Quantum Mechanical Hamiltonian Research Articles

Modal analysis and coupling in metal-insulator-metal waveguides

This paper shows how to analyze plasmonic metal-insulator-metal waveguides using the full modal structure of these guides. The analysis applies to all frequencies, particularly including the near infrared and visible spectrum, and to a wide range of sizes, including nanometallic structures. We use the approach here specifically to analyze waveguide junctions. We show that the full modal structure of the metal-insulator-metal MIM waveguides—which consists of real and complex discrete eigenvalue spectra, as well as the continuous spectrum—forms a complete basis set. We provide the derivation of these modes using the techniques developed for Sturm-Liouville and generalized eigenvalue equations. We demonstrate the need to include all parts of the spectrum to have a complete set of basis vectors to describe scattering within MIM waveguides with the mode-matching technique. We numerically compare the mode-matching formulation with finite-difference frequency-domain analysis and find very good agreement between the two for modal scattering at symmetric MIM waveguide junctions. We touch upon the similarities between the underlying mathematical structure of the MIM waveguide and the PT symmetric quantum-mechanical pseudo-Hermitian Hamiltonians. The rich set of modes that the MIM waveguide supports forms a canonical example against which other more complicated geometries can be compared. Our work here encompasses the microwave results but extends also to waveguides with real metals even at infrared and optical frequencies.

The use of internal cartesian coordinates for describing molecular vibrations

An analysis of the influence of isotope substitution on the system of electronic-nuclear equations for an arbitrary molecular system was used as a basis for formulating invariance conditions with respect to isotope substitution of the potential energy surface written in the Cartesian coordinates rigidly bound with the center of mass of the molecule (internal Cartesian coordinates). This property of the potential function obviates the necessity of using curvilinear natural coordinates, which can be replaced by Cartesian coordinates, in theoretical studies of the vibrational spectra of molecules and their isotopomers and in solving the direct and inverse anharmonic problems. An equation for the quantum-mechanical Hamiltonian of a normal molecule in internal Cartesian coordinates was obtained.

Strongly coupled largeNspectrum of two matrices coupled via a Yang-Mills interaction

We consider the large N spectrum of the quantum mechanical Hamiltonian of two Hermitian matrices coupled via a Yang-Mills interaction. In a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity, the difference equation associated with the Yang-Mills interaction is derived and solved exactly for two impurities. In this case, the full string tension corrected spectrum depends on two momenta. For a specific value of one of these momenta, the spectrum has the same structure as that of giant magnon bound states. States with general number of impurities are also discussed.

Time asymmetry in quantum mechanics: a pure mathematical point of view

In the paper it is pointed out that ‘time asymmetry in quantum mechanics’ (TAQM) is an intrinsic element of the mathematical apparatus of quantum mechanics for semibounded Hamiltonians H with absolutely continuous spectrum coinciding with the positive half-line and of constant multiplicity. It is shown that the TAQM-semigroups are unsuitable for a spectral characterization of the resonances in terms of H.

GENERALIZED BÄCKLUND–DARBOUX TRANSFORMATIONS IN ONE-DIMENSIONAL QUANTUM MECHANICS

We consider an action of the group of curves in GL(2,ℝ) on the set of linear systems and therefore on the set of Schrödinger equations in full similarity with the action of the group of curves in SL(2,ℝ) on the set of Riccati equations considered in previous articles. We also consider the transformations defined by a first-order differential expression which carry solutions of a Schrödinger equation into solutions of another one. We find then two non-trivial situations: transformations which can be described by the previous transformation group, generalizing previous work by us, and transformations which are singular. We show that both situations appear, e.g., in the usual problem of partner Hamiltonians in quantum mechanics. We show that the difference Bäcklund algorithm, both in the finite and confluent versions, can be understood in terms of the above mentioned transformation group, the case of two exactly equal factorization energies being an instance of the singular case. We apply the generalized theorem relating three eigenfunctions of three different Hamiltonians to the generation of new potentials with a known (excited state) eigenfunction, starting from potentials of Coulomb, Morse and Rosen–Morse type. The potentials found are new and non-trivial.

Study of Spectral and thermal properties of Selenium Diatomic Halides by Semi-empirical Treatment

The spectroscopic properties, potential energy curve, dipole moments, total charge density, Electrostatic potential as well as the thermodynamic properties of selenium diatomic halides have been studied using code Mopac.7.21 and hyperchem, semi-empirical molecular orbital of MNDO-method (modified neglected of differential overlap) of parameterization PM3 involving quantum mechanical semi-empirical Hamiltonian. The relevant molecular parameters like interatomic distance, bond angle, dihedral angle and net charge were also calculated.

Study of Spectral and thermal properties of Selenium Diatomic Halides by Semi-empirical Treatment

The spectroscopic properties, potential energy curve, dipole moments, total charge density, Electrostatic potential as well as the thermodynamic properties of selenium diatomic halides have been studied using code Mopac.7.21 and hyperchem, semi-empirical molecular orbital of MNDO-method (modified neglected of differential overlap) of parameterization PM3 involving quantum mechanical semi-empirical Hamiltonian. The relevant molecular parameters like interatomic distance, bond angle, dihedral angle and net charge were also calculated.

Third-order differential ladder operators and supersymmetric quantum mechanics

Hierarchies of one-dimensional Hamiltonians in quantum mechanics admitting third-order differential ladder operators are studied. Each Hamiltonian has associated three-step Darboux (pseudo)-cycles and Painlevé IV equations as a closure condition. The whole hierarchy is generated applying some operations on the cycles. These operations are investigated in the frame of supersymmetric quantum mechanics and mainly involve algebraic manipulations. A consistent geometric representation for the hierarchy and cycles is built that also helps in understanding the operations. Three kinds of hierarchies are distinguished and a realization based on the harmonic oscillator Hamiltonian is supplied, giving an interpretation for the spectral properties of the Hamiltonians of each hierarchy.

Deriving Vibration Modes of Semi-infinite Chain Model by "Invariant Eigen-operator" Method

For the first time, we introduce a fully quantum mechanical Hamiltonian for a semi-infinite chain model of atoms. We then derive the vibration modes of this model by virtue of the "invariant eigen-operator" method in two different cases, which is concise and revealing.

Multiparticle quasiexactly solvable difference equations

Several explicit examples of multiparticle quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable multiparticle Hamiltonians, the Ruijsenaars-Schneider-van Diejen systems. These are difference analogs of the quasiexactly solvable multiparticle systems, the quantum Inozemtsev systems obtained by deforming the well-known exactly solvable Calogero-Sutherland systems. They have a finite number of exactly calculable eigenvalues and eigenfunctions. This paper is a multiparticle extension of the recent paper by one of the authors [R. Sasaki, J. Math. Phys. 48, 122104 (2007)] on deriving quasiexactly solvable difference equations of single degree of freedom.

Quasiexactly solvable difference equations

Several explicit examples of quasiexactly solvable “discrete” quantum mechanical Hamiltonians are derived by deforming the well-known exactly solvable Hamiltonians of one degree of freedom. These are difference analogs of the well-known quasiexactly solvable systems, the harmonic oscillator (with∕without the centrifugal potential) deformed by a sextic potential, and the 1∕sin2x potential deformed by a cos2x potential. They have a finite number of exactly calculable eigenvalues and eigenfunctions.

Interpolation of SUSY quantum mechanics

Interpolation of two adjacent Hamiltonians in SUSY quantum mechanics is discussed together with related operators. For a wide variety of shape-invariant degree-one quantum mechanics and their ‘discrete’ counterparts, the interpolation Hamiltonian is also shape-invariant, that is, it takes the same form as the original Hamiltonian with shifted coupling constant(s).

Criterion for dynamical chiral symmetry breaking

The Bethe-Salpeter equation is related to a generalized quantum-mechanical Hamiltonian. Instability of the presumed vacuum, indicated by a tachyon, is related to a negative energy eigenstate of this Hamiltonian. The variational method shows that an arbitrarily weak long-range attraction leads to chiral symmetry breaking, except in the scale-invariant case when the instability occurs at a critical value of the coupling. In the case of short-range attraction, an upper bound for the critical coupling is obtained.

Spin-charge separation and the Pauli electron

The separation between the spin and the charge converts the quantum mechanical Pauli Hamiltonian into the Hamiltonian of the non-Abelian Georgi-Glashow model, which is notorious for its magnetic monopoles and confinement. The independent spin and charge fluctuations both lead to the Faddeev model, suggesting the existence of a deep duality structure and indicating that the fundamental carriers of the spin and charge are knotted solitons.

A mathematical theory for deterministic quantum mechanics

Classical, i.e. deterministic theories underlying quantum mechanics are considered, and it is shown how an apparent quantum mechanical Hamiltonian can be defined in such theories, being the operator that generates evolution in time. It includes various types of interactions. An explanation must be found for the fact that, in the real world, this Hamiltonian is bounded from below. The mechanism that can produce exactly such a constraint is identified in this paper. It is the fact that not all classical data are registered in the quantum description. Large sets of values of these data are assumed to be indistinguishable, forming equivalence classes. It is argued that this should be attributed to information loss, such as what one might suspect to happen during the formation and annihilation of virtual black holes.The nature of the equivalence classes follows from the positivity of the Hamiltonian. Our world is assumed to consist of a very large number of subsystems that may be regarded as approximately independent, or weakly interacting with one another. As long as two (or more) sectors of our world are treated as being independent, they all must be demanded to be restricted to positive energy states only. What follows from these considerations is a unique definition of energy in the quantum system in terms of the periodicity of the limit cycles of the deterministic model.

SCALING LIMIT FOR A GENERALIZATION OF THE NELSON MODEL AND ITS APPLICATION TO NUCLEAR PHYSICS

We study a mathematically rigorous derivation of a quantum mechanical Hamiltonian in a general framework. We derive such a Hamiltonian by taking a scaling limit for a generalization of the Nelson model, which is an abstract interaction model between particles and a Bose field with some internal degrees of freedom. Applying it to a model for the field of the nuclear force with isospins, we obtain a Schrödinger Hamiltonian with a matrix-valued potential, the one pion exchange potential, describing an effective interaction between nucleons.

The Spherically Symmetric a2–dynamo and Some of its Spectral Peculiarities

A brief overview is given of recent results on the spectral properties of spherically symmetric MHD α2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized α-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization ofthe dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated.

Path-integral derivation of the anomaly for the Hermitian equivalent of the complexPT-symmetric quartic Hamiltonian

It can be shown using operator techniques that the non-Hermitian PT-symmetric quantum mechanical Hamiltonian with a 'wrong-sign' quartic potential -gx{sup 4} is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naieve derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.

Adaptive perturbation theory: quantum mechanics and field theory

Adaptive perturbation is a new method for perturbatively computing the eigenvalues and eigenstates of quantum mechanical Hamiltonians that are widely believed not to be solvable by such methods. The novel feature of adaptive perturbation theory is that it decomposes a given Hamiltonian, H , into an unperturbed part and a perturbation in a way which extracts the leading non-perturbative behavior of the problem exactly. In this talk I will introduce the method in the context of the pure anharmonic oscillator and then apply it to the case of tunneling between symmetric minima. After that, I will show how this method can be applied to field theory. In that discussion I will show how one can non-perturbatively extract the structure of mass, wavefunction and coupling constant renormalization.

Symmetric squares of graphs

We consider symmetric powers of a graph. In particular, we show that the spectra of the symmetric square of strongly regular graphs with the same parameters are equal. We also provide some bounds on the spectra of the symmetric squares of more general graphs. The connection with generic exchange Hamiltonians in quantum mechanics is discussed in Appendix A.