In this thesis, we present a theoretical investigation of reactive processes involving the HeH$^+$ molecular ion, with applications in laboratory and astrophysical plasma physics. We consider in particular two processes, which are the charge transfer in H + He$^+$ collisions at low energy from a molecular approach and the photodissociation of HeH$^+$. At the molecular level, the cross section is the basic quantity that has to be determined in order to achieve an understanding of reactive processes. Its calculation will be based on the description of the reactions using an emph{ab initio}, quantum mechanical approach. In this work, we will rely on the Born-Oppenheimer approximation, which allows the molecular motion to be separated into an electronic and a nuclear motion. The evaluation of cross sections then follows two steps. The first is the determination of the electronic structure of the molecule. We will calculate the adiabatic potential energy curves of the excited electronic states as well as the dipole matrix elements between these states. The non-adiabatic radial and rotational couplings, which result from the breakdown of the Born-Oppenheimer approximation, are also estimated. The second step is to solve the nuclear motion, which we achieve using a time-dependent method based on the propagation of wave packets on the coupled electronic states. A particular emphasis will be put on the importance of the excited states and of the non-adiabatic couplings in the description of reactive processes. In the treatment of the charge transfer reaction between H and He$^+$ in excited states, it is well known that the non-adiabatic radial couplings cannot be neglected. However, we will show that the inclusion of the non-adiabatic rotational couplings is also necessary in order to obtain accurate state-to-state cross sections. In the description of the photodissociation of HeH$^+$ from its ground state, we will show the influence of the excited states on the rate constant and the role of the non-adiabatic radial couplings in the determination of partial cross sections. We will also consider the possible astrophysical applications of the first triplet state of HeH$^+$. We will show that this state is metastable by evaluating its lifetime, and calculate the cross sections and rate constants for the photodissociation and radiative association of HeH$^+$ in this state.

Shor’s algorithm shows that circuit-model quantum computers can factorize integers in polynomial time – exponentially more efficiently than classical computers. There is currently no analogous algorithm for Adiabatic Quantum Computers(AQCs). We illustrate through a number of factorization problems that a naive AQC implemen- tation fails to reveal an exponential speed up. An exponential speed up does become evident with the optimization of the AQC evolution path utilizing existing optimisa- tion approaches. We reduce the computation time even further by optimization over heuristically-derived parametrised functions. Finally, we improve our own results by exploring two-dimensional paths, and give arguments that using more dimensions in the search space can enhance the computational power to an even greater extent.

The fingerprint of deformation in materials at large strains and at high strain rates is the formation of adiabatic shear bands. Adiabatic shear bands lead to unexpected failure of materials during service. This study investigated the possibility of eliminating adiabatic shear bands from materials subjected to severe deformation at high strain rates by post impact heat treatment. Five groups of cylindrical AISI 4340 steel samples were impacted using the Direct Impact Hopkinson Pressure Bar (DIHPB) developed at the University of Manitoba. Selected impacted samples with distinct transformed shear bands were soaked at 350⁰C to 850⁰C for periods ranging from 30 minutes to 4 hours to investigate how temperature and time affects the properties and structure of the shear bands. Annealing the shear bands at 350⁰C resulted in an increase in hardness of the shear bands and the surrounding material outside the shear bands regardless of the heat treatment before impact, amount of deformation, and the time of annealing. Significant decrease in hardness of the shear bands occurred after post impact annealing at 650⁰C for 30 minutes and 2 hours. Hardness of the shear bands reduced to the same level as that of the impacted material outside the shear bands. However, the initial path of the shear bands in the impacted steel samples could be traced through a “signature” left after the annealing. Post-impact annealing of the steel samples at 750⁰C and 850⁰C resulted in a homogenous microstructure with no trace of the shear bands. The “signatures” which were used to trace the path of the shear bands in impacted samples annealed at 650⁰C disappeared and the hardness across the samples became uniform. The observations from this study show that adiabatic shear bands in typical steel can be eliminated by annealing heat treatment. The temperature of annealing is the most critical parameter and the annealing should be performed above 650⁰C.

The fingerprint of deformation in materials at large strains and at high strain rates is the formation of adiabatic shear bands. Adiabatic shear bands lead to unexpected failure of materials during service. This study investigated the possibility of eliminating adiabatic shear bands from materials subjected to severe deformation at high strain rates by post impact heat treatment. Five groups of cylindrical AISI 4340 steel samples were impacted using the Direct Impact Hopkinson Pressure Bar (DIHPB) developed at the University of Manitoba. Selected impacted samples with distinct transformed shear bands were soaked at 350⁰C to 850⁰C for periods ranging from 30 minutes to 4 hours to investigate how temperature and time affects the properties and structure of the shear bands. Annealing the shear bands at 350⁰C resulted in an increase in hardness of the shear bands and the surrounding material outside the shear bands regardless of the heat treatment before impact, amount of deformation, and the time of annealing. Significant decrease in hardness of the shear bands occurred after post impact annealing at 650⁰C for 30 minutes and 2 hours. Hardness of the shear bands reduced to the same level as that of the impacted material outside the shear bands. However, the initial path of the shear bands in the impacted steel samples could be traced through a “signature” left after the annealing. Post-impact annealing of the steel samples at 750⁰C and 850⁰C resulted in a homogenous microstructure with no trace of the shear bands. The “signatures” which were used to trace the path of the shear bands in impacted samples annealed at 650⁰C disappeared and the hardness across the samples became uniform. The observations from this study show that adiabatic shear bands in typical steel can be eliminated by annealing heat treatment. The temperature of annealing is the most critical parameter and the annealing should be performed above 650⁰C.

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One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi <i>et. al. </i> [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. UPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.

One of the most important quantum algorithms is Grover's search algorithm [G96]. Quantum searching can be used to speed up the search for solutions to NP-complete problems e. g. 3SAT. Even so, the best known quantum algorithms for 3SAT are considered inefficient. Soon after Grover's discovery, Farhi and Gutmann [FG96] devised a "continuous-time analogue" of quantum searching. More recently Farhi <i>et. al. </i> [FGGS00] proposed a continuous-time 3SAT algorithm which invokes the adiabatic approximation [M76]. Their algorithm is difficult to analyze, hence we do not know whether it can solve typical 3SAT instances faster than Grover's search algorithm can. I begin with a review of the discrete- and continuous-time models of quantum computation. I then make precise the notion of "efficient quantum algorithms", motivating sufficient conditions for discrete- and continuous-time algorithms to be considered efficient via discussion of standard techniques for discrete-time simulation of continuous-time algorithms. After reviewing three quantum search algorithms [F00,FG96,G96], I develop the adiabatic 3SAT algorithm as a natural extension of Farhi and Gutmann's search algorithm. Along the way, I present the adiabatic search algorithm [vDMV01] and remark on its discrete-time simulation. Finally I devise a generalization of the adiabatic algorithm and prove some lower bounds for various cases of this general framework. UPDATE (February 2003): Please see article http://arxiv. org/abs/quant-ph/0302138 for a resolution to the problem of simulating the continuous-time adiabatic search algorithm with a quantum circuit using only O(sqrt(N)) resources.

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