I. Quantum mechanics of one-dimensional two-particle models. II. A quantum mechanical treatment of inelastic collisions

Part I

Solutions of Schrödinger’s equation for system of two particles bound in various stationary one-dimensional 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 (1-10 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 one-dimensional Coulomb potential. This degeneracy is verified numerically by the finite-difference method. The properties of the square-well system are compared with those of the free-electron and hard-sphere models; perturbation and variational treatments are also carried out using the hard-sphere Hamiltonian as a zeroth-order approximation. The lowest several finite-difference eigenvalues converge from below with decreasing mesh size to energies below those of the “best” linear variational function consisting of hard-sphere eigenfunctions. The finite-difference 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 center-of-mass motion and hence to effect a considerable numerical simplification. It is shown that the pathological one-dimensional 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 one-dimensional 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) infinite-square-well potential surface, in which all three particles interact as hard spheres and each two-particle subsystem (i.e. BC and AB) is bound by an attractive infinite-square-well potential; (2) truncated parabolic potential surface, in which the two-particle 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.

Invisível, sutil e palpável: Shén神 nas dimensões diagnose e terapêutica da Racionalidade Médica Chinesa

Shén神 é uma categoria chinesa de sentidos múltiplos e abrangentes, presente na filosofia e na medicina. Direcionaria o processo de surgimento de todas as coisas no universo, como uma força configuradora. Na pessoa, atuaria sobre todos os estágios de desenvolvimento corporal e pessoal, do nascimento até a morte, caracterizando-a como pessoa única no universo. Teve seus sentidos reduzidos pela Medicina Chinesa Contemporânea, hegemônica na China e instituída após a Revolução comunista. Costuma ser traduzida como Espírito ou Mente no ocidente e é indissociável do corpo físico. Está presente em todas as dimensões da Racionalidade Médica Chinesa, observando-se na literatura convencional, grande abrangência de sentidos quando tratada pela dimensão cosmológica, porém costuma ser reduzida aos sentidos de mente, segundo definições ocidentais, quando tratada pelas dimensões de ordem prática: diagnose e terapêutica. O desafio desta tese foi elaborar uma síntese entre os sentidos cosmológicos e as dimensões práticas. Para isso, procedeu a uma pesquisa teórico conceitual, a partir de leituras e interpretações ocidentais de textos clássicos chineses, elaboradas por autores com critério filológico apurado. Observou-se que os sentidos de Shén神 se fazem presentes em ressonância recíproca com diversas outras categorias da Doutrina Médica Chinesa, como Qì氣, Xuè血, Jīng精, Qìng情, Emoções, Zàng-Fǔ贓腑, Órgãos e Vísceras, entre outras. Cada uma delas com atribuições específicas, que, porém, se reorganizariam em ressonância com as outras. Assim, ao proceder a diagnose e instituir uma terapêutica direcionada a cada uma dessas categorias, o terapeuta estaria interferindo diretamente sobre Shén神 e vice versa. Shén神 poderia assumir sentidos diversos, numa visão de totalidade. A partir daí, percebeu-se a necessidade de estudar essas categorias em ressonância com Shén神, além da própria diagnose e da terapêutica, à luz das premissas do Pensamento Chinês. Categorias como processo, totalidade, potencial ou eficácia ajudaram a apreender, não só os amplos sentidos de Shén神, e suas ressonâncias, mas também as peculiaridades do ato de diagnosticar e tratar na Racionalidade Médica Chinesa. Foi, então, elaborada uma visão da Diagnose e da Terapêutica capaz de contemplar Shén神 e suas ressonâncias, que se espera poder utilizar nos serviços de saúde, contribuindo para estratégias de promoção da saúde, estreitamento de vínculos terapeuta-paciente e maior eficácia terapêutica na prática da Racionalidade Médica Chinesa.

Dynamic response of hysteretic systems with application to a system containing limited slip

A general class of single degree of freedom systems possessing rate-independent hysteresis is defined. The hysteretic behavior in a system belonging to this class is depicted as a sequence of single-valued functions; at any given time, the current function is determined by some set of mathematical rules concerning the entire previous response of the system. Existence and uniqueness of solutions are established and boundedness of solutions is examined.

An asymptotic solution procedure is used to derive an approximation to the response of viscously damped systems with a small hysteretic nonlinearity and trigonometric excitation. Two properties of the hysteresis loops associated with any given system completely determine this approximation to the response: the area enclosed by each loop, and the average of the ascending and descending branches of each loop.

The approximation, supplemented by numerical calculations, is applied to investigate the steady-state response of a system with limited slip. Such features as disconnected response curves and jumps in response exist for a certain range of system parameters for any finite amount of slip.

To further understand the response of this system, solutions of the initial-value problem are examined. The boundedness of solutions is investigated first. Then the relationship between initial conditions and resulting steady-state solution is examined when multiple steady-state solutions exist. Using the approximate analysis and numerical calculations, it is found that significant regions of initial conditions in the initial condition plane lead to the different asymptotically stable steady-state solutions.