Harmonic Propagator Research Articles

Evaluating Harmonic Oscillator Propagator with Stochastic Decoupling

Evaluating Harmonic Oscillator Propagator with Stochastic Decoupling

Hybrid derivation of the 1D harmonic oscillator propagator

We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

Computationally fast EM field propagation through axi-symmetric media using cylindrical harmonic decomposition.

We describe and provide a systematic procedure for computationally fast propagation of arbitrary vector electromagnetic (EM) fields through an axially symmetric medium. A cylindrical harmonic field propagator is chosen for this purpose and in most cases, this is the best and the obvious choice. Firstly, we describe the cylindrical harmonic decomposition technique in terms of both scalar and vector basis for a given input excitation field. Then we formulate a generalized discrete Fourier-Hankel transform to achieve efficient vector basis decomposition. We allow a slower, pre-computation step, that finds a representation of the axi-symmetric medium as a transfer matrix in a discrete, cylindrical-harmonic basis. We find this matrix from a series of axi-symmetric (2D) finite element simulations (also known as the 2.5D technique). This transfer matrix approach significantly reduces the computational load when the transverse size or range exceeds about 30 wavelengths. This matrix is independent of the input excitation field for a given space-bandwidth product and hence makes it reusable for different excitation fields. We numerically validate the above approaches for different axi-symmetric EM scattering media which include a hemispherical gradient-index Maxwell's fish-eye lens, a transformation optics designed spherical invisibility cloak, a thin aspheric lens, and a cylindrical perfect lens.

Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schroedinger equation can be mapped to solutions of the Schroedinger equation for harmonic potentials, both the trapping oscillator and the inverted 'oscillator'. This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-a 1,920 characters

A simple evaluation of a determinant in a path integral calculation of a harmonic oscillator propagator

The calculation of a one-dimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical significance. We focus on the evaluation of a determinant resulting from the action integral in the discretized form of the path integral. As a consequence we can make the commonly accepted calculations in the literature simpler and clearer.

Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations

We present the simplest and most straightforward derivation of the one-dimensional harmonic oscillator propagator, using the Feynman path integral and recursive relations. Our calculations have pedagogical benefits for those undergraduate students beginning to learn the path integral in quantum mechanics, in that they can follow its calculations very simply with only elementary mathematical manipulation. Further, our calculations do not require cumbersome matrix algebra.

Suppression of anharmonicities in crystalline membranes by external strain

In practice, physical membranes are exposed to a certain amount of external strain (tension or compression), due to the environment where they are placed. As a result, the behavior of the phonon modes of the membrane is modified. We show that anharmonic effects in stiff two-dimensional membranes are highly suppressed under the application of tension. For this, we consider the anharmonic coupling between bending and stretching modes in the self-consistent screening approximation (SCSA), and compare the obtained height-height correlation function in the SCSA to the corresponding harmonic propagator. The elasticity theory results are compared to atomistic Monte Carlo simulations for a graphene membrane under tension. We find that, while rather high values of strain are needed to avoid anharmonicity in soft membranes, strain fields less than 1% are enough to suppress all the anharmonic effects in stiff membranes, as graphene.

Noncommutative field theory from quantum mechanical space–space noncommutativity

We investigate the incorporation of space noncommutativity into field theory by extending to the spectral continuum the minisuperspace action of the quantum mechanical harmonic oscillator propagator with an enlarged Heisenberg algebra. In addition to the usual ⋆-product deformation of the algebra of field functions, we show that the parameter of noncommutativity can occur in noncommutative field theory even in the case of free fields without self-interacting potentials.

Applying reproducing kernels to the evaluation and approximation of the simple and time-dependent imaginary time harmonic oscillator path integrals

Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n 3).

An elementary derivation of the harmonic oscillator propagator

The harmonic oscillator propagator is found straightforwardly from the free particle propagator within the imaginary-time Feynman path integral formalism. The derivation is simple, and requires only elementary mathematical manipulations and no clever use of Hermite polynomials, annihilation and creation operators, cumbersome determinant evaluations, or involved algebra.

Comment on “The harmonic oscillator propagator” by Barry R. Holstein [Am. J. Phys. 66 (7), 583–589 (1998)

First Page

Influence of isotopic disorder on phonon frequencies and phonon linewidths of an anharmonic crystal

Based on a perturbation theory using Matsubara functions and a diagrammatic treatment, it is shown that the phonon linewidths of an anharmonic crystal with isotopic impurities include two different contributions, a disorder-induced temperature-independent part, caused by harmonic scattering processes of phonons at isotopic impurities, and a disorder-influenced temperature-dependent part, caused by anharmonic decay processes. Whereas in case of an anharmonic ideal crystal it is usual to restrict the corresponding self-energy on diagonal elements only, this is no longer sufficient in the presence of impurities. For that reason it is shown that only nondiagonal elements of the effective vertex function of the harmonic phonon propagator contribute to the disorder-induced contribution to the phonon linewidth, whereas the corresponding harmonic and the anharmonic frequency shifts are determined by diagonal and nondiagonal elements as well. As an application, numerical results of the disorder-induced temperature-independent as well as the disorder-influenced temperature-dependent contribution to the phonon linewidths of a diatomic linear chain with a single mass defect are presented. {copyright} {ital 1996} {ital The American Physical Society}

Extensions to the distributed approximating functional: The harmonic propagator.

The formalism of the distributed approximating functional for integrating the time-dependent Schr\odinger equation is extended to allow an analytic treatment of all Hamiltonians of quadratic order. The resulting propagator matrix is highly banded in both position and momentum representations. For certain choices of the parameters, the action of this matrix can be cast in the form of a Toeplitz operator. We demonstrate that banded Toeplitz matrix-vector multiplication can be performed in significantly fewer numerical operations than is required by conventional split-operator fast Fourier transform algorithms. Numerical examples verify the efficiency of this approach in both storage and speed. Extensions of the method to Monte Carlo path integration, and the implementation on massively parallel computer architectures, are discussed.

Improved propagators for the path integral study of quantum systems

A local harmonic (LH) propagator has been derived through an adequate splitting of the Hamiltonian operator. This propagator has been applied to study several quantum systems including a quartic oscillator, a double well potential, and the Morse potential. We have calculated equilibrium and dynamical properties using the transfer matrix path integral method. It has been verified that the LH propagator is significantly more accurate than other propagators proposed by several studies. For the LH propagator, higher order corrections to the Trotter formula lead to an excellent short time approximation. It is suggested that the LH propagator can be of wide interest for path integral studies of quantum systems.

Path integration of an action related to an electron gas in a random potential

Path integration of an action related to an electron gas in a random potential is performed within the framework of Feynman’s polygonal path approach. The exact propagator obtained is simply related to the harmonic oscillator propagator. The integration is direct and does not require the knowledge of an auxiliary measure or the artificial coupling of the system to the external forces.

Converged values of second-order core-polarization diagrams with orthogonalized-plane-wave intermediate states

We carry out an extended study of the Vary-Sauer-Wong effect on the second-order core-polarization diagrams ${G}_{3\mathrm{p}1\mathrm{h}}$ and ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ in the effective interaction between two valence nucleons. As in Vary-Sauer-Wong, we first calculate ${G}_{3\mathrm{p}1\mathrm{h}}$ using a harmonic oscillator propagator for the intermediate particle states $p$, including particle-hole excitations with energies up to $22\ensuremath{\hbar}\ensuremath{\omega}$. Results are in close agreement with those of Vary-Sauer-Wong. We then calculate ${G}_{3\mathrm{p}1\mathrm{h}}^{T}$ where a free particle propagator for $p$ is used. This is obtained by including the $\ensuremath{-}U$, the oscillator one-body potential, insertions of all orders to $p$ of ${G}_{3\mathrm{p}1\mathrm{h}}$. Although the resulting matrix elements are generally smaller in magnitude than those of Vary-Sauer-Wong, the qualitative feature of the Vary-Sauer-Wong effect is clearly maintained. Namely there are strong cancellations between the contributions from the low and high energy $p$ states. This makes the net effect from the core polarization diagrams significantly weaker than from ${G}_{3\mathrm{p}1\mathrm{h}}$ calculated with $2\ensuremath{\hbar}\ensuremath{\omega}$ excitations alone. We also study some intermediate choices for the propagator of $p$, where the free particle propagator is used only for high energy valence states. The Brueckner reaction matrix elements in a mixed representation where one particle is in a harmonic oscillator state and the other in a plane wave state are needed in our calculations. By using the vector transformation brackets of Wong and Clement and of Balian and Brezin and the Tsai-Kuo treatment of the Pauli exclusion operator, we have developed a technique for accurately calculating these matrix elements.NUCLEAR STRUCTURE Contributions to effective interaction for $A=18$ nuclei calculated from converged values of lowest order core polarization diagrams as function of intermediate state propagator using new momentum space techniques.

Subduction zone dip angles and flow driven by plate motion

Kinematic models of the large scale flow in the mantle accompanying the observed plate motions are calculated by neglecting thermal buoyancy forces. The large scale flow is therefore determined by the mass flux imposed by the moving plates. The energy and momentum equations decouple, and with the assumption of a radially symmetric Newtonian viscosity, the flow accompanying the plate motions can be obtained using harmonic analysis and propagator matrices. The resulting flow models predict remarkably well the observed dips of subducted slabs if the flow extends into the lower mantle. The plates drag along a thick boundary layer which should be included in models of the heating of subducted slabs.