Schrodinger Wave Equation Research Articles

Quantum probability and unified approach to quantization and dynamics

A simplified derivation of the Gudder-Hemion quantum probability formula is proposed. Defining configurations as the classical (q, p) deterministic states and generalized action as the (quantum) generating function of a canonical transformation, we obtain the usual quantization rules (for arbitrary polynomial quantities) and derive the Schrodinger wave equation on the same grounds. This approach suggests a statistical interpretation of the wave function in terms of the classical canonical transformations.

Discrete phase space I. Finite difference operators and lattice Schrödinger wave equation

A new representation of quantum mechanics involving finite difference operators is presented. The time-dependent Schrodinger wave equation is furnished as a partialdifference-differential equation. This wave equation is solved exactly for a three-dimensional oscillator.It is emphasized that this representation is exact and not a numerical approximation.

Discrete symmetries in quantum scattering

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrodinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.

Electronic band structure for two‐dimensional periodic lattice quantum configurations by the finite element method

AbstractA finite element method formulation is given for solving Schrodinger's wave equation for a single electron in a crystal lattice cell subject to a known periodic potential. The formulation has been implemented for a two‐dimensional lattice, with an arbitrary potential profile, modelled by quadratic isoparametric elements, The FEM solver returns a specified number of electronic energy states, En, and nodal values of the complex wavefunctionψn Input data is generated by a standard FEM mesh generator. The postprocessing, given n, for reproducing a full 2‐D E‐k Brillouin diagram and given k, the electronic distribution, has been implemented. Tests on a 2‐D generalized Kronig‐Penney energy band model showed excellent agreement; between FEM results and analysis. The solver was further satisfactorily checked against published augmented plane wave calculations for a circular potential well within a square lattice. Specimen results are presented for the same circular well but with graded potential distributions and for a rectangular potential barrier set askew in a square lattice. Two‐dimensional energy band solvers have application to superlattice nanostruc‐ tures, whilst a general, full 3‐D FEM quantum solver seems feasible.

Exact bound-state solutions of the cut-off Coulomb potential inN-dimensional space

Exact solutions of the potentialV(r)=−Ze 2/(r+β),β>0 are obtained in theN-dimensional space for certain values ofβ by means of factorization of infinite Hill determinant. We discuss some features of the radial wave Schrodinger equation in theN-dimensional space.

Nonlinear wave mechanics, information theory, and thermodynamics

A logarithmic nonlinear term is introduced in the Schrodinger wave equation, and a physical justification and interpretation are provided within the context of information theory and thermodynamics. From the resulting nonlinear Schrodinger equation for a system at absolute temperatureT>0, the energy equivalence,kT 1n 2, of a bit of information is derived.

Fisher information and the complex nature of the Schr�dinger wave equation

We show that the minimum Fisher information (MFI) approach to estimating the probability law p(x) on particle position x, over the class of all two-component laws p(x), yields the complex Schrodinger wave equation. Complexity, in particular, traces from an “efficiency scenario” (demanded by MFI) where the two components of p(x) are so separated that their informations add.

Quantum mechanics of noncentral harmonic and anharmonic potentials in two-dimensions

With a view to obtaining an exact analytic solution of the Schrodinger wave equation for a variety of potentials we investigate further applications of an already suggested ansatz (Phys. Lett. 142A (1989) 57) for the eigenfunction. The problems related to noncentral harmonic and anharmonic potentials in two-dimensions are investigated. While the difficulties that arise in the study of these potentials are explicitly shown, several new potentials are suggested which admit an exact solution of the wave equation and sometimes at the cost of certain constraints on their parameters. The method is also applied to some other types of potentials; particularly, the cases of singular- and exponential-type central potentials in three-dimensions are discussed through the examples of the potential, V( r) = ar 2 + br −2 + cr −6, and the Morse potential.

A new interpretation of the wave function: Reply

I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.

Approximate relativistic quantum Hamiltonians for N interacting particle systems

Time evolution of relativistic particle systems can be accomplished by means of a Schrödinger wave equation, provided the Hamiltonian of the N particle system satisfies some commutator equations involving the other generators of a suitable representation of the Poincaré group on the initial data space of the Schrödinger equation. This set of operator equations is solved in the N free particle case. Further, the structure of N interacting particle Hamiltonians is worked out to include second-order relativistic corrections, i.e., (1/c2)-terms. It is shown that Breit and Barker–O’Connell Hamiltonians (nonzero spin particles) and the Bažański Hamiltonian (zero-spin particles) are particular cases of this more general Hamiltonian.

Applications to Optics and Wave Mechanics of the Criterion of Maximum Cramer-Rao Bound

Abstract A wide class of physical problems requires the estimation of probability laws. Diffraction patterns and quantum mechanical probability laws on position are examples. Minimum Fisher information is one approach to estimating such laws; maximum entropy is another. In this paper, we show that the minimum Fisher information approach may be derived from a prior principle of maximum Cramer-Rao bound (MCRB). The MCRB-Fisher approach is then applied to some fundamental physical problems, including diffraction theory and quantum mechanics. It is found to give the correct physical solutions, that is the Helmholtz and Schrodinger wave equations respectively. By comparison, the maximum entropy approach gives incorrect solutions to these problems, except in the special (thermodynamic) case of a harmonic oscillator potential.

The original schröddinger wave equation revisited

We consider a nonlinear generalization of the Schrodinger wave equation in its original form and investigate its relationship with the corresponding time dependent Schrodinger equation

Possible mass limits for quantum double galaxies

When cosmic quantum mechanics is applied to a double galaxy, the result is mass limits in order for the two galaxies to form a quantum binary system. For a non-relativistic theory (based on the Schrodinger wave equation), the mass limits are: (mg)max ≈1012M⊙ and (mg)min ≈1010M⊙. One possible consequence appears to be a Newtonian gravitational constant that varies with cosmic time, with its value larger in the cosmic past.

Inversion of a normally incident reflection seismogram by the Gopinath--Sondhi integral equation

Summary. The one-dimensional acoustic wave equation has been transformed to two coupled first-order equations whose inverse solution is obtained through application of the Gopinath and Sondhi integral equation. A scatter- ing solution of the Schrodinger wave equation for an explosive source leads us to express the kernel of the Gopinath-Sondhi integral equation in terms of a seismic reflection response. A numerical solution of the integral equation obtained by a trapezoidal rule yields a continuous impedance profile whose derivative has step-like discontinuities. The method is illustrated with computer model studies.

PERIODIC BOUNDARY PROBLEMS FOR SEMI LINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER

Abstract It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order ∂ u ∂ t = ∑ m - 1 M ( - 1 ) m + 1 A m ( x ) ∂ 2 m u ∂ x 2 m + ∑ r = 0 R B r ( t ) ∂ 2 r + 1 u ∂ x 2 r + 1 + f ( u ) , where the matrices A(t), m = 1, …, M are nonnegative definite and the matrices B(t), r = 1 · ·, M are symmetric. One can show that the nonlinear Schrodinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrodinger equation and wave equation etc. may be changed into the above type.

ON THE PERIODIC BOUNDARY PROBLEMS FOR SEMILINEAR DEGENERATE EVOLUTION SYSTEMS OF HIGHER ORDER

It is the purpose of this paper to consider some semilinear evolution systems of partial differential equations, which are found in physics, chemical reactions and biology. The existence, uniqueness and regularity of solutions for periodic boundary problems in global are proved for following degenerate parabolic and hyperbolic systems of higher order ∂u∂t = ∑m-1M(-1)m+1Am(x)∂2mu∂x2m + ∑r=0RBr(t)∂2r+1u∂x2r+1+f(u),where the matrices A(t), m = 1, …, M are nonnegative definite and the matrices B(t), r = 1 · ·, M are symmetric. One can show that the nonlinear Schrödinger equation or system, which may be treated as a real degenerate parabolic system, the Sine-Gordon equation, Klein-Gordon equation, some simultaneous equations of Schrodinger equation and wave equation etc. may be changed into the above type.

On the additional invariance of arbitrary spin relativistic wave equations

The authors demonstrate that the recently derived relativistic Schrodinger wave equations for particles of arbitrary spin and non-vanishing mass also exhibit additional invariance originally pointed out by Fushchich (1974) in the case of the Dirac and Maxwell equations.

Simulation of the Schrodinger equation on SHAC

Describes a simulation of the Schrodinger wave equation for the hydrogen atom on SHAC, a simple homogeneous analogue computer. The program is part of a scheme of research aimed at showing the value of a simple general purpose analogue computer as a teaching aid in chemistry and biology as well as in physics, and in higher education as well as in schools.

Long journey into tunneling

N 1923, during the infancy of the quantum theory, de Broglie [ 11 introducted a new fundamental hypothesis that matter may be endowed with a dualistic natureparticles may also have the characteristics of waves. This hypothesis, in the hands of Schrodinger [2] found expression in the definite form now known as the Schrodinger wave equation, whereby an electron or a particle is assumed to be represented by a solution to this equation. The continuous nonzero nature of such solutions, even in classically forbidden regions of negative kinetic energy, implies an ability to penetrate such forbidden regions and a probability of tunneling from one classically allowed region to another. The concept of tunneling, indeed, arises from this quantum-mechanical result. The subsequent experimental manifestations of this concept can be regarded as one of the early triumphs of the quantum theory. In 1928, theoretical physicists believed that tunneling could occur by the distortion, lowering or thinning, of a potential barrier under an externally applied high electric field. Oppenheimer [3] attributed the autoionization of excited states of atomic hydrogen to the tunnel effect: The coulombic potential well which binds an atomic electron could be distorted by a strong electric field so that the electron would see a finite potential barrier through which it could tunnel. Fowler and Nordheim [4] explained, on the basis of electron tunneling, the main features of the phenomenon of electron emission from cold metals by high external electric fields, which had been unexplained since its observation by Lilienfeld [SI in 1922. They proposed a one-dimensional model. Metal electrons are confined by a potential wall whose height is determined by the work function 9 plus the Fermi energy E/, and the wall thickness is substantially decreased with an externally applied high electric field, allowing electrons to tunnel through the potential wall, as shown in Fig. 1. They successfully derived the well-known Fowler-Nordheim formula for the current as a function of electric field F: J = AF* exp [-4(2nr)1'*t$a'*/3kF]. An application of these ideas which followed almost immediately came in the model for a decay as a tunneling process put forth by Gamow [6] and Gurney and Condon [7]. Subsequently, Rice [8] extended this theory to the description of molecular dissociation. The next important development was an attempt to invoke tunneling in order to understand transport properties of electrical contacts between two solid conductors. The problems of metal-to-metal and semiconductor-to-metal contacts are important technically, because they are directly related to electrical switches and rectifiers or detectors.

Long journey into tunneling

N 1923, during the infancy of the quantum theory, de Broglie [ 11 introducted a new fundamental hypothesis that matter may be endowed with a dualistic natureparticles may also have the characteristics of waves. This hypothesis, in the hands of Schrodinger [2] found expression in the definite form now known as the Schrodinger wave equation, whereby an electron or a particle is assumed to be represented by a solution to this equation. The continuous nonzero nature of such solutions, even in classically forbidden regions of negative kinetic energy, implies an ability to penetrate such forbidden regions and a probability of tunneling from one classically allowed region to another. The concept of tunneling, indeed, arises from this quantum-mechanical result. The subsequent experimental manifestations of this concept can be regarded as one of the early triumphs of the quantum theory. In 1928, theoretical physicists believed that tunneling could occur by the distortion, lowering or thinning, of a potential barrier under an externally applied high electric field. Oppenheimer [3] attributed the autoionization of excited states of atomic hydrogen to the tunnel effect: The coulombic potential well which binds an atomic electron could be distorted by a strong electric field so that the electron would see a finite potential barrier through which it could tunnel. Fowler and Nordheim [4] explained, on the basis of electron tunneling, the main features of the phenomenon of electron emission from cold metals by high external electric fields, which had been unexplained since its observation by Lilienfeld [SI in 1922. They proposed a one-dimensional model. Metal electrons are confined by a potential wall whose height is determined by the work function 9 plus the Fermi energy E/, and the wall thickness is substantially decreased with an externally applied high electric field, allowing electrons to tunnel through the potential wall, as shown in Fig. 1. They successfully derived the well-known Fowler-Nordheim formula for the current as a function of electric field F: J = AF* exp [-4(2nr)1'*t$a'*/3kF]. An application of these ideas which followed almost immediately came in the model for a decay as a tunneling process put forth by Gamow [6] and Gurney and Condon [7]. Subsequently, Rice [8] extended this theory to the description of molecular dissociation. The next important development was an attempt to invoke tunneling in order to understand transport properties of electrical contacts between two solid conductors. The problems of metal-to-metal and semiconductor-to-metal contacts are important technically, because they are directly related to electrical switches and rectifiers or detectors.