Exact solutions of the Schrödinger equation for a class of hyperbolic potential well

We propose a new scheme to study the exact solutions of a class of hyperbolic potential well. We first apply different forms of function transformation and variable substitution to transform the Schrödinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate. And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant, we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well. Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function. The linearly dependent relation between two eigenfunctions is also studied.     

A Schrödinger Potential Conditionally Integrable in Terms of the Hermite Functions

We introduce a biconfluent Heun potential well for the one-dimensional stationary Schrödinger equation which is composed of a confining fraction-power term and a repulsive centrifugal-barrier core. This is a conditionally integrable potential in that the strength of the centrifugal barrier is fixed to a constant. The potential supports a countable infinite number of bound states. We present the general solution of the Schrödinger equation, deduce the exact equation for the energy spectrumand derive a highly accurate approximation for energy levels. The bound state wave functions are written as irreducible linear combinations with constant coefficients of two Hermite functions of a scaled and shifted argument.

     

Spin-dependent potentials in the linearized collective Schrödinger equation for nuclear quadrupole vibrations

The linearized collective Schrödinger equation for nuclear quadrupole surface vibrations incorporates a new spin degree of freedom with a spin value of 3/2. We use this equation to describe the low energy spectrum of certain even-odd Ir nuclei which have a spin 3/2 in their ground state. For that purpose we explicitly introduce collective spin-dependent potentials which simulate the interaction of the valence nucleon with the core. The linearized Schrödinger equation is transformed into an effective Schrödinger equation with collective spin-dependent potentials. Already collective spin-orbit couplings of SO(3) and SO(5) type are sufficient to reproduce the lowest excited states of even-odd Ir nuclei.     

Exact hole-bound states for semiconductor quantum wells with arbitrary potential profiles

An efficient numerical-analytical method for finding confined and continuum states in quantum-well systems with arbitrary potential profiles, described by coupled Schrödinger equations, is presented. The method is based on the analytical properties of the wave functions, in particular, the power series representation of solutions of the corresponding coupled differential equations. Using only the general properties of the coefficients of a system of an arbitrary number of coupled Schrödinger equations, and imposing for definiteness the simplest boundary conditions, we derive exact expressions for the wave functions and present methods for accurate calculations of the energies and wave functions of confined states and of the wave functions of continuum states in quantum wells. The method is applied to the calculation of the dispersion of hole bound states in a single GaAs quantum well with truncated parabolic confining potentials of different strengths. The results are compared with data available from previous calculations.     

Reconstruction of quantum well potentials

The intertwining operator technique is applied to the generalized Schrödinger equation with a position-dependent effective mass. It is shown on concrete examples how to construct the quantum well potential with a desired spectrum for the Schrödinger equation with a nonhermitian kinetic energy operator.     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

Ground State of a Quantum Particle in a Potential Field

A solution of the Schrödinger equation for the ground state of a particle in a potential field is analyzed. Since the wavefunctions of the ground state are nodeless, potentials of various kinds can be unambiguously determined. It turns out that the ground state corresponds to zero energy for a wide class of model potentials. Moreover, the zero level can be a single one at the boundary of the continuous spectrum. Crater-like potentials monotonically dependent on coordinates in one-, two-, and three-dimensional cases are studied. Instanton-type potentials with two local minima are of interest in the one-dimensional case. For the Coulomb potential, the energy of the ground state is stable with respect to both long- and short-range screening of this potential. Two-soliton solutions of the nonlinear Schrödinger equation are found. It is demonstrated that the proposed version of the inverse scattering transform is efficient in the analysis of solutions of differential equations.

     

Approximate analytical solution of continuous states for the l-wave Schrödinger equation with a diatomic molecule potential

The continuous states of the l-wave Schrödinger equation for the diatomic molecule represented by the hyperbolical function potential are carried out by a proper approximation scheme to the centrifugal term. The normalized analytical radial wave functions of the l-wave Schrödinger equation for the hyperbolical function potential are presented and the corresponding calculation formula of phase shifts is derived. Also, we interestingly obtain the corresponding bound state energy levels by analyzing analytical properties of scattering amplitude.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Rigorous Solutions for the One-Dimensional Hartmann Potential in the Quantum Phase-Space Representation

The rigorous solutions of the Schrödinger equation with the one-dimensional Hartmann potential for a particle are solved and discussed within the framework of the quantum phase space representation established by Torres-Vega and Frederick. For a simple example, the uncertainty principle for the quantum probability density functions is revealed in phase space representation.     

Spectrum of localized states in graphene quantum dots and wires

We developed semiclassical method and show that any smooth potential in graphene describing elongated a quantum dot or wire may behave as a barrier or as a trapping well or as a double barrier potential, Fabry–Perot structure, for 1D Schrödinger equation. The energy spectrum of quantum wires has been found and compared with numerical simulations. We found that there are two types of localized states, stable and metastable, having finite life time. These life times are calculated, as is the form of the localized wave functions which are exponentially decaying away from the wire in the perpendicular direction.     

Scattering of a dipole by a dipole

The spectrum of eigenvalues of the Schrödinger equation with a dipole-dipole potential is not bounded below. An appropriate cut-off leads to a correct low energy problem. Furthermore, it is shown that in polar coordinates the corresponding Schrödinger equation is separable for two spins 1/2.     

Energy levels of a particle confined in an ellipsoidal potential well

The Schrödinger equation is solved for a particle confined within the ellipsoidal potential well using the perturbation theory and the Hamiltonian diagonalization method. The explicit expressions are obtained for the energy levels that are size and shape dependent and appropriate wave functions. The calculated energy levels are in a good qualitative and quantitative agreement with the result obtained by numerical solution of the Schrödinger equation. It is revealed that for the lowest states of a given symmetry the region of validity of the perturbation approximation is much larger than it follows from the usual condition of applicability of the perturbation theory. The optical properties of nanoparticles of a prolate and oblate ellipsoidal shape are discussed.     

Violation of the Bloch-Nordsieck mechanism in general non-abelian theories and SUSY QCD

We discuss the ordering of energy levels of bound states of various relativistic wave equations with local potentials. We concentrate on wave equations for two spinless particles. We point out differences and similarities to the case of the nonrelativistic Schrödinger equation.     

A General Approach for the Exact Solution of the Schrödinger Equation

The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.     

On the relation between Schrödinger and von Neumann equations

The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrödinger equation is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given, and its relation to the addition law of vectors in the Hilbert space of states and the role of a constant phase factor of the wave function is elucidated. The superposition law of density matrices, Wigner functions, and tomographic probabilities describing quantum states in the probability representation of quantum mechanics is studied. Examples of spin-1/2 and Schrödinger-cat states of the harmonic oscillator are discussed. The connection of the addition law with the entanglement problem is considered.     

Supersymmetry and Darboux transformations for the generalized Schrödinger equation

We construct Darboux transformations for a generalized Schrödinger equation by means of the intertwining operator method. We establish a relation between first-order Darboux transformations, supersymmetry, and factorization of the Hamiltonians that are associated with our generalized Schrödinger equation. Furthermore, our methods allow for the generation of isospectral potentials, where one of the potentials has additional or less bound states than its partner. In the particular case of a conventional Schrödinger equation our generalized Darboux transformations reduce correctly to the well-known expressions.