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- All Subjects: Quantum Mechanics
- All Subjects: Monte Carlo method
- Creators: Lebed, Richard
- Resource Type: Text
- Status: Published
This thesis attempts to explain Everettian quantum mechanics from the ground up, such that those with little to no experience in quantum physics can understand it. First, we introduce the history of quantum theory, and some concepts that make up the framework of quantum physics. Through these concepts, we reveal why interpretations are necessary to map the quantum world onto our classical world. We then introduce the Copenhagen interpretation, and how many-worlds differs from it. From there, we dive into the concepts of entanglement and decoherence, explaining how worlds branch in an Everettian universe, and how an Everettian universe can appear as our classical observed world. From there, we attempt to answer common questions about many-worlds and discuss whether there are philosophical ramifications to believing such a theory. Finally, we look at whether the many-worlds interpretation can be proven, and why one might choose to believe it.
In thesis we will build up our operator theory for finite and infinite dimensional systems. We then prove that Heisenberg and Schrodinger representations are equivalent for systems with finite degrees of freedom. We will then make a case to switch to a C*-algebra formulation of quantum mechanics as we will prove that the Schrodinger and Heisenberg pictures become inadequate to full describe systems with infinitely many degrees of freedom because of inequivalent representations. This becomes important as we shift from single particle systems to quantum field theory, statistical mechanics, and many other areas of study. The goal of this thesis is to introduce these mathematical topics rigorously and prove that they are necessary for further study in particle physics.
representation theory of the Lorentz and Poincare groups, and a review of some basic rela- ´
tivistic wave equations that will play an important role in the work to follow. In Chapter 2,
a complex covariant form of the classical Maxwell’s equations in a moving medium or at
rest is introduced. In addition, a compact, Lorentz invariant, form of the energy-momentum
tensor is derived. In chapter 3, the concept of photon helicity is critically analyzed and its
connection with the Pauli-Lubanski vector from the viewpoint of the complex electromag- ´
netic field, E+ iH. To this end, a complex covariant form of Maxwell’s equations is used.
Chapter 4 analyzes basic relativistic wave equations for the classical fields, such as Dirac’s
equation, Weyl’s two-component equation for massless neutrinos and the Proca, Maxwell
and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir ´
operators of the Poincare group. A connection between the spin of a particle/field and ´
consistency of the corresponding overdetermined system is emphasized in the massless
case. Chapter 5 focuses on the so-called generalized quantum harmonic oscillator, which
is a Schrodinger equation with a time-varying quadratic Hamiltonian operator. The time ¨
evolution of exact wave functions of the generalized harmonic oscillators is determined
in terms of the solutions of certain Ermakov and Riccati-type systems. In addition, it is
shown that the classical Arnold transform is naturally connected with Ehrenfest’s theorem
for generalized harmonic oscillators. In Chapter 6, as an example of the usefulness of the
methods introduced in Chapter 5 a model for the quantization of an electromagnetic field
in a variable media is analyzed. The concept of quantization of an electromagnetic field
in factorizable media is discussed via the Caldirola-Kanai Hamiltonian. A single mode
of radiation for this model is used to find time-dependent photon amplitudes in relation
to Fock states. A multi-parameter family of the squeezed states, photon statistics, and the
uncertainty relation, are explicitly given in terms of the Ermakov-type system.
The auxiliary field diffusion Monte Carlo is an effective and accurate method for calculating the ground state and low-lying exited states in nuclei and nuclear matter. It has successfully employed the Argonne v6' two-body potential to calculate the equation of state in nuclear matter, and has been applied to light nuclei with reasonable agreement with experimental results. However, the spin-orbit interactions were not included in the previous simulations, because the isospin-dependent spin-orbit potential is difficult in the quantum Monte Carlo method. This work develops a new method using extra auxiliary fields to break up the interactions between nucleons, so that the spin-orbit interaction with isospin can be included in the Hamiltonian, and ground-state energy and other properties can be obtained.