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1.
A new form of the degenerate perturbation series is proposed. Its construction is inspired by the standard numerical algorithm of inverse iterations: As input, we only assume knowledge of an approximate non-diagonal propagator matrix at some trial zero-order energy. The convergence and a few other technicalities are illustrated by the anharmonic oscillator.Dedicated to Ladislav Trlifaj on the occasion of his 65th birthday.  相似文献   

2.
The Rayleigh-Schrödinger perturbation theory is applied to calculation of the energy levels of excited vibrational states of the HD16O molecule. The model of coupled anharmonic oscillators is considered, with the anharmonic part of potential energy being taken into account as the perturbation. The calculations are carried out for the vibrational states that correspond to three-to seven-fold vibrational excitations. Since the perturbation series diverge in the case of strong resonance interactions and their approximations by the Padé and Padé-Hermite methods do not yield sufficiently correct results, a calculation technique is applied that allows the zero-order approximation to be modified. The zero-order Hamiltonian is modified by shifting the vibrational frequencies, which decreases the mixing of states. The new Rayleigh-Schrödinger series can be summed using the quadratic Padé-Hermite approximation method.  相似文献   

3.
A Rayleigh-Schr?dinger perturbation theory approach based on the adiabatic (Born-Oppenheimer) separation of vibrational motions was previously developed and used to evaluate for a system of coupled oscillators the adiabatic energy levels and their nonadiabatic corrections. This method is applied here to calculate rotation-vibration energies of the triatomic molecular ions HeH(+)(2) and ArNO(+) consisting of a strongly bound diatomic fragment and a relatively loosely bound rare gas atom. In these systems the high-frequency stretching motion of the diatomic fragment can be separated from the other two low-frequency motions without substantial loss of accuracy. Treating the diatomic fragment as a rigid rotor, the low-frequency stretching motion is decoupled from the bending motion in analogy to the concept of the adiabatic (Born-Oppenheimer) separation of motions and the strong nonadiabatic couplings between these two motions are accounted for perturbationally. Although the resulting perturbation series may show poor convergence, they turn out to be accurately summable by applying standard techniques for the summation of divergent series. Comparison with the results obtained from full-dimensional calculations for the two ions shows that the approach is capable of providing accurate energies for quite a few of the bound rotation-vibration states and that in the case of the HeH(+)(2) ion it is even able to predict the positions and widths of some low-lying resonance states with good accuracy. The perturbation approach yields zeroth-order energies and corrections in terms of the relevant quantum numbers. It thus allows a direct assignment of the energy levels without any reference to the corresponding eigenfunctions. The weak couplings between the high- and low-frequency motions can easily be treated by the same perturbative approach and numerically exact energies can finally be obtained. Copyright 2000 Academic Press.  相似文献   

4.
《Physics letters. A》1987,121(5):221-223
A numerical method for solving the Schrödinger equation for a one-dimensional potential expressed as a function which increases in both directions away from its minimum is proposed. The basic assumption relies on the asymptotic properties of the solution. We exemplify the method calculating energies and expectation values for the quartic anharmonic oscillator.  相似文献   

5.
Lagrangian formulation of quantum mechanical Schrödinger equation is developed in general and illustrated in the eigenbasis of the Hamiltonian and in the coordinate representation. The Lagrangian formulation of physically plausible quantum system results in a well defined second order equation on a real vector space. The Klein–Gordon equation for a real field is shown to be the Lagrangian form of the corresponding Schrödinger equation.  相似文献   

6.
A fermionic perturbation theory is developed for the statistical mechanics of the nonlinear Schrödinger model. The theory is based on an interacting-fermion picture of the Bethe wave function. The inner product of the Bethe wave function is explicitly evaluated, and a simple graphical representation of it is given. The basic equations obtained for the free energy agree with those of Yang and Yang. In particular, the present theory gives a clear-cut meaning to the function of Yang and Yang: It represents a fermion energy at finite temperatures.  相似文献   

7.
We present a Rayleigh-Schrödinger-Goldstone perturbation formalism for many Fermion systems. Based on this formalism, variational perturbation scheme which goes beyond the Gaussian approximation is developed. In order to go beyond the Gaussian approximation, we identify a parent Hamiltonian which has an effective Gaussian vacuum as a variational solution and carry out further perturbation with respect to the renormalized interaction using Goldstones expansion. Perturbation rules for the ground state wavefunctional and energy are found, thus, opening a way for general use of the Schrödinger picture method for many Fermion systems. Useful commuting relations between operators and the Gaussian wavefunctional are also found, which could reduce the calculational efforts substantially. As examples, we calculate the first order correction to the Gaussian wavefunctional and the second order correction to the ground state of an electron gas system with the Yukawa-type interaction.  相似文献   

8.
Within the framework of the second-order Rayleigh-Schr?dinger perturbation theory, we investigate the effects of the interaction of the electron and longitudinal-optical phonons in two-dimensional semiconductive quantum dots with respect to a general potential. We propose a simple expression for the ground state energy, and compare it with those obtained by Landau-Pekar strong coupling theory. It is shown both analytically and numerically that the results obtained from the second-order Rayleigh-Schr?dinger perturbation theory could be better than those from Landau-Pekar strong coupling theory when the coupling constant is sufficiently small. Moreover, some interesting problems, such as polarons in quasi-one-dimensional quantum wires, and quasi-zero-dimensional asymmetric or symmetric quantum dots can be easily discussed only by taking different limits. After the numerical calculations, we find that there exists a simple dimensional scaling and symmetry relation for the ground state polaron energy. Furthermore, we apply our results to some weak-coupling polar semiconductors such as GaAs, CdS. It is shown that the polaronic effects are found to be quiet appreciable if the confinement lengths and smaller than a few nanometers. Received: 3 December 1997 / Revised: 6 July 1998 / Accepted: 17 September 1998  相似文献   

9.
《Physics letters. A》1988,130(3):141-146
Relativistic hypervirial and Hellmann-Feynman theorems are used to construct Rayleigh-Schrödinger expansions for eigenvalues of perturbed radial Dirac equations to aribitrary order. The method is very simple and flexible, requiring no matrix elements. Only the unperturbed energy is required as input. Any difficulties due to the presence of unperturbed continuum states are bypassed. Particular attention is paid to hydrogenic atoms with confining scalar potentials of the form W(r) = λrq, q = 0, 1, 2, …. Continued fraction representations of these expansions reveal their Stieltjes behavior for q ⩾ 1 and Padé summability for q = 1, 2.  相似文献   

10.
In this paper we shall re-visit the well-known Schrödinger equation of quantum mechanics. However, this shall be realized as a marginal dynamics of a more general, underlying stochastic counting process in a complex Minkowski space. One of the interesting things about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the Belavkin formalism; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be formally represented by a counting process in a second-quantized Minkowski space. The Minkowski space arises as a canonical quantization of the clock, and this is derived naturally from the matrix-algebra representation [1, 2] of the Newton-Leibniz differential time increment, dt. And so the unitary dynamics of a quantum object, described by the Schrödinger equation, may be obtained as the expectation of a counting process of object-clock interactions.  相似文献   

11.
Abstract

We have calculated the high-order ac Stark Shift, multiphoton ionization rates, and nonlinear susceptibilities for high harmonic generation for the hydrogen atom in a radiation field. The calculations are done in the framework of Rayleigh-Schrödinger perturbation theory applied to a complex-rotated Hamiltonian. Our intention is to investigate the limitations of perturbation theory in calculations of multiphoton processes. Comparisons are made with results from nonperturbative calculations. For some frequencies the results of lowest-order perturbation theory are found to disagree with nonperturbative calculations even at moderate to low intensities (I ~ 1011 W cm?2) and in the absence of resonances. We find that the high-order perturbation expansion theory is not a reliable predictor of the behavior of hydrogen atoms in radiation fields with intensities greater than ~ 1012 W cm?2.  相似文献   

12.
The soliton solutions of the form=A/coshkx and=B tanhkx of the nonlinear Schrödinger equation have been considered with respect to many problems. In this paper, it is shown that the nonlinear Schrödinger equation also possesses a solution manifold that generalizes the above soliton functions and provides a discrete spectrum of eigenfunctions and eigenvalues. With the help of a slight modification of these eigenfunctions, it is possible to extend them to the relativistic case, where they become solutions of a nonlinear Klein-Gordon equation associated with a discrete mass spectrum.  相似文献   

13.
LetT 0(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH 0()+V, whereH 0() is thed-dimensional harmonic oscillator with non-resonant frequencies =(1, ... , d ) and the potentialV(q 1, ... ,q d) is an entire function of order (d+1)–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT 0(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues n (, ) ofT 0+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .  相似文献   

14.
The Rayleigh-Schr?dinger perturbation theory is applied to calculation of vibrational energy levels of triatomic molecules with the C 2v and C s symmetries: SO2, H2S, F2O, HOF, HOCl, and DOCl. Particular attention is given to the states coupled by anharmonic resonances; for such states, the perturbation theory series diverge. To sum these series, the known methods of Padé, Padé-Borel, and Padé-Hermite and the method of power moments are used. For low-lying levels, all the summation methods give satisfactory results, while the method of quadratic Padé-Hermite approximants appears to be more efficient for high-excited states. Using these approximants, the structure of singularities of the vibrational energy, as a function in the complex plane, is studied.  相似文献   

15.
16.
《Physics letters. A》2002,306(1):25-34
We study the solution of the focusing nonlinear Schrödinger equation in the semiclassical limit. Numerical solutions are presented for four different kinds of initial data, of which three are analytic and one is nonanalytic. We verify numerically the weak convergence of the oscillatory solution by examining the strong convergence of the spatial average of the solution.  相似文献   

17.
P. Yabosdee 《Optik》2010,121(5):442-445
We propose a new design of the nano-scale strain monitoring system, which consists of a fiber Bragg grating and a standard vibration source. The measurement resolution is increased from μ-strain to n-strain, using the perturbation method on the fiber grating stretching length. The change in Bragg wavelength due to the change in fiber stretching length is simulated. Results obtained have shown the feasibility of using such a proposed system to monitor small strain and vibration in the nano-scale range. The relationships between temperature and strain on the one hand, and Bragg wavelength on the other hand are plotted. This is shown as it has potential of being used in simultaneous measurement.  相似文献   

18.
A new estimate for the groundstate energy of Schrödinger operators on L2(n) (n 1) is presented. As a corollary, it is shown that the groundstate energy of the Schrödinger operator with a scalar potential V is more than the classical lower bound ess.infx__V(x) if V is essentially bounded from below in a certain manner (enhancement of the groundstate energy due to quantization). As an application, it is proven that the groundstate energy of the Hamiltonian of the hydrogen-like atom is enhanced under a class of perturbations given by scalar potentials (vanishing at infinity).  相似文献   

19.
We present a class of confining potentials which allow one to reduce the one-dimensional Schrödinger equation to a named equation of mathematical physics, namely either Bessel’s or Whittaker’s differential equation. In all cases, we provide closed form expressions for both the symmetric and antisymmetric wavefunction solutions, each along with an associated transcendental equation for allowed eigenvalues. The class of potentials considered contains an example of both cusp-like single wells and a double-well.  相似文献   

20.
It was shown in a previous communication that the nonlinear Schrödinger equation exhibits a spectrum of eigenfunctions of the form = k,A k (coshkx) –k and = k B k (coshkx) –k–1sinhkx, and the corresponding eigenvalues of the energy are related to a band structure with a characteristic energy gap as a significant feature. In the present paper, it is shown that a further spectrum exists exhibiting the general structure = k=0 A k(cosh kx)–k–1/2and = k=0 Bk(cosh kx)–k–3/2sinhkx and yielding also a band structure. An extension of the solution spectrum to a nonlinear Klein-Gordon equation and a nonlinear Dirac equation does not imply essential difficulties, and the corresponding characteristic band structure has to be related to a mass spectrum.  相似文献   

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