Minimum-uncertainty coherent states of the hyperbolic and trigonometric Rosen–Morse oscillators

A mixed supersymmetric-algebraic approach is employed to generate the minimum uncertainty coherent states of the hyperbolic and trigonometric Rosen–Morse oscillators. The method proposed produces the superpotentials, ground state eigenfunctions and associated eigenvalues as well as the Schrödinger equation in the factorized form amenable to direct treatment in the algebraic or supersymmetric scheme. In the standard approach the superpotentials are calculated by solution of the Riccati equation for the given form of potential energy function or by differentiation of the ground state eigenfunction. The procedure applied is general and permits derivation the exact analytical solutions and coherent states for the most important model oscillators employed in molecular quantum chemistry, coherent spectroscopy (femtochemistry) and coherent nonlinear optics.

     

The eigenvalue problem in phase space

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c‐function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi‐representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc.     

Approximate Eigenvalue and Eigenfunction Solutions for the Generalized Hulthén Potential with any Angular Momentum

An approximate solution of the Schr?dinger equation for the generalized Hulthén potential with non-zero angular quantum number is solved. The bound state energy eigenvalues and eigenfunctions are obtained in terms of Jacobi polynomials. The Nikiforov–Uvarov method is used in the computations. We have considered the time-independent Schr?dinger equation with the associated form of Hulthén potential which simulate the effect of the centrifugal barrier for any l-state. The energy levels of the used Hulthén potential gives satisfactory values for the non-zero angular momentum as the generalized Hulthén effective potential.      

Laplace transform wave functions

The wave function defining a quantum-mechanical system is considered as the Laplace transform of some distribution and the consequent form of the Variational Principle derived; an integral equation defines the eigenfunctions of a certain subclass. The model of the hydrogen-like atom is used to test the theory; the eigenfunctions and associated energy levels of the ground and excited states are obtained for arbitrary values of the orbital quantum number.     

Exponential type complex and non-Hermitian potentials within quantum Hamilton–Jacobi formalism

PT-/non-PT-symmetric and non-Hermitian deformed Morse and P?schl-Teller potentials are studied first time by quantum Hamilton–Jacobi approach. Energy eigenvalues and eigenfunctions are obtained by solving quantum Hamilton–Jacobi equation.     

一维中心势V(q)=Bq2+A/q2(A, B>0)中运动的粒子在量子相空间表示中的解

在Torres-Vega 和 Frederick(T-F)量子相空间理论的框架下, 求解了相空间中一维中心势场V(q)=Bq2+A/q2(A, B>0),中运动的单粒子的本征函数, 并对相空间中的概率密度函数进行了讨论.     

Unified derivation of Bohmian methods and the incorporation of interference effects

We present a unified derivation of Bohmian methods that serves as a common starting point for the derivative propagation method (DPM), Bohmian mechanics with complex action (BOMCA), and the zero-velocity complex action method (ZEVCA). The unified derivation begins with the ansatz psi = eiS/Planck's where the action (S) is taken to be complex, and the quantum force is obtained by writing a hierarchy of equations of motion for the phase partial derivatives. We demonstrate how different choices of the trajectory velocity field yield different formulations such as DPM, BOMCA, and ZEVCA. The new derivation is used for two purposes. First, it serves as a common basis for comparing the role of the quantum force in the DPM and BOMCA formulations. Second, we use the new derivation to show that superposing the contributions of real, crossing trajectories yields a nodal pattern essentially identical to that of the exact quantum wavefunction. The latter result suggests a promising new approach to deal with the challenging problem of nodes in Bohmian mechanics.     

Supersymmetric treatment of a particle subjected to a ring-shaped potential

The ring-shaped Hartmann potential was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM. © 1996 John Wiley & Sons, Inc.     

Ab initio calculation of anisotropic magnetic properties of complexes. I. Unique definition of pseudospin Hamiltonians and their derivation

A methodology for the rigorous nonperturbative derivation of magnetic pseudospin Hamiltonians of mononuclear complexes and fragments based on ab initio calculations of their electronic structure is described. It is supposed that the spin-orbit coupling and other relativistic effects are already taken fully into account at the stage of quantum chemistry calculations of complexes. The methodology is based on the establishment of the correspondence between the ab initio wave functions of the chosen manifold of multielectronic states and the pseudospin eigenfunctions, which allows to define the pseudospin Hamiltonians in the unique way. Working expressions are derived for the pseudospin Zeeman and zero-field splitting Hamiltonian corresponding to arbitrary pseudospins. The proposed calculation methodology, already implemented in the SINGLE_ANISO module of the MOLCAS-7.6 quantum chemistry package, is applied for a first-principles evaluation of pseudospin Hamiltonians of several complexes exhibiting weak, moderate, and very strong spin-orbit coupling effects.     

On the random walk of energy to traps in disordered systems

Adams recently derived a theorem concerning solutions of the Schrödinger equation for an N-electron system. From this theorem he concludes that exact eigenvalues and (antisymmetric) eigenfunctions can be obtained by solving a different type of equation whose eigenfunctions are not antisymmetric. We critically discuss aspects of Adams' formalism.     

Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Nonadiabatic effects in the nuclear dynamics of the H₂⁺ molecular ion, initiated by ionization of the H₂ molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H₂⁺ and D₂⁺ molecules are analyzed. In particular, it is shown that the position-momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.     

Molecular translational-rovibronic hamiltonian. I. Non-linear molecules

The translational-rovibronic hamiltonian for a non-linear polyatomic molecule is derived by using the Schrödinger equation in tensor form and employing the Eckart conditions (determining the nuclear-framework rotational variables). The present derivation is a unified comprehensive one by a quantum-mechanical pathway and contrasts with fragmentary previous derivations via a classical-intermediate path. The method presented affords a firm conceptual picture of the nature of the transformation and the origin of coupling terms, and avoids mathematical complexities with their residue of obscurity. The correct form of the total angular momentum operators is also derived quantum mechanically.     

Supersymmetry and the Hartmann potential of theoretical chemistry

An exactly solvable ring-shaped potential in quantum chemistry given by $$V = \eta \sigma ^2 \varepsilon _{\text{o}} \left( {\frac{{2a_{\text{O}} }}{r} - \frac{{\eta a_{\text{O}}^2 }}{{r^2 {\text{sin}}^{\text{2}} \theta }}} \right)$$ was introduced by Hartmann in 1972 to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed, We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line [0, ∞). A reformulation of SUSYQM in the full line (? ∞, ∞) is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number L but different values of ησ2 and quantum number N. This is in contrast to the first formulation, which relates states with identical values of the quantum number N and ησ2 but different values of the quantum number L.     

Non-Markovian finite-temperature two-time correlation functions of system operators: beyond the quantum regression theorem

An extremely useful evolution equation that allows systematically calculating the two-time correlation functions (CF's) of system operators for non-Markovian open (dissipative) quantum systems is derived. The derivation is based on perturbative quantum master equation approach, so non-Markovian open quantum system models that are not exactly solvable can use our derived evolution equation to easily obtain their two-time CF's of system operators, valid to second order in the system-environment interaction. Since the form and nature of the Hamiltonian are not specified in our derived evolution equation, our evolution equation is applicable for bosonic and/or fermionic environments and can be applied to a wide range of system-environment models with any factorized (separable) system-environment initial states (pure or mixed). When applied to a general model of a system coupled to a finite-temperature bosonic environment with a system coupling operator L in the system-environment interaction Hamiltonian, the resultant evolution equation is valid for both L = L(?) and L ≠ L(?) cases, in contrast to those evolution equations valid only for L = L(?) case in the literature. The derived equation that generalizes the quantum regression theorem (QRT) to the non-Markovian case will have broad applications in many different branches of physics. We then give conditions on which the QRT holds in the weak system-environment coupling case and apply the derived evolution equation to a problem of a two-level system (atom) coupled to the finite-temperature bosonic environment (electromagnetic fields) with L ≠ L(?).     

Asymptotic series for the quantum quartic anharmonic oscillator

We propose a numerical method to find solutions of the one-dimensional Schrödinger equation when the potential is symmetric and can be expanded in a polynomial form. We used a non-perturbative method, in which we include explicitly the correct asymptotic behavior of the wave function computed by the WKB method. The numerical convergence is very fast and allows to compute the energy eigenvalues and eigenfunctions simultaneously. The method is applied to the quartic anharmonic oscillator with one and two wells, we compute the energy eigenvalues for the ground state and for the first six excited states, the results obtained are in agreement with those reported previously in the literature.     

Everyman's derivation of the theory of atoms in molecules

This paper demonstrates that it is straightforward to develop the theory of an atom in a molecule--the extension of quantum mechanics to an open system--by deriving the necessary equations of motion from Schr?dinger's equation, followed by a comparison of the predicted properties with experiment to determine the correct boundary condition. Although less fundamental than the variational derivation of the quantum theory of atoms in molecules, this heuristic approach makes the quantum mechanics of an atom in a molecule accessible to "everyman" possessing a knowledge of Schr?dinger's equation, aiding its general acceptance by experimental chemists.     

Sinc collocation in quantum chemistry: Solving the planar coulomb Schrödinger equation

Dirac bra-ket notation is introduced for the Whittaker cardinal (Sinc) functions and a previously unreported completeness relation for these quantities is presented and derived. With the use of this completeness relation it becomes simple to transform to a Sinc-basis the eigenvalue equations arising from a light-cone quantization of field theory or the similar equations occurring in nonrelativistic quantum mechanics. The simplicity and power of Sinc-function expansions is illustrated by computation of the eigenvalues and eigenfunctions of the position-space planar Coulomb equation, a problem for which convergence has not been achieved by a variety of other computational methods. © 1996 John Wiley & Sons, Inc.     

Approximate analytical solutions for arbitrary l-state of the Hulthén potential with an improved approximation of the centrifugal term

An approximate analytical solution of the radial Schr?dinger equation for the generalized Hulthén potential is obtained by applying an improved approximation of the centrifugal term. The bound state energy eigenvalues and the normalized eigenfunctions are given in terms of hypergeometric polynomials. The results for arbitrary quantum numbers n _r and l with different values of the screening parameter δ are compared with those obtained by the numerical method, asymptotic iteration, the Nikiforov-Uvarov method, the exact quantization rule, and variational methods. The results obtained by the method proposed in this work are in a good agreement with those obtained by other approximate methods.      

Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V(r) = V 0(r/a 0)2ν−2, ν ≥ 1

The solution of a generalized non-relativistic Schrödinger equation with radial potential energy V(r)=V ₀(r/a ₀)^2–2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).