Department of Mathematics
http://hdl.handle.net/1957/13737
2014-04-25T04:18:54ZCanonical states in quantum statistical mechanics
http://hdl.handle.net/1957/47523
Canonical states in quantum statistical mechanics
Kvarda, Robert Edward
This report presents a characterization of the quantum mechanical
analog of the Gibbs canonical density. The approach is
based on a method developed by D.S. Carter for the case of classical
statistical mechanics, which considers composite mechanical systems
composed of mechanically and statistically independent components.
After a brief introductory chapter, Chapter II outlines how
the case of classical mechanics may be described in terms of the
usual measure theoretic treatment of probability. The necessary
statistical background of quantum mechanics is then discussed in
Chapter III, relying on the classic treatment of J. von Neumann and
the more recent work of G. W. Mackey. The basic idea of probability
measure in quantum mechanics differs from that in classical measure
theory, for the measure is defined on a non-Boolean lattice consisting
of all closed linear subspaces of a Hilbert space. Because of this
difference, the classical theory of product measures does not apply. Chapter IV presents a detailed treatment of probability measures for
composite quantum systems.
The analog of the Gibbs canonical density is characterized in
Chapter V, by considering a large collection Q of noninteracting
quantum systems, each of which is in an equilibrium statistical state.
The set Q, the Hamiltonian operator for each system, and the
equilibrium states are assumed to have certain properties which are
given as axioms.
The axioms require each Hamiltonian operator to have a pure
point spectrum. It is assumed, without loss of generality, that the
lowest characteristic value of each Hamiltonian is zero. The set Q
is assumed to be closed under the formation of pairwise mechanically
independent composite systems. This implies that the set D of all
Hamiltonian spectra is closed under addition. It is further assumed
that D is closed under positive differences. The final requirement
on the set Q is that it contain certain "harmonic oscillators".
More precisely, for each positive λεD, Q must contain a system
whose Hamiltonian has the spectrum {nλ : n=0,1,2,[superscript ...]}. The usual
assumption is made that each density operator is a function of the
system Hamiltonian. Finally, it is assumed that for each composite
system in Q, with two mechanically independent components, the
component systems are statistically independent.
It is shown that these assumptions imply that each member of Q is in a canonical state at a temperature which is the same for all
systems. The possibility of zero absolute temperature is included.
Graduation date: 1966
1965-08-26T00:00:00ZA construction of the real numbers using nested closed intervals
http://hdl.handle.net/1957/47396
A construction of the real numbers using nested closed intervals
Huang, Nancy Mang-ze
It is well- known that a real number can be defined as an equivalence
class of fundamental rational sequences. In fact, it is also possible
to define a real number as an equivalence class of sequences of
nested closed rational intervals. This paper is devoted to the latter
case.
Graduation date: 1968
1968-04-26T00:00:00ZA program for the fundamental theorum on symmetric functions
http://hdl.handle.net/1957/47395
A program for the fundamental theorum on symmetric functions
Upatisringa, Visutdhi
The computation of symmetric functions can be tedious.
For this reason, the object of this paper is to devise a computer
program so that these symmetric functions can be handled automatically.
The contribution of this investigation is a heuristic for
finding the polynomials proved to exist by the Fundamental Theorem
on Symmetric Functions. This program is heuristic, since a proof
of the necessary existence theorem is lacking. Examples and
Alcom program are given in the appendix.
Graduation date: 1967
1967-03-13T00:00:00ZSummation formulas for the greatest integer part function
http://hdl.handle.net/1957/47390
Summation formulas for the greatest integer part function
Haertel, Raymond Delbert
This thesis contains a collection of summation formulas for
the greatest integer part function. Proofs are supplied for original
results and for those formulas which are stated without proof in the
literature. References are given for formulas and proofs which
appear in the literature.
Graduation date: 1967
1966-08-11T00:00:00Z