Green's operator for Hamiltonians with Coulomb plus polynomial potentials

<p> We study quantum theory as an example of entropic inference. Our goal is to remove conceptual difficulties that arise in quantum mechanics. Since probability is a common feature of quantum theory and of any inference problem, we briefly introduce probability theory and the entropic methods to update probabilities when new information becomes available. Nelson's stochastic mechanics and Caticha's derivation of quantum theory are discussed in the subsequent chapters. </p><p> Our first goal is to understand momentum and angular momentum within an entropic dynamics framework and to derive the corresponding uncertainty relations. In this framework momentum is an epistemic concept &ndash; it is not an attribute of the particle but of the probability distributions. We also show that the Heisenberg's uncertainty relation is an osmotic effect. Next we explore the entropic analog of angular momentum. Just like linear momentum, angular momentum is also expressed in purely informational terms. </p><p> We then extend entropic dynamics to curved spaces. An important new feature is that the displacement of a particle does not transform like a vector. It involves second order terms that account for the effects of curvature . This leads to a modified Schr&ouml;dinger equation for curved spaces that also take into account the curvature effects. We also derive Schrodinger equation for a charged particle interacting with external electromagnetic field on general Riemannian manifolds. </p><p> Finally we develop the entropic dynamics of a particle of spin 1/2. The particle is modeled as a rigid point rotator interacting with an external EM field. The configuration space of such a rotator is <i>R</i>³ &times; <i>S</i>³ (<i>S</i>³ is the 3-sphere). The model describes the regular representation of <i>SU</i>(2) which includes all the irreducible representations (spin 0, 1/2, 1, 3/2,...) including spin 1/2.</p>

<p> The Baranger model is used to compute collisional broadening and shift of the D1 and D2 spectral lines of M + Ng, where M = K, Rb, Cs and Ng = He, Ne, Ar, using scattering phase shift differences which are calculated from scattering matrix elements. Scattering matrix elements are calculated using the Channel Packet Method where the collisions are treated non-adiabatically and include spin-orbit and Coriolis couplings. Non-adiabatic wavepacket dynamics are determined using the split-operator method together with a unitary transformation between adiabatic and diabatic representations. Scattering phase shift differences are thermally weighted and integrated over energies ranging from E = 0 Hartree up to E = 0.0075 Hartree and averaged over values of total angular momentum that range from J = 0.5 up to J = 400.5. Phase shifts are extrapolated linearly to provide an approximate extension of the energy regime up to E = 0.012 Hartree. Broadening and shift coefficients are obtained for temperatures ranging from T = 100 K up to T = 800 K and compared with experiment. Predictions from this research find application in laser physics and specifically with improvement of total power output of Optically Pumped Alkali Laser systems.</p>