The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum

On two different kinds of resonances in one-dimensional quantum-mechanical models

We apply the Riccati-Padé method and the Rayleigh-Ritz method with complex rotation to the study of the resonances of a one-dimensional well with two barriers. The model exhibits two different kinds of resonances and we calculate them by means of both approaches. While the Rayleigh-Ritz method reveals each set at a particular interval of rotation angles the Riccati Padé method yields both of them as roots of the same Hankel determinants.

Determination of Resonances by the Optimized Spectral Approach

The Rayleigh-Ritz procedure for determining bound-states of the Schrödinger equation relies on spectral representation of the solution as a linear combination of the basis functions. Several possible extensions of the method to resonance states have been considered in the literature. Here we propose the application of the optimized Rayleigh-Ritz method to this end. The method uses a basis of the functions containing adjustable nonlinear parameters, the values of which are fixed so as to make the trace of the variational matrix stationary. Generalization to resonances proceeds by allowing the p…

Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with $O_{h}$ symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh-Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions . We show how to obtain suitable symmetry-adapted basis sets.

Measurement of single nanoparticle anisotropy by laser induced optical alignment and Rayleigh scattering for determining particle morphology

We demonstrate the measurement of nanoparticle shape by angularly resolved Rayleigh scattering of single optical levitated particles that are oriented in space via the trapping light in vacuum. This technique is applied to a range of particle geometries, from perfect spherical nanodroplets to octahedral nanocrystals. We show that this method can resolve shape differences down to a few nanometers and be applied in both low-damping environments, as demonstrated here, and in traditional overdamped fluids used in optical tweezers.

Bound states for the quantum dipole moment in two dimensions

We calculate accurate eigenvalues and eigenfunctions of the Schrödinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh-Ritz variational method with a basis set of Slater-type functions is considerably more efficient than the same approach with the basis set of point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in earlier calculations.

Simple one-dimensional quantum-mechanical model for a particle attached to a surface

We present a simple one-dimensional quantum-mechanical model for a particle attached to a surface. We solve the Schrödinger equation in terms of Weber functions and discuss the behavior of the eigenvalues and eigenfunctions. We derive the virial theorem and other exact relationships as well as the asymptotic behaviour of the eigenvalues. We calculate the zero-point energy for model parameters corresponding to H adsorbed on Pd(100) and also outline the application of the Rayleigh-Ritz variational method.

限制在圆形盒中的带电场二维谐振子

On the two-dimensional harmonic oscillator with an electric field confined to a circular box

针对不可穿透圆形盒中带电场的二维谐振子，本文利用多项式与高斯基组的 Rayleigh-Ritz 法求能谱，并与已有结果及微扰计算比较。

一类双阱振子的变分处理研究

On the variational treatment of a class of double-well oscillators

本文比较了Rayleigh-Ritz变分法与基于超对称量子力学和Gram-Schmidt正交化的方法在双阱谐振子问题上的表现，发现前者计算量更少且精度更高，并证明后者声称的上界不成立。

论电场存在下圆形盒中的二维氢原子

On the two-dimensional hydrogen atom in a circular box in the presence of an electric field

本文研究了电场中圆形盒内的二维氢原子，采用Rayleigh-Ritz方法和多项式基组计算能谱，探讨了大小盒半径极限与解的对称性，并揭示了电场引起的偶然简并及简并能级分裂现象。

WKB formalism and a lower limit for the energy eigenstates of bound states for some potentials

In the present work the conditions appearing in the WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the text books on quantum mechanics, whereas the second one is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a …

CANOE：经典辅助非正交本征解算器

CANOE: Classically Assisted Non-Orthogonal Eigensolver

CANOE通过结合量子和经典资源，利用量子态的表达能力与经典基态的低成本补充，提升早期容错量子模拟的地面态表征精度。