Wave Function Of Free Particle Research Articles

One- and two-dimensional higher-point conformal blocks as free-particle wavefunctions in {textrm{AdS}}_3^{otimes m}

We establish that all of the one- and two-dimensional global conformal blocks are, up to some choice of prefactor, free-particle wavefunctions in tensor products of AdS3 or limits thereof. Our first core observation is that the six-point comb-channel conformal blocks correspond to free-particle wavefunctions on an AdS3 constructed directly in cross-ratio space. This construction generalizes to blocks for a special class of diagrams, which are determined as free-particle wavefunctions in tensor products of AdS3. Conformal blocks for all the remaining topologies are obtained as limits of the free wavefunctions mentioned above. Our results show directly that the integrable models associated with all one- and two-dimensional conformal blocks can be seen as limits of free theory, and manifest a relation between AdS and CFT kinematics that lies outside of the standard AdS/CFT dictionary. We complete the discussion by providing explicit Feynman-like rules that can be used to work out blocks for all topologies, as well as a Mathematica notebook that allows simple computation of Casimir equations and series expansions for blocks, by requiring just an OPE diagram as input.

First passage time of laser-driven tunneling.

The notion of the first passage time is shown to offer a meaningful extension to quantum tunneling, providing a closed-integral-form analytical unification of the tunneling rate and the tunneling passage time. We demonstrate that, in suitable potential settings, the quantum first passage time, found as a solution to the Fokker-Planck and backward Kolmogorov's equations for the quantum probability density, recovers the hallmark results for the Kramers escape rate, the lifetime of tunneling quasi-stationary wave packets, leads to a classical, distance-over-speed passage time for a free-particle wave function, and offers useful insights into Keldysh's intimation on the electron barrier-traversal time in field-induced ionization.

Common Features of Free Particle Wave Functions in Curved Space-times

Quantum equations for massless particles of any spin and for massive spin one-half particles are considered in curved space-times. It is demonstrated that in stationary axially symmetric space-times the angular wave functions up to a normalization function are the same as in a Minkowski space-time. The radial wave functions satisfy second order nonhomogeneous differential equations with three nonhomogeneous terms which depend in a unique form on the ratio of time to space curvatures. For a Dirac spin one-half particle, in addition to these three terms a fourth term which depends on the particle rest mass is added.

CsAr, CsXe, and RbXe B2Σ1/2+ Interatomic Potentials Determined from Absorption Spectra and Calculations of Franck–Condon Factors for Free–Free Optical Transitions of Atomic Collision Pairs

Interatomic potentials for the B2Σ1/2+ states of CsAr, CsXe, and RbXe have been determined through comparisons of experimental B ← X absorption spectra for alkali vapor-rare gas mixtures with calculations of the Franck-Condon factors (FCFs) associated with free-free transitions of thermal atomic pairs. Simulations of optical transitions of alkali-rare gas atomic pairs between the thermal and vibrational continua of the X2Σ1/2+ and B2Σ1/2+ states of the molecule, responsible for the blue satellites of the Cs and Rb D2 resonance lines in a rare gas background, require the incorporation of ground-state J values above ∼400 into the FCF calculations and proper normalization of the free-particle wave functions. Absorption spectra computed on the basis of several X and B state interatomic potentials available in the literature were found to be sensitive to the height of the B2Σ1/2+ state barrier, as well as the X2Σ1/2+ state repulsive wall contour and the location of the van der Waals minimum. Other spectral simulations entailed iterative modifications to a selected B2Σ1/2+ interatomic potential, again coupled with comparison to experimental B ← X spectra. Comparisons of calculated spectra with experiment yield a CsXe B2Σ1/2+ potential, for example, exhibiting a barrier height of 76 cm-1 at 5.2 Å and yet is nearly flat at smaller values of internuclear separation (R). The latter contrasts with previous theoretical calculations of VB(R) in the vicinity of the barrier maximum. For the CsAr molecule, the B2Σ1/2+ barrier height was found to be 221 cm-1, which is within 3% of the value determined from pseudopotential calculations incorporating the spin-orbit effect. Reproducing Cs-rare gas experimental absorption spectra also requires the existence of a broad, shallow potential well lying beyond the B2Σ1/2+ barrier that, for CsAr, has a dissociation energy (De ∼ 24 cm-1) a factor of 3 larger than values predicted by theory. Similar results are obtained for the RbXe and CsXe complexes.

Ground state energy of the hydrogen atom inside penetrable spherical cavities; variational approach

In this work we calculate the ground state energy of the hydrogen atom confined in a sphere of penetrable walls of radius Rc. Inside the sphere the system is subject to a Coulomb potential, whereas outside of it the potential is a finite constant V0. The energy is obtained as a function of Rc and V0 by means of the Rayleigh-Ritz variational method, in which, the trial function is proposed as a free particle wave function within a finite square well potential but including an exponential factor that takes into account the electron-nucleus Coulomb attraction. For an impenetrable sphere, , the energy grows fast as Rc approaches zero. On the other hand, when the height of the barrier V0 is finite, the energy increases slowly as Rc goes to zero. We also compute the Fermi contact term, nuclear magnetic screening, polarizability, pressure and tunneling as a function of Rc and V0. As expected, these physical quantities approach the corresponding values of the free hydrogen atom as Rc grows. We also discuss the pressure-induced ionization of the hydrogen atom. The present results are found in good agreement with those previously published in the literature.

Quantum Interference without Quantum Mechanics

A recently proposed model of the Dirac electron, which describes observed properties of the particle correctly, is in the present paper shown to be also able to explain quantum interference by classical probabilities. According to this model, the electron is not point-like, but rather an "entity" formed by a fast periodic motion of a quantum whose energy is equal to the rest energy of the electron. Only after a time span equal to the period of that periodic motion the "entity" becomes the electron, with its properties, mass, spin, charge, etc.. When in motion with respect to the observer, the "dynamic substructure" of the electron described in this way, leads to a certain time structure of its detection probability, if the space-time point of detection is taken as the space-time point of the quantum. In the typical "two slit" experimental situation, this leads to a periodic detection probability with a frequency of twice the De Broglie frequency. This result is identical to the result obtained by the quantum mechanical description of the moving electron by the free particle wave function. The different interpretations of the established interference pattern, inherent in the two alternative theoretical descriptions is outlined, and the relation between the two descriptions discussed. It is concluded that quantum interference is well explained with classical probabilities, without quantum mechanics and without paradoxes. In view of the demonstrated merits of the model on the one hand, and the new aspects regarding the established theories it implies on the other, a more thorough investigation of its role in relation to relativistic quantum mechanics, and to quantum field theory is suggested. Some of the interesting aspects are summarized.

High-Order Energy and Linear Momentum Conserving Methods for the Klein-Gordon Equation

The Klein-Gordon equation is a model for free particle wave function in relativistic quantum mechanics. Many numerical methods have been proposed to solve the Klein-Gordon equation. However, efficient high-order numerical methods that preserve energy and linear momentum of the equation have not been considered. In this paper, we propose high-order numerical methods to solve the Klein-Gordon equation, present the energy and linear momentum conservation properties of our numerical schemes, and show the optimal error estimates and superconvergence property. We also verify the performance of our numerical schemes by some numerical examples.

Origin of Quantum Mechanical Results and Life: A Clue from Quantum Biology

Although quantum mechanics can accurately predict the probability distribution of outcomes in an ensemble of identical systems, it cannot predict the result of an individual system. All the local and global hidden variable theories attempting to explain individual behavior have been proved invalid by experiments (violation of Bell’s inequality) and theory. As an alternative, Schrodinger and others have hypothesized existence of free will in every particle which causes randomness in individual results. However, these free will theories have failed to quantitatively explain the quantum mechanical results. In this paper, we take the clue from quantum biology to get the explanation of quantum mechanical distribution. Recently it was reported that mutations (which are quantum processes) in DNA of E. coli bacteria instead of being random were biased in a direction such that the chance of survival of the bacteria is increased. Extrapolating it, we assume that all the particles including inanimate fundamental particles have a will and that is biased to satisfy the collective goals of the ensemble. Using this postulate, we mathematically derive the correct spin probability distribution without using quantum mechanical formalism (operators and Born’s rule) and exactly reproduce the quantum mechanical spin correlation in entangled pairs. Using our concept, we also mathematically derive the form of quantum mechanical wave function of free particle which is conventionally a postulate of quantum mechanics. Thus, we prove that the origin of quantum mechanical results lies in the will (or consciousness) of the objects biased by the collective goal of ensemble or universe. This biasing by the group on individuals can be called as “coherence” which directly represents the extent of life present in the ensemble. So, we can say that life originates out of establishment of coherence in a group of inanimate particles.

Free-particle wave function and Niederer's transformation

The solutions to the free Schr\"odinger equation discussed by P. Strange (arXiv:1309.6753) and A. Aiello (arXiv:1309.7899) are analyzed. It is shown that their properties can be explained with the help of Niederer's transformation.

Free particle wave function in light of the minimum-length deformed quantum mechanics and some of its phenomenological implications

At a fundamental level the notion of particle (quantum) comes from quantum field theory. From this point of view we estimate corrections to the free particle wave function due to minimum-length deformed quantum mechanics to the first order in the deformation parameter. Namely, in the matrix element $<0 |\varPhi(t,\,\mathbf{x}) |\mathbf{p} >$ that in the standard case sets the free particle wave function $\propto \exp(i[\mathbf{p}\mathbf{x} - \epsilon(\mathbf{p}) t])$ there appear three kinds of corrections when the field operator is calculated by using the minimum-length deformed quantum mechanics. Starting from the standard (not modified at the classical level) Lagrangian, after the field quantization we get a modified dispersion relation, and besides that we find that the particle's wave function contains a small fractions of an antiparticle wave function and the backscattered wave. The result leads to interesting implications for black hole physics.

Initial phase and free-particle wave packet evolution

The evolution of the free-particle wave function in one dimension is the same as scalar Fresnel diffraction from a one-dimensional structure. Quantum mechanics courses often explore the propagation of Gaussian wave packets, but the diffractionlike mathematics is sufficiently tractable to investigate the propagation of other wave packets, both numerically and analytically. More importantly, the diffraction analogy facilitates the development of an intuitive understanding of the role that the initial phase plays in free-particle wave packet evolution. This article considers some of the effects of the initial phase function on the subsequent evolution of free-particle wave packets in the position representation. These considerations reinforce the idea that the classical mechanics limit embodied in the correspondence principle and formalized in the Ehrenfest theorem is necessarily an incomplete representation of quantum behavior.

Quantum mechanical models in fractional dimensions

We formulate an algebraic approach to quantum mechanics in fractional dimensions in which the momentum and position operators P, Q satisfy the R-deformed Heisenberg relations, which depend on an operator ν. We find representations of P, Q in which the dimension d and angular momentum l appear as parameters related to the eigenvalues of ν. We analyse the domain of P and find conditions which ensure that P is Hermitian. We investigate plane wave solutions and also free particle wavefunctions in fractional dimensions, and show that as a consequence of wavefunction continuity l is quantized. The representations of P, Q also lead to the corresponding representations of paraboson operators which are used to solve the harmonic oscillator in dimension d, both algebraically and analytically. We demonstrate that the formalism extends also to time-dependent Hamiltonians by solving the time-dependent harmonic oscillator in any dimension d > 0 using the method of Lewis and Riesenfeld.

Curvature-induced toroidal bound states

Curvature-induced bound-state eigenvalues and eigenfunctions for a particle constrained to move on the surface of a torus are calculated. A limit on the number of bound states that a torus with minor radius a and major radius R can support is obtained. A condition for mapping constrained particle wave functions on the torus into free particle wave functions is established.

Deep inelastic neutron scattering in condensed hydrogen

The neutron cross-section of molecular hydrogen that is measured by deep inelastic neutron scattering (DINS) is compared with two distinct models. One is a generalization of the molecular Young and Koppel model (1964) that takes into account the modification to the translational kinetic energy that is induced by quantum effects. The second model assumes a free particle wave function for the final state of the proton (C. Andreani et al., 1995). The comparison between these two models, and with the experimental results, provides information on the crossover between the molecular and atomic regime of hydrogen in DINS.

Homeomorphism connecting the hydrogenic system and the free-particle system

The topological equivalence between the hydrogenic system and the free-particle system is established by a homeomorphic embedding of ${\mathrm{R}}_{3}$ into ${\mathrm{R}}_{4}$. The oscillator representation of the generators of the dynamical group SO(4,2) is realized for both systems in a unified way. It is shown that the free-particle wave functions form the basis of the unitary irreducible representation D(0,0) of the homogeneous Lorentz group instead of $\mathrm{D}{(0,(\ensuremath{-}\frac{\frac{1}{2}Z}{E})}^{\frac{1}{2}})$, $E&gt;0$, in the case of the hydrogen atom.

Theory of Low-Energy Electron Scattering by Polyatomic Molecules. I

The variational method is developed to attack the scattering of low energy electrons by polyatomic molecules. The form chosen for a trial function is F ( r )= F s h o r t ( r )+ F l o n g ( r ), where F s h o r t ( r ) is the linear combination of quadratically integrable functions with their centers at constituent atoms of the molecule, r the position vector of the scattered electron, and F l o n g ( r ) chosen so as to include distortion of the scattered wave caused by long range forces is smoothly connected to the free particle wave function. The applicability of this theory is tested by a numerical example, in which the calculation is made for the elastic scattering of electrons by the static field of H 2 in the two-center coordinate system and the results are compared with the Hara's (J. Phys. Soc. Japan 27 (1969) 1009).

The time of arrival in quantum mechanics I. Formal considerations

It is argued that the complementarity of time and energy in one-dimensional wave mechanics finds its proper application in the measurement of the time at which a moving particle arrives at (or departs from) a fixed spatial point. A rigorous proof is given that the ordinary Hilbert space of a freely moving particle does not contain a set of measurement eigenstates for this measurement problem. The proof depends upon the semi-infinite extent of the energy spectrum. Both this result and the physical nature of an arrival-time situation suggest that the most suitable and general wave function for the problem might be obtained by arranging that the particle be emitted by a source situated at a finite distance from the point of detection. Such sources are introduced, and the motion over the intervening space is treated by using the one-dimensional nonrelativistic Schrödinger equation. The resultant free-particle wave functions contain components with both positive and negative energies, and constitute a linear space different from the ordinary Hilbert space. The arrival-time problem then becomes one of interpreting these unfamiliar wave functions, with a view to obtaining a formula for the arrival probability per unit time. It is proved with complete generality and rigour that this modified problem also has no solution. The mathematical properties of the time-dependent Schrödinger equation make it impossible to construct any operationally meaningful and apparatus-independent probability formula. This demonstrates the existence of genuine measurement problems which lie beyond the scope of the usual measurement formalism. Such problems can only be satisfactorily discussed by making reference to the mechanical parameters of the measuring apparatus.

K-Ionization Cross Sections for Relativistic Electrons

Experimental values of the $K$-shell ionization cross section are determined from scintillation spectrometer measurements of the $K$ x rays emitted when thin targets of tin and gold are bombarded by 50-, 100-, 200-, and 500-keV electrons. For these energies and atomic numbers, the experimental results show differences from the cross sections given by Burhop's nonrelativistic calculations and by Perlman's relativistic calculations, but show good agreement with the predictions of Arthurs and Moiseiwitsch, who employed relativistic free-particle wave functions for the projectile electron and nonrelativistic wave functions for the atomic electron.