Generalization of Supersymmetric Quantum Mechanics

A generalization of supersymmetric quantummechanics can be obtained in two different ways usingthe theory of the q-deformation of the oscillatoralgebra, according to whether q is a root of unity ornot. In the first case the fractional supersymmetricquantum mechanic is between bosons and q-bosons. In thesecond case we obtain the deformed supersymmetricquantum mechanics by considering bosons and a deformed truncated oscillator algebra.     

Quaternionic Formulation of Supersymmetric Quantum Mechanics

Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, super partner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions. Supercharges, super partner Hamiltonians and energy eigenvalues are discussed and it has been shown that the results are consistent with the results of quantum mechanics.     

Supersymmetric Quantum Mechanics and Super-Lichnerowicz Algebras

We present supersymmetric, curved space, quantum mechanical models based on deformations of a parabolic subalgebra of osp(2p+2|Q). The dynamics are governed by a spinning particle action whose internal coordinates are Lorentz vectors labeled by the fundamental representation of osp(2p|Q). The states of the theory are tensors or spinor-tensors on the curved background while conserved charges correspond to the various differential geometry operators acting on these. The Hamiltonian generalizes Lichnerowicz’s wave/Laplace operator. It is central, and the models are supersymmetric whenever the background is a symmetric space, although there is an osp(2p|Q) superalgebra for any curved background. The lowest purely bosonic example (2p, Q) = (2, 0) corresponds to a deformed Jacobi group and describes Lichnerowicz’s original algebra of constant curvature, differential geometric operators acting on symmetric tensors. The case (2p, Q) = (0, 1) is simply the superparticle whose supercharge amounts to the Dirac operator acting on spinors. The (2p, Q) = (0, 2) model is the supersymmetric quantum mechanics corresponding to differential forms. (This latter pair of models are supersymmetric on any Riemannian background.) When Q is odd, the models apply to spinor-tensors. The (2p, Q) = (2, 1) model is distinguished by admitting a central Lichnerowicz-Dirac operator when the background is constant curvature. The new supersymmetric models are novel in that the Hamiltonian is not just a square of super charges, but rather a sum of commutators of supercharges and commutators of bosonic charges. These models and superalgebras are a very useful tool for any study involving high rank tensors and spinors on manifolds. Dedicated to the memory of Tom Branson     

N=4超对称量子力学

利用Clifford代数构造了N=4超对称量子力学的一般形式,并讨论了它的实现.     

Resonances in A=6 Nuclei: Use of Supersymmetric Quantum Mechanics

We propose a novel theoretical technique for the calculation of resonances at low excitation energies in weakly bound systems. Starting from an effective potential, supersymmetric quantum mechanics can be successfully used to generate families of isospectral potentials having desirable and adjustable properties. For resonance states, for which there is no bound ground state of the same spin-parity, one can construct an isospectral potential with a bound state in the continuum (BIC). The potential looks quite different but is strictly isospectral with the original one. The quasi-bound state in the original shallow potential will be effectively trapped in the deep well of the isospectral family facilitating an easier and more accurate calculation of the resonance energy. Application to ⁶He, ⁶Be, and ⁶Li systems yields quite accurate results. The beauty of our technique: We get both the bound ground state and the resonances by a single technique and using the same potential.     

On the Gravitization of Quantum Mechanics 1: Quantum State Reduction

This paper argues that the case for “gravitizing” quantum theory is at least as strong as that for quantizing gravity. Accordingly, the principles of general relativity must influence, and actually change, the very formalism of quantum mechanics. Most particularly, an “Einsteinian”, rather than a “Newtonian” treatment of the gravitational field should be adopted, in a quantum system, in order that the principle of equivalence be fully respected. This leads to an expectation that quantum superpositions of states involving a significant mass displacement should have a finite lifetime, in accordance with a proposal previously put forward by Diósi and the author.     

Generating of Conditionally Exactly Soluble Potentials by Method of Supersymmetric Quantum Mechanics

This paper suggests a systematic method based on supersymmetric quantum mechanics for generating conditionally exactly soluble potentials,and uses the variational supersymnetric WKB method to obtain the approximate wlues of the energy spectrum of the whole class.     

N=2超对称量子力学的新实现和形状不变性

构造新的超荷和定义权重函数,并研究了N=2一维超对称量子力学.在新的实现中讨论了若干实例.     

New Theory of Periodic Systems by Supersymmetric Quantum Mechanics on Finite Interval

We show that the mechanism of gap formation has a resonance nature. The special real solutions were discovered which have knot distribution with a period coinciding with that of potential at all energies of the whole gap interval. In terms of these solutions resonance gap appearance gets the most direct explanation: ever repeating influence of the potential upon wave function results in exponential increase (decrease) of the wave amplitudes in the forbidden zones. Periodical continuation of exactly solvable models of SUSY QM on finite interval gives rise to algorithms of fixing zone properties - shifting chosen boundaries of spectral bands.     

Logarithmic Quantum Mechanics: Existence of the Ground State

We study nonlinear logarithmic Schrödinger equation in three dimensions. We examine energy levels in this setting, we are especially interested in the ground state. We also show some topological properties of the spectrum. The main contribution of this paper is the first rigorous proof of existence of the ground state in logarithmic quantum mechanics.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

Tripartite Nonlinear Entangled State Representations in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators we construct a kind of tripartite nonlinear entangled states, which can make up a complete set. As its application, we also derive nonlinear 3-mode charge-related entangled state. The essential point for constructing these states lies in choosing the appropriate charge operator.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state |α,γ〉_λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

New Tripartite Nonlinear Entangled State Representation in Quantum Mechanics

Based on the technique of integral within an ordered product of nonlinear bosonic operators, we construct a new kind of tripartite nonlinear entangled state ｜α,γ）λ in 3-mode Fock space, which can make up a complete set. We also simply discuss its properties and application.     

Calculating the C Operator in PT-symmetric Quantum Mechanics

It has recently been shown that a non-Hermitian Hamiltonian H possessing an unbroken PT symmetry (i) has a real spectrum that is bounded below, and (ii) defines a unitary theory of quantum mechanics with positive norm. The proof of unitarity requires a linear operator C, which was originally defined as a sum over the eigenfunctions of H. However, using this definition it is cumbersome to calculate C in quantum mechanics and impossible in quantum field theory. An alternative method is devised here for calculating C directly in terms of the operator dynamical variables of the quantum theory. This new method is general and applies to a variety of quantum mechanical systems having several degrees of freedom. More importantly, this method can be used to calculate the C operator in quantum field theory. The C operator is a new time-independent observable in PT-symmetric quantum field theory.     

Quantum Trajectories, State Diffusion, and Time-Asymmetric Eventum Mechanics

We show that the quantum stochastic Langevin model for continuous in time measurements provides an exact formulation of the von Neumann uncertainty error-disturbance principle. Moreover, as it was shown in the 1980s, this Markov model induces all stochastic linear and nonlinear equations of the phenomenological informational dynamics such as quantum state diffusion and spontaneous localization by a simple quantum filtering method. Here we prove that the quantum Langevin equation is equivalent to a Dirac-type boundary-value problem for the second quantized input offer waves from future in one extra dimension, and to a reduction of the algebra of the consistent histories of past events to an Abelian subalgebra for the trajectories of the output particles. This result supports the wave-particle duality in the form of the thesis of Eventum Mechanics that everything in the future is constituted by quantized waves, everything in the past by trajectories of the recorded particles. We demonstrate how this time arrow can be derived from the principle of quantum causality for nondemolition continuous in time measurements.     

The Resonance Phenomena Associated with the Time Asymmetry in Non-Hermitian Quantum Mechanics

Resonances are defined as the poles of the scattering matrix. The poles are associated with the complex eigenvalues of the Hamiltonian which are embedded in the lower half of the complex plane. The asymptotes of the corresponding eigenfunctions are exponentially diverged. Therefore, the resonance eigenfunctions are not embedded in the Hermitian domain of the Hamiltonian. The time asymmetric problem is discussed for these types of non-Hermitian Hamiltonians and several solutions of this problem are proposed.     

Double Coil Resonance Experiment and Partial Reduction of Wave Packet in Quantum Mechanics

Interference and measurement aspects for the double coil resonance experiment are reanalyzed. The resulting situation is analogous to partial reduction of wave packet in quantum mechanics. Using convergence results of relative frequencies, magnitudes of the intensity are calculated when prior probabilities are assigned to the coefficients associated with the states.