Bohr Quantization Research Articles

A Rydberg-Bohr-de Broglie Algebraic Model of Hydrogen Atoms

Bohr ignored the electromagnetic interaction of moving charges in atoms and only considered Coulomb forces, thus encountering half frequency difficulties. This article considers the Lorentz force to make the classical Bohr model consistent with modern quantum theory, uses de Broglie's standing wave principle instead of Bohr's quantization assumption, derives the energy level formula for hydrogen atoms, and explains the radiation mechanism of hydrogen atomic spectra. The hydrogen atom in a spectral tube emits a photon wave train during one free path, and the wavelength of the photon is the eigen wavelength of its wave train. The head wavelength of the wave train is four thirds of its eigen wavelength, and the tail wavelength of the wave train is two-thirds of its eigen wavelength.

Квантование энергетических уровней в 1D квантовом колодце в случае мгновенных стационарных состояний при нерелятивистском движении стенки и микрочастицы

The paper considers the problem of finding energy levels in the 1D quantum well in case of its width alteration at the nonrelativistic rate. According to the reviewed literature, the exact solution is known only in the case of nonrelativistic motion of the 1D quantum well wall at the constant rate. It is shown that motion with the constant rate is physically unrealizable. Therefore, it is necessary to find at least small areas of the Schrodinger equation solution for a wider range of nonrelativistic alterations in the 1D quantum well width. Analysis results presented in the study show existence of such areas. The found areas correspond to the instantaneous stationary states satisfying the Bohr quantization condition. In this case, the Dirichlet condition is also satisfied on the moving wall. It means that in this case energy of the level with the n number also becomes a function of the k second quantum number, which takes into account dynamic alteration in the 1D quantum well width. Variants were found of the k second quantum number spectrum and of the quantum level spectrum in various cases of the wall continuous motion with zero initial speed and finite acceleration. Within the framework of the analysis used, formulas were obtained to change the difference between energies of the two arbitrary levels. An analysis was made for the boundaries of the wall speed and the 1D quantum well width in considering the nonrelativistic problem. The obtained results and their possible applications are under discussion, including analysis of the problems related to nanotechnology

Naïve Bohr-type quantization for power-law potentials

The naïve Bohr quantization condition mvr=nℏ is applied to arbitrary spherically symmetric power-law potentials. The dependence of the energy eigenvalues on the principal quantum number n agrees with fully quantum mechanical results either exactly (for the hydrogen atom, the harmonic oscillator, and the infinite square well) or asymptotically (the linear potential). This naïve treatment can be used in precalculus, high-school, or “quantum theory for poets” expositions.

Full-Dimensional Theory of Pair-Correlated HNCO Photofragmentation.

Full-dimensional semiclassical dynamical calculations combining classical paths and Bohr quantization of product internal motions are reported for the prototype photofragmentation of isocyanic acid in the S1 state. These calculations allow one to closely reproduce for the first time key features of state-of-the-art imaging measurements at photolysis wavelengths of 201 and 210 nm while providing insight into the underlying dissociation mechanism. Quantum scattering calculations being beyond reach for most polyatomic fissions, pair-correlated data on these processes are much more often measured than predicted. Our theoretical approach can be used to fill this gap.

A secondary solution of the Schrödinger and Klein-Gordon equations in modeling atomic, molecular and electrodynamic systems

We study atomic and molecular (AM) stationary systems and electrodynamic (ED) systems, composed of an electron interacting with an electromagnetic field. We show that the Schrödinger and Klein-Gordon equations written for these systems have a secondary solution, which is the wave function associated to a classical system. For AM systems, the wave surfaces (S surfaces) and their normals (C curves) are solutions of the Hamilton-Jacobi equation, written for the same system. The S surfaces have a periodical motion and the C curves are closed. The integral relation of the Schrödinger equation on the C curve has a solution identical to the wave function associated to the classical motion. This solution leads to the generalized Bohr quantization relation and to the generalized de Broglie relations, which are valid in the space of the electron coordinates. An identical wave function verifies the Klein-Gordon equation, in the case of the EM systems. The above properties lead to a central field method for calculation of the energetic values and symmetry properties for AM systems, whose accuracy is comparable to the accuracy of the Hartree-Fock method. They also lead to an accurate method for modeling EM systems, which is verified by experimental data from the literature. The above properties are deduced without using any approximation.

A NEW GEOMETRICAL FRAMEWORK FOR THE DE BROGLIE–BOHM QUANTUM THEORY

A geometrical framework for the de Broglie–Bohm quantum theory is presented, in which the trajectories of an N-particle system are interpretable as the integral curves of a particular vector field defined on a 3N-dimensional manifold [Formula: see text] constructed from physical space M. It is mathematically valid even when M is curved. If M is flat, the usual theory is recovered and automatically expressed in whatever curvilinear coordinates one may wish to choose. The general construction is illustrated by the case of a free particle moving on the surface of a sphere. (A modified Bohr quantization condition for angular momentum is obtained, with a first correction proportional to the curvature.) The Zeeman effect and some bound states on the sphere are also considered.

Normalization of the Gaussian binning trajectory method for indirect reactions

The method of Gaussian weighted trajectories combines the classical description of gas-phase chemical reactions and Bohr quantization of final fragment vibrations. In practice, trajectories are assigned Gaussian statistical weights such that the closer the final vibrational actions to integer values, the larger the weights. This approach, called classical trajectory method with Gaussian binning (CT-GB) in the following, is a practical implementation of classical S matrix theory (CSMT) in the random phase approximation. However, CSMT is non unitary and consequently, the most utilized version of CT-GB is not strictly normalized to unity. In other words, the sum of product and re-formed reagent state populations is not exactly equal to one. The purpose of this work is to show that normalizing these populations to unity should significantly improve the quality of the predictions for indirect reactions. This finding is illustrated from calculations on the D++H2 reaction.

Classical photodissociation dynamics with Bohr quantization: Application to the fragmentation of a van der Waals cluster

The recent classical dynamical approach of photodissociations with Bohr quantization [L. Bonnet, J. Chem. Phys. 133 (2010) 174108] is applied for the first time to a realistic process, the photofragmentation of the van der Waals cluster NeBr2. We illustrate the fact that this approach, formally equivalent to the standard one, may be numerically much more efficient.

Classical photodissociation dynamics with Bohr quantization

The standard classical expression of the state-resolved photodissociation cross section is not consistent with an efficient Bohr quantization of product internal motions. A new and strictly equivalent expression not suffering from this drawback is proposed. This expression opens the way to more realistic classical simulations of direct polyatomic photodissociations in the quantum regime where only a few states are available to the products.

Bohr's semiclassical model of the black hole thermodynamics

We propose a simple procedure for evaluating the main attributes of a Schwarzschild's black hole: Bekenstein-Hawking entropy, Hawking temperature and Bekenstein's quantization of the surface area. We make use of the condition that the circumference of a great circle on the black hole horizon contains finite and whole number of the corresponding reduced Compton's wavelength. It is essentially analogous to Bohr's quantization postulate in Bohr's atomic model interpreted by de Broglie's relation. It implies the standard meaning of the black hole entropy corresponding to surface of the quantum variation of the great circles on the black hole horizon surface area. We present black hole radiation in the form conceptually analogous to Bohr's postulate on the photon emission by discrete quantum jump of the electron within the Old quantum theory. This enables us, in accordance with Heisenberg's uncertainty relation and Bohr's correspondence principle, to make a rough estimate of the time interval for black hole evaporation, which turns out very close to time interval predicted by the standard Hawking's theory. Our calculations confirm Bekenstein's semiclassical result for the energy quantization, in variance with Frasca's (2005) calculations. Finally we speculate about the possible source-energy distribution within the black hole horizon.

Quantum structure of space near a black hole horizon

We describe a midi-superspace quantization scheme for generic single horizon black holes in which only the spatial diffeomorphisms are fixed. The remaining Hamiltonian constraint yields an infinite set of decoupled eigenvalue equations: one at each spatial point. The corresponding operator at each point is the product of the outgoing and ingoing null convergences, and describes the scale invariant quantum mechanics of a particle moving in an attractive 1/X2 potential. The variable X that is analogous to particle position is the square root of the conformal mode of the metric. We quantize the theory via Bohr quantization, which by construction turns the Hamiltonian constraint eigenvalue equation into a finite difference equation. The resulting spectrum gives rise to a discrete spatial topology exterior to the horizon. The spectrum approaches the continuum in the asymptotic region.

Wave model for conservative bound systems

In the hidden variable theory, Bohm proved a connection between the Schrodinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrodinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.

Relation between stable orbits and quantum transmission resonance in ballistic cavities

Classical and quantum-mechanical transport properties in chaotic cavities are investigated to establish a link between them. Because of the stickiness at the boundary between stochastic seas and islands of regular orbits in phase space, classical trajectories spend a long time in the vicinity of a few regular orbits. The trapping results in an exclusive excitation of these stable orbits even when the cavity is terminated by classical leads. The wave-function pattern at quantum transmission resonances is found to be identical with one of the stable orbits. The correspondence implies that the transmission resonance takes place when the stable orbit satisfies Bohr and Sommerfeld's quantization rule, and hence explains why conductance fluctuations in ballistic cavities contain only several frequency components.

PRINCIPLES OF A NEW QUANTUM THEORY

The principles of a random quantum theory (R-QT) which is alternatively time-asymmetric or time-symmetric if the quantization is Fermi–Dirac or Bose–Enstein, respectively are presented. Bohr's quantization rule is applied on the field-action integral. A time topological space, [Formula: see text], is mathematically defined in which the paradoxes in standard quantum theory are solved. The time "quantum", is created as a regular, positive into-map of an observed observable's change resulting from a fundamental interaction process. [Formula: see text] is constructed as the union of time elements and can be embedded disconnectedly in the continuous Newtonian universal time, [Formula: see text]. Six axioms are formulated characterizing the space–time and R-QT. The disconnectedness of the (κ×λκ)-fold time-space, [Formula: see text], imparts a kind of disconnectedness to the κ×λκ-fold space–times, [Formula: see text], and induces the chrono-topology. In chrono-topology the unitary, U, or non-measure preserving, R, dynamics, is implemented by means of a time evolution, "complex" operator, [Formula: see text]. It breaks down by means of Bohr quantization into: [Formula: see text][Formula: see text] coincides formally — apart from the spontaneous renormalization — with the time evolution operator in the standard QFT. [Formula: see text] is a novum and produces the Maxwell–Boltzmann energy level distribution in a non-Euclidean QFT. Compatibility between time-reversal invariance of the standard QT equations and irreversibility of some phenomena both in microcosmos and macrocosmos is obtained. The [Formula: see text]-evolution leads to a time's arrow on quantum-scale systems.

Nuclear force strength: a simple estimate

The Bohr quantization rule for angular momentum is applied to one of the simplest nuclear systems, the deuteron. The interaction energy between the neutron and the proton is taken to be of the Yukawa type, with a strength g2. Circular orbits are considered and g2 is estimated using the known binding energy of the deuteron. The resulting value of g2 compares favourably with accepted values. Another feature which the model predicts correctly is that the deuteron has a single stable state, the ground state. This simple model may thus present an interest, from a pedagogical viewpoint, at the elementary level.

Modified Bohr quantization of charmonium

Bohr quantization of circular orbits is modified by the addition of radial excitations treated in the harmonic approximation, and by the use of half-interger angular momentum quantum numbers. Applied to the hydrogen spectrum, the modification leads to a correct assignment of angular momenta, but fails to reproduce the ground state. Applied to charmonium this simple method yields a spectrum in fairly good agreement with the results of other more complicated calculations.

New states of positronium and the neutrino

It is argued that non-radiating stationary orbits imply rotating fields. For positronium the interparticle force is found to be (1 - β 2) 1 2 e 2/ r 12 2, where charges ± e follow a circular orbit of diameter r 12 with velocity β c. Bohr quantization of canonical angular momentum leads to two sets of stationary states, one familiar and the other new. The properties of the latter are such that they offer a model for the neutrino.

Semiclassical quantization of the electron-dipole system

A recent note by Fox called attention to a semiclassical treatment by Fermi in 1925 of the motion of an electron in the field of two stationary positive charges. The similarity of this problem to that of an electron moving in a dipole field prompted the author to investigate the latter system under semiclassical quantization. It is shown that Bohr quantization leads to the value Dmin = (3√3/4) ea0=1.30ea0 for the minimum dipole moment required to bind an electron. The implied separation of unit dipole charges is identical with a special value for unit positive-charge separation given by Fermi. Furthermore, this semiclassical value of Dmin is more than twice the quantum-mechanical value Dmin=0.639ea0.

What is the relativistic generalization of a linearly rising potential?

Recent progress in summing graphs of non-Abelian gauge theories focuses attention on certain manifestly Lorentz-invariant classical action-at-a-distance theories, whose solution allows for the construction of field-theoretic Green functions in the WKB approximation. The sum of graphs is of QED type, except that the gluon propagator is modified to incorporate the renormalization-group-invariant charge g(k) . The purely phenomenological choice g 2 ∼ k −2 is equivalent non-relativistically to a linearly rising potential and yields a fully relativistic classical action-at-a-distance theory with exactly soluble circular orbits, whose Bohr quantization yields an asymptotic approximation to the poles of the Green function. One finds asymptotically linear Regge trajectories, but other phenomenological aspects are not as promising when only a linearly rising potential is used (in common with other phenomenological studies). As a redeeming feature, there do not appear to be any obvious pathologies of the sort familiar from string theories and ad hoc generalizations of linearly rising potentials.

Quasiclassical selection of initial coordinates and momenta for a rotating Morse oscillator

The classical orbits of a rotating Morse oscillator are calculated by means of Hamilton–Jacoby theory after truncating the Hamiltonian to permit analytical solution. Except at very high J, the approximate analytic orbit for the radial coordinate is in good agreement with that obtained by numerical integration of the exact equations of motion. Bohr quantization gives an expression for the rotation–vibration energy correct through quadratic terms in (v+1/2) and J (J+1), where v and J are the vibrational and rotational quantum numbers, respectively. The principal result is an analytic prescription for obtaining values of the coordinates and momenta, given v, J, and a set of random numbers, that facilitates properly weighted quasiclassical selection of initial states of diatomic molecules in trajectory calculations.