Quantum Mechanical Theorem Research Articles

Pauli Repulsions Exist Only in the Eye of the Beholder

This paper presents a rebuttal to the preceding paper in this issue entitled "Hydrogen-Hydrogen Bonding in Planar Biphenyl, Predicted by Atoms-In-Molecules Theory, Does Not Exist". The arguments presented therein are based on an arbitrary partitioning of the energy into contributions from physically unrealizable states of the system. The response given here is presented in terms of the Feynman, Ehrenfest, and virial theorems of quantum mechanics and the observable properties of a system. A reader is thus free to choose between subjectivity or physics.

Size-dependent band gap of colloidal quantum dots

The size-dependent band gap of semiconductor quantum dots is a well-known and widely studied quantum confinement effect. In order to understand the size-dependent band gap, different theoretical approaches have been adopted, including the effective-mass approximation with infinite or finite confinement potentials, the tight-binding method, the linear combination of atomic orbitals method, and the empirical pseudopotential method. In the present work we calculate the size-dependent band gap of colloidal quantum dots using a recently developed method that predicts accurately the eigenstates and eigenenergies of nanostructures by utilizing the adiabatic theorem of quantum mechanics. We have studied various semiconductor (CdS, CdSe, CdTe, PbSe, InP, and InAs) quantum dots in different matrices. The theoretical predictions are, in most cases, in good agreement with the corresponding experimental data. In addition, our results indicate that the height of the finite-depth well confining potential is independent of the specific semiconductor of the quantum dot and exclusively depends on the matrix energy-band gap by a simple linear relation.

Quantum Bayesian methods and subsequent measurements

The use of Bayes theorem in quantum mechanics is discussed. It is shown that the quantum Bayes theorem follows from the ordinary quantum measurement theory, when applied to density operators that represent our knowledge of a system. The examples studied involve measurements on multiple copies of the same (unknown) state. The theorem is used to determine the unknown state by successive measurements on several of the copies of the state. An idealized information-theoretic description of propagating CW laser beams is treated in detail. It is shown how photon detections on part of the beams can be used to determine the phase of the rest of the beams. Also discussed, are the limitations on the accuracy of the phase determination that follow from the fact that it is accomplished by the detection of a finite number of photons. Explicit expressions are derived for the conditional probabilities of detecting photons at different locations, given the numbers of photons detected in the past. The quantitative predictions could be used, in principle, to test proposed quantum states of propagating laser beams.

Superconducting critical temperature and the isotope exponent versus total electron concentration for two-band superconductors: Effect of the band structure

We consider the superconducting critical temperature ${T}_{c}$ and the isotope exponent $\ensuremath{\alpha}$ versus carrier concentration $\ensuremath{\rho}$ using the one-particle Green functions of the simplest coupled Hamiltonian for the two-band model with a single ${T}_{c}$ [N. Kristoffel and P. Rubin, Physica C $356,$ 171 (2001)]. One of the bands, ${\ensuremath{\varepsilon}}_{2}(\stackrel{\ensuremath{\rightarrow}}{k}),$ is taken to be two dimensional, while the second band ${\ensuremath{\varepsilon}}_{3}(\stackrel{\ensuremath{\rightarrow}}{k})$ is taken to be three dimensional. Taking only nearest-neighbor hopping for each one of the bands, we obtain that ${T}_{c}$ versus $\ensuremath{\rho}$ is symmetric around half filling in agreement with a general theorem of quantum mechanics. We also obtained the chemical potential $\ensuremath{\mu}$ versus $\ensuremath{\rho}$ for several values of attractive interaction. We find that (1) the superconducting critical temperature ${T}_{c}$ is maximum at $\ensuremath{\rho}=1/2;$ (2) the chemical potential remains basically inside the smaller band, for the range of carrier concentration and Debye frequencies considered; (3) the isotope exponent has several symmetric minima around half filling depending on the anisotropy of the three-dimensional tight-binding band, namely, $\ensuremath{\gamma}\ensuremath{\equiv}{t}_{3,x}{/t}_{3,z}\ensuremath{\ne}1;$ (4) the chemical-potential curves, namely, $\ensuremath{\mu}$ versus $\ensuremath{\rho}$ are completely independent of pairing interaction and Debye frequency; and (5) the number of symmetric minima and ${T}_{c}^{\mathrm{max}}$ goes down with increasing ${t}_{3}{/t}_{2}$ ratio. We have briefly discussed the possible relevance of our results with recent experimental data of the two-band compound ${\mathrm{MgB}}_{2}.$

Simple applications of Noether’s first theorem in quantum mechanics and electromagnetism

Internal global symmetries exist for the free nonrelativistic Schrödinger particle, whose associated Noether charges, space integrals of the wavefunction and wavefunction multiplied by the spatial coordinate, are exhibited. Analogous symmetries in classical electromagnetism also are demonstrated. These results illustrate a number of subtleties associated with Noether’s first theorem.

Nature: What Matters in Nature-DNA, Light-Driven Ion Pumps, and Antimatter

This report from Nature points the reader to a series of articles commemorating the 50th anniversary of Watson and Crick's elucidation of DNA structure. In addition, a light-powered membrane pump for Ca2+ ions is described that can generate a membrane potential and is reminiscent of photoinduced biological proton pumps. Finally, the column covers an article on the successful generation of cold anti-hydrogen as part of the collaborative ATHENA project at CERN. Scientists are hoping that this simplest of anti-matter will allow them to test the validity of the central theorem in quantum mechanics in the near future.

Simple algebraic recursion formulas for calculation of propagation characteristics of graded-index waveguides

Using the perturbation method with the aid of the hypervirial theorem and Hellmann–Feynman theorem in quantum mechanics, simple algebraic recursion formulas for calculating propagation constants of graded-index planar optical waveguides in successive orders of approximation are derived. Algebraic formulas for calculation of the modal field functions are also presented. Numerical results for typical graded-index waveguides (Gaussian, exponential, complementary error function and symmetric Epstein profiles) demonstrate that the present method can be applied to an accurate calculation of the propagation characteristics to an arbitrary order of perturbation approximation.

On the adiabatic theorem in quantum mechanics

The adiabatic and postadiabatic approximations, as well as the adiabatic and nonstationary perturbation theories, are constructed using a canonical averaging method under the assumptions which are more general than those used in the existing theories. An asymptotic evaluation of the proximity of rigorous and approximate solutions is performed.

A Note on the Adiabatic Theorem Without Gap Condition

We simplify the proof of the adiabatic theorem of quantum mechanics without gap condition of Avron and Elgart by providing an elementary solution of the ‘commutator equation’. In addition, a minor modification of their argument allows for more direct treatment of eigenvalue crossings. We also obtain simple, explicit conditions that yield information on the rate of convergence in the adiabatic limit.

On the Virial Theorem in Quantum Mechanics

We review the various assumptions under which abstract versions of the quantum mechanical virial theorem have been proved. We point out a relationship between the virial theorem for a pair of operators H, A and the regularity properties of the map \(\). We give an example showing that the statement of the virial theorem in [CFKS] is incorrect.

Function-mixing hypothesis and quantum chaos

Billiards are studied whose boundaries comprise two or more segments which allow local (but not global) separation of the Helmholtz equation. Related local solutions are labeled “sep” functions. Such billiards comprise the Σ and Σ sets. A definition of quantum chaos for these sets of billiards is presented based on the ratio of the “fluctuation length” of the wave function nodal pattern to the “c-diameter” of the billiard. The “function-mixing hypothesis” states that a sufficient condition for a billiard to be chaotic is that the billiard be an element of one of these sets. It further ascribes such chaotic behavior to be due to mixing of dissimilar sep functions. Examples of the application of this hypothesis are described. A set-theoretic formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave-sided square billiard of sufficiently small concavity. It is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers.

Virial theorem in quantum statistical mechanics for a charged particle in an electromagnetic field

The virial theorem is derived in quantum statistical mechanics for a time-dependent Hamiltonian that describes a charged particle in a time-dependent electromagnetic field. The time derivative of the expectation value of the Clausius operator satisfies an equation which is the same form as the classical equation for the Clausius function (r⋅mv). By taking a time average, the virial theorem of quantum mechanics for time-dependent states is obtained. By taking a statistical average and a time average, the virial theorem of quantum statistical mechanics is obtained.

Hidden variables and Bell's theorem in quantum mechanics

In the present paper we give a precise definition of a hidden-variable theory for quantum mechanics, whereby we adopt the weakest possible definition of a hidden-variable theory, which is compatible with the assumption that the bounded observables of a quantum mechanical system are represented by the elements of the real part Ar of a W*-algebra A (of the most general type) and the states are represented by the “normal states” (in the mathematical sense) of A. We then go on to show that an example put forward by Bell in 1966 satisfies our definition (Sec. 2). Finally we make use of Bell's famous theorem to show that for a sufficiently non-commutative W*-algebra A no hidden-variable theory in our sense exists (Theorem 3.3 and its corollaries).

QUANTUM GROUPS AND VON NEUMANN THEOREM

We discuss the q-deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in quantum mechanics. We show that the q-deformation parameter labels the Weyl systems in quantum mechanics and the unitarily inequivalent representations of the canonical commutation relations in quantum field theory.

On the calculation of alloy scattering relaxation time for ternary III–V and II–VI semiconductors

Abstract An approach to the calculation of disorder scattering in ternary semiconductor alloys is proposed. It is based on the Hellman-Feynman theorem of quantum mechanics and enables the calculation of alloy scattering relaxation times provided that the dependences of the band edges and the effective masses on the alloy composition are known. The result obtained incorporates nonparabolicity effects not only via the electron spectrum but also the p -wave mixing matrix element renormalization. As an example the calculation of the alloy scattering relaxation times for three main valleys in GaAlAs is considered. It is shown that the model existing in the literature strongly overestimates the alloy relaxation rate in the Γ-valley at high energies and gives an incorrect relation between the alloy relaxation rates in Γ, X and L valleys.

NONLINEAR KLEIN-GORDON SOLITON MECHANICS

Nonlinear Klein-Gordon solitary waves — or “solitons” in a loose sense — in n+1 dimensions, driven by very general external fields which must only satisfy continuity — together with regularity conditions at the boundaries of the system, obey a quite simple equation of motion. This equation is the exact generalization to this dynamical system of infinite number of degrees of freedom — which may be conservative or not — of the second Newton’s law setting the basis of material point mechanics. In the restricted case of conservative nonlinear Klein-Gordon systems, where the external driving force is derivable from a potential energy, we recover the generalized Ehrenfest theorem which was itself the extension to such systems of the well-known Ehrenfest theorem in quantum mechanics (G. Reinisch and J.C. Fernandez, Phys. Rev. Lett.67, 1968 (1991)). This review paper first displays a few (of one-dimensional sine-Gordon type) typical examples of the basic difficulties related to the trial construction of solitary-wave mechanics. Then the general equation of motion of nonlinear Klein-Gordon solitary waves is proved and the derivation of the previous sine-Gordon examples from this theorem is displayed. Two-dimensional nonlinear solitary-wave patterns are considered, as well as a special emphasis is put on the applications to space-time complexity of 1-dim. sine-Gordon systems.

Refining the comparison theorem of quantum mechanics

If two potentials V1(x) and V2(x) are ordered V1<V2, then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E1<E2. He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.

Quantum-mechanical representations of canonical transformations and adiabatic invariance

A representation of the canonical transformation to action-angle variables is obtained using the (classical) adiabatic invariance of the actions. It is shown that in one dimension this is in fact the exact quantum-mechanical formula for this representation and hence provides a formally exact expression for the eigenfunctions. For higher dimensions some interesting limitations of the method are also discussed and shown to be related to the adiabatic theorem in quantum mechanics. They cause the representation and thus the eigen-functions of integrable systems in more than one dimension to be given correctly to all orders of ħ.

Ehrenfest theorem for nonlinear Klein-Gordon solitary waves.

A theorem which describes the Newtonian dynamics or the center or mass (as defined with respect to the energy density) or nonlinear Klein-Gordon-type solitary waves is proved. If appears as a generalization or Ehrenrest's theorem in quantum mechanics, which describes the Newtonian dynamics or the center or mass (as defined with respect to the probability density) or the linear wave functions

A counterexample and a modification to the adiabatic approximation theorem in quantum mechanics

A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.