Application of two-spinor calculus in quantum mechanical and field calculations

This paper describes Lorentz two-spinors and proposes using them in calculations with Dirac four-spinors and quaternions.     

An analytical procedure to evaluate electronic integrals for molecular quantum mechanical calculations

The main contribution of this paper is to provide an alternative strategy to reduce the number of bi-electronic integrals in ab initio quantum mechanics calculations. An analytical procedure to evaluate the energy of a molecule as well as two-electron integrals is proposed. This approach is based on the generalized exponential function (q-exponential) and is particularly advantageous because it reduces substantially the CPU time in quantum mechanical calculations. Some examples of the effectiveness of this methodology are presented. It is important to point out that the new methodology is applicable to any kind of molecular system including relatively large molecular systems in the context of the Hartree–Fock (HF) and density functional theories (DFT).     

The quantum mechanical toda lattice

The aim of this paper is to establish the exact quantization conditions for the three-body Toda lattice. The Hamiltonian consists of the kinetic energy for three particles in one dimension, and of the potential energy which couples each particle to its two companions through an exponential spring. After eliminating the center of mass motion, one is left with a system of two degrees of freedom and two constants of motion, the total energy E and a third integral A which commute. Nevertheless, no transformation has been found to separate the classical equations of motion or Schrödinger's equation. The wave function is written as a double Laurent series. Its coefficients have to satisfy two sets of recursion relations on a triangular grid where each set insures that we have a simultaneous eigenfunction of E and A. The condition for the convergence of this series can be expressed as the vanishing of a tridiagonal infinite determinant with 1 in the diagonal and the inverse of a third-order polynomial in the first off-diagonals. The coefficients in this polynomial are E and A, and the variable corresponds to a component of the wave vector associated with the wave function. This determinant can be treated exactly as Hill's, and yields the 3 components. The condition for the square integrability of the wave function requires the phase angle of the principal minors to be equal to 0, $\text{π}\text{3}$, or $\text{2π}\text{3}$ according as the representation of the cyclic groups, for each component of the wave vector. But the third condition follows from the two others. The analogy with the corresponding two-body problem is pointed out.     

The quantum mechanical Toda lattice,II

The periodic Toda lattice consists of N particles which move along a closed line and are coupled with an exponential spring to their immediate neighbors. This system, in contrast to the open Toda lattice, has only bound states. In the method of Kac and Van Moerbeke, the classical periodic Toda chain is transformed to a new of set of canonically conjugate variables, μ and ν, which are closely related to the natural coordinates of an open Toda chain with one particle less. The quantum mechanical eigenfunctions for this reduced system are constructed explicitly, and this allows the quantum mechanical analogs of μ and ν to be defined. The bounds states for the periodic Toda chain are then written as linear combinations of functions resembling the wave functions of the reduced open chain. These functions satisfy a set of remarkably simple recursion formulas, and the coefficients in the expansion can be written essentially as a product of as many factors as pairs of conjugate variables μ and ν. Each factor is given as a solution of a second order difference equation which can be recognized as a quantum analog for the equations of motion of one pair μ and ν. The quantization conditions result from cancelling out the exponential growth in the overall wave function, and are phrased in terms of certain phase angles being submultiples of π according as the representation of the group of cyclic permutations. The calculations are simple for N = 3, and moderately tricky for N = 4 although the results are always fairly obvious.     

The solution of the quantum mechanical many-body problem bound by harmonic forces as a translationally invariant basis for nuclear structure calculations

A very simple method to obtain the translationally invariant wave functions of anA-particle system of bosons or fermions bound by harmonic forces is derived and a formalism to apply these solutions as a basis for nuclear structure calculations is presented. The simplicity of this formalism is demonstrated for⁴He. The method is not restricted toA≦4.     

Numerical calculations in quantum field theory

The relevance of numerical calculations for quantum field theories in general, and in particular for quantum chromodynamics, is illustrated. The evaluation of the force between static quark and antiquark is presented in some detail as exemplification.     

AlC,SiC基态分子结构与分析势能函数的量子力学计算

用密度泛函理论的B3LYP方法和二次组态相互作用(QCISD(T))方法,选择6-31G(d,p)、6-311 G(2df,2pd)、6-311 G(3df,3pd)、cc-PVTZ、AUG-cc-PVTZ基组,优化计算了AlC和SiC分子基态的能量,平衡结构,谐振频率.根据原子分子反应静力学原理,导出了AlC和SiC分子基态的合理离解极限和离解能.通过优化计算结果和实验数据的对比,选择QCISD(T)/6-311 G(3df,3pd)方法对AlC和SiC分子基态的势能面进行了单点能扫描.采用最小二乘法拟合得到了AlC和SiC分子基态的Murell-Sor-bie势能函数.同时计算了光谱参数(Be,eα,ωe,ωeχe)和力常数(f2,f3,f4),并与实验结果进行比较.结果表明,计算结果与实验数据吻合的较好.     

The classical limit for quantum mechanical correlation functions

For quantum systems of finitely many particles as well as for boson quantum field theories, the classical limit of the expectation values of products of Weyl operators, translated in time by the quantum mechanical Hamiltonian and taken in coherent states centered inx- andp-space around? ^?1/2 (coordinates of a point in classical phase space) are shown to become the exponentials of coordinate functions of the classical orbit in phase space. In the same sense,? ^?1/2 [(quantum operator) (t) — (classical function) (t)] converges to the solution of the linear quantum mechanical system, which is obtained by linearizing the non-linear Heisenberg equations of motion around the classical orbit.     

经典菲涅耳变换的量子力学对应

介绍经典菲涅耳变换的量子力学对应,它是相干态在相空间中的代表点做辛运动所对应的量子菲涅耳算符.     

The entropy of an ensemble of quantum mechanical systems

A mathematical result about a positive self-sdjoint operator of unit trace is proved whose physical interpretation is that when an observation is made on a mixed state of a quantum system leading to a change in its state, the entropy increases as it must.     

The interplay of different symmetries in quantum mechanical potentials

We construct an so (2,2) potential algebra and discuss how it is influenced when PT symmetry is imposed on the potential. We illustrate the procedure with the PT symmetric Scarf II potential.     

Collisional transfer of roto-vibrational energy from quantum calculations: The He-HF system

Summary The problem of simultaneous vibrational and rotational excitations of HF molecules in collision with helium has been approached via a quantum-mechanical treatment of the full dynamics and by employing a very accurate potential-energy surface suggested earlier in the literature. The complicated coupling of rotational and vibrational channels has been partly simplified by taking advantage of their different time scales, thus allowing a reduction of dimensionality for the corresponding multichannel scattering couple equations (IOS approximation). The relative influence of the various modes on preferential energy depositions upon collision is discussedvis à vis specific features of the interaction and specific regions of the range of collision energies employed. The essential inefficiency of (R, T) processes as apposed to (V, T) or (V, R, T) processes is once more pointed out for the present system. To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.     

Monte-carlo calculations for some problems of quantum mechanics

The Monte-Carlo technique for the calculations of functional integral in two one-dimensional quantum-mechanical problems had been applied. The energies of the bound states in some potential wells were obtained using this method. Also some peculiarities in the calculation of the kinetic energy in the ground state had been studied.     

Collisional transfer of rotovibrational energy from quantum calculations

Summary A newly obtained anisotropic interaction for the He−N₂ system is here employed to analyse in detail the inelastic flux distribution between the many rotational states of the target molecule and the few of its vibrational states that are accessible at near-thermal ranges of collisional energy. Both integral and differential inelastic cross-sections are obtained from a quantal formulation of the energy transfer process, within which the orientational dependence of rotational and vibrational couplings is examined for its bearing on the interference structures that are exhibited by the dynamical observables as functions of final rotational states and of centre-of-mass scattering angles. Similarities and differences with previously studied polar molecules also interacting with helium are underlined and explained.

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Riassunto Una superficie di potenziale recentemente ottenuta da noi per il sistema He−N₂ è utilizzata per studiare il processo quantico di transferimento d'energia interna, per via collisionale, fra i numerosi livelli rotazionali della molecola bersaglio ed alcuni dei livelli vibrazionali che risultano energeticamente accessibili alle velocità traslazionali qui esaminate. La natura anisotropa dell'accoppiamento (V, T) è così analizzata dal punto di vista del suo effettodiretto su osservabili dinamiche come sezioni d'urto inelastiche, parziali integrali e differenziali. Le strutture interferenziali che si originano in tale sistema sono scarsamente influenzate dalla presenza o meno di accoppiamento vibrazionale e sono qui discusse in relazione alla loro diversità con altri sistemi polari (LH, KF) precedentemente da noi studiati con tecniche teoriche simili.

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Резюме Недавно полученное анизотропное взаимодействие для He−N₂ систем испоиьзуется в зтой работе для поробного анализа квантового переноса внутренней знергии меЗду многочисленными ротационными состояниями молекул мишени и несколькими колебатэльными состояниями. Анализируется природа анизотропии вращательіх и колебательньіх связей с точки зрения влияния зтой анизотропии на динамичэскиэ наблюдаемье величины, такиэ как интэграиьіе и дифференциалъные неупругие поперечные сеч∈ния. Интерферендионные структуры, которы∈ возникют в таких системах, слабо зависят от вибрадионной связх. Проводится сравнениэ полученных резульіх резулвтатоб с резулвтатми для других полярных молекул, ранее рассмотренных в рамках того Зе теоретическочо подхода.

     

Quantum mechanical calculations of the HCN-HNC isomerization

Ab initio calculations using a minimum basis of Slater orbitals have been carried out on the molecules HCN and HNC with optimization of geometry. The predicted geometry of HNC is compatible with an observed radio-emission line which has been attributed to this molecule. The reaction profile for the internal migration of the proton has been calculated. The transition state is approximately T shaped with an activation energy from HNC of 251 kJ/mole.     

The interaction of a quantum mechanical oscillator with gravitational radiation

Within the framework of the linearized field equations of gravitation, the interaction operators between a quantum mechanical system and an external gravitational field are derived from the general-covariant Klein-Gordon and Dirac equation. In the case of linearly polarized plane gravitational waves the transition probabilities for absorption and induced and spontaneous emission of gravitational radiation by a quantum mechanical harmonic oscillator are calculated with the help of the time-dependent perturbation method. The results coincide with the classical ones according to the correspondence principle.     

Capacity and quantum mechanical tunneling

We connect the notion of capacity of sets in the theory of symmetric Markov process and Dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics. In particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero. We also discuss several examples in ^d .