Bound State Wave Functions through the Quantum Hamilton - Jacobi Formalism

The bound state wave functions for a wide class of exactly solvable potentials are found utilizing the quantum Hamilton-Jacobi formalism. It is shown that, exploiting the singularity structure of the quantum momentum function, until now used only for obtaining the bound state energies, one can straightforwardly find both the eigenvalues and the corresponding eigenfunctions. After demonstrating the working of this approach through a number of solvable examples, we consider Hamiltonians, which exhibit broken and unbroken phases of supersymmetry. The natural emergence of the eigenspectra and the wave functions, in both the unbroken and the algebraically non-trivial broken phase, demonstrates the utility of this formalism.Comment: replaced with the journal versio

Shape invariance and the exactness of quantum Hamilton-Jacobi formalism

Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schr\"odinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory.Comment: Accepted for publication in Phys. Lett.

The Quantum Effective Mass Hamilton-Jacobi Problem

In this article, the quantum Hamilton- Jacobi theory based on the position dependent mass model is studied. Two effective mass functions having different singularity structures are used to examine the Morse and Poschl- Teller potentials. The residue method is used to obtain the solutions of the quantum effective mass- Hamilton Jacobi equation. Further, it is shown that the eigenstates of the generalized non-Hermitian Swanson Hamiltonian for Morse and Poschl-Teller potentials can be obtained by using the Riccati equation without solving a differential equation

The Generalized PT-Symmetric Sinh-Gordon Potential Solvable within Quantum Hamilton-Jacobi Formalism

The generalized Sinh-Gordon potential is solved within quantum Hamiltonian Jacobi approach in the framework of PT symmetry. The quasi exact solutions of energy eigenvalues and eigenfunctions of the generalized Sinh-Gordon potential are found for n=0,1 states.Comment: 10 pages appear to in IJT

Breaking and Entering: Verb Semantics and Event Structure

Any event can be construed from a variety of perspectives. While this flexibility is fundamental to human ingenuity, it poses a challenge for language learners who must discern which meanings are encoded in their language and by which forms. The papers in this dissertation focus on verbs encoding directed motion (e.g., a girl runs into a house) and caused change-of-state events (e.g., a boy blows out candles). Both classes of events can be expressed by verbs that lexicalize different components of the event, namely Manner-of-motion (e.g., run) or Path (e.g., enter), and Means (e.g., blow) or Effect (e.g., extinguish), respectively. Papers 1 and 2 examine the representation of higher-order generalizations about the meanings of directed motion and novel caused change-of-state verbs. Both studies use a novel verb-learning paradigm to manipulate the meanings of novel verbs in the input and then assess how learners interpret subsequently encountered novel verbs (measure lexicalization bias). The results indicate that learners rapidly use semantic regularities to form higher-order generalizations about verb meaning. In Paper 1, adults taught Manner verbs construed new directed motion verbs as lexicalizing Manner more often than those taught Path verbs. Moreover, changes in verb learning bias were accompanied by shifts in visual attention: Manner-verb learners fixated on Manner-related elements of visually-presented events more than Path-verb learners. These results indicate that previously observed cross-linguistic differences in verb lexicalization biases are unlikely to stem from the restructuring of semantic representations along language-specific lines and more likely reflect the operation of a flexible, inferential learning mechanism that monitors the input and updates beliefs accordingly. Likewise, in Paper 2, adults taught Means verbs interpreted unknown verbs for caused change-of-state events as encoding the Means more often than those taught Effect verbs. Unlike directed motion verbs, the encoding of these events is not characterized by marked typological variation and the availability of Means and Effect verbs does not appear to vary appreciable within or across languages. Our results, then, suggest that the formation of higher-level generalizations about meaning is a fundamental property of the processes that undergird lexical acquisition. Paper 3 focuses on the representation of the event concepts that underlie verb meanings. Specifically, we examine the possibility that Manner-of-motion and Means are actually instances of a broader semantic category, MANNER, whereas Path and Effect are instances of a different semantic category, RESULT. Adults were taught novel verbs for either directed motion or caused changes of state and subsequently presented with novel verbs from the other semantic class. The results revealed that adults transfer newly-learned higher-order generalizations about the meanings of directed motion verbs to caused change-of-state verbs (and vice versa), providing support for the psychological reality of superordinate event concepts.Psycholog

A Study of Quasi-Exactly Solvable Models within the Quantum Hamilton-Jacobi Formalism

A few quasi-exactly solvable models are studied within the quantum Hamilton-Jacobi formalism. By assuming a simple singularity structure of the quantum momentum function, we show that the exact quantization condition leads to the condition for quasi-exact solvability