation. The lowest several finitedifference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hardsphere eigenfunctions. The finitedifference solutions in general yield expectation values and matrix elements as accurate as those obtained using the “best” variational function.
The system of two electrons in a parabolic well is also treated by finite differences. In this system it is possible to separate the centerofmass motion and hence to effect a considerable numerical simplification. It is shown that the pathological onedimensional Coulomb potential gives rise to doubly degenerate eigenstates for the parabolic well in exactly the same manner as for the infinite square well.
Part II
A general method of treating inelastic collisions quantum mechanically is developed and applied to several onedimensional models. The formalism is first developed for nonreactive “vibrational” excitations of a bound system by an incident free particle. It is then extended to treat simple exchange reactions of the form A + BC →AB + C. The method consists essentially of finding a set of linearly independent solutions of the Schrödinger equation such that each solution of the set satisfies a distinct, yet arbitrary boundary condition specified in the asymptotic region. These linearly independent solutions are then combined to form a total scattering wavefunction having the correct asymptotic form. The method of finite differences is used to determine the linearly independent functions.
The theory is applied to the impulsive collision of a free particle with a particle bound in (1) an infinite square well and (2) a parabolic well. Calculated transition probabilities agree well with previously obtained values.
Several models for the exchange reaction involving three identical particles are also treated: (1) infinitesquarewell potential surface, in which all three particles interact as hard spheres and each twoparticle subsystem (i.e. BC and AB) is bound by an attractive infinitesquarewell potential; (2) truncated parabolic potential surface, in which the twoparticle subsystems are bound by a harmonic oscillator potential which becomes infinite for interparticle separations greater than a certain value; (3) parabolic (untruncated) surface. Although there are no published values with which to compare our reaction probabilities, several independent checks on internal consistency indicate that the results are reliable.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Chemistry 
Degree Grantor:  California Institute of Technology 
Division:  Chemistry and Chemical Engineering 
Major Option:  Chemistry 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  7 August 1967 
Record Number:  CaltechTHESIS:12142015085217224 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12142015085217224 
DOI:  10.7907/70MF3B21 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9322 
Collection:  CaltechTHESIS 
Deposited By:  INVALID USER 
Deposited On:  14 Dec 2015 19:04 
Last Modified:  20 Dec 2019 19:33 
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