The difficulties associated with the quantization of the gravitational field suggests a modification of space-time is needed. For example at suffici~ly small length scales the geometry of space-time might better discussed in terms of a noncommutative algebra. In this thesis we discuss a particular example of a noncommutative algebra, namely the symplectic Schonberg algebra, which we treat as a geometric algebra. Thus our investigation has some features in common with recent work that explores how geometry can be formulated in terms of noncommutative structures. The symplectic Schonberg algebra is a geometric algebra associated with the covariant and the contravariant vectors of a general affine space. The "embedding" of this space in a noncommutative algebra leads us to a structure which we regard as a noncommutative affine geometry. The theory in question takes us naturally to stochastic elements without the usual ad-hoc assumptions concerning measurements in physical ensembles that are made in the usual interpretation of quantum mechanics. The probabilistic nature of space is obtained purely from the structure of this algebra. As a consequence, geometric objects like points, lines and etc acquire a kind of fuzzy character. This allowed us to construct the space of physical states within the algebra in terms of its minimum left-ideals as was proposed by Hiley and Frescura [1J. The elements of these ideals replace the ordinary point in the Cartesian geometry. The study of the main inner-automorphisms of the algebra gives rise to the representation of the symplectic group of linear classical canonical transformations. We show that this group acts on the minimum left-ideal of the algebra and in this case manifests itself as the metaplectic group, i.e the double covering of the symplectic group. Thus we are lead to the theory of symplectic spinors as minimum left-ideals in exactly the same way as the orthogonal spinors can be formulated in terms of minimum left-ideals in the Clifford algebra .. The theory of the automorphisms of the symplectic Schonberg algebra allows us to give a geometrical meaning to integral transforms such as: the Fourier transform, the real and complex Gauss Weierstrass transform, the Bargmann (3) transform and the Bilateral Laplace transform. We construct a technique for obtaining a realization of these algebraic transformations in terms of integral kernels. This gives immediately the Feynmann propagators of conventional non-relativistic quantum mechanics for Hamiltonians quadratic in momentum and position. This then links our approach to those used in quantum mechanics and optics. The link between the theory of this noncommutative geometric algebra and the theory of vector bundles is also discussed.

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In this work we reconstruct Quantum Mechanical wave functions for some confining potentials, using the moment-wavelet method of Handy and Murenzi. This method consists in transforming the Schrodinger equation into an equivalent continuous wavelet transform (CWT) representation involving scaled and translated moments, µ1/ α,b (p)=∫χpe-Q\( Ψ(x +b), where ϐ -Q becomes the mother wavelet. The discrete bound states are determined through a multiscale process involving the integration of a finite number of coupled linear first order differential equation in the moments µ1/ α,b (p). The underlying initial value problem depends on moment quantization methods to determine the infinite scale (a = ∞) moments and energy. Using this method we calculate the energies and wavefunctions for the first two quantum states of quartic and dectic anharmonic oscillator potentials, V(x) = mχ2 + gχ4, V(χ) = χ 2+χ10 respectively.

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