Determination of the mass spectrum of the bound state in the framework of the relativistic hamiltonian approach

We propose one a variant of calculation of the energy spectrum of bound state systems with relativistic corrections. In the framework of quantum field theory, an expression that takes into account relativistic corrections to the mass of the bound state with a known nonrelativistic pair interaction potential is proposed on the basis of calculating the asymptotic behavior of correlation functions of the corresponding field currents with the necessary quantum numbers. Excluding the time variables allows one to determine nonperturbative corrections to the interaction potential. The following results have been obtained in the framework of this approach. The nonperturbative corrections arising due to the relativistic nature of a system to the interaction Hamiltonian are determined. The dependence of the constituent mass of bound-state forming particles on the free state mass and on the orbital and radial quantum numbers is analytically derived. The energy level shift of muonic hydrogen taking into account relativistic corrections is calculated. The energy spectrum of a wide class of potentials that describe the Coulomb bound state is analytically derived with relativistic corrections. The mass spectrum of the glueballs and the constituent masses of the gluons are analytically calculated taking into account spin-orbit, spin—spin, and tensor interactions. Our numerical results have shown very good agreement with the lattice data. Taking into account nonperturbative and nonlocality characters of interactions, the mass spectrum of the mesons consisting of light-light and light-heavy quarks with orbital and radial excitations is determined. Our results show that good agreement with the experimental data for the slope and the intercept of the Regge trajectory can be obtained only taking into account the nonperturbative and the nonlocal characters of interactions. The dependences of the constituent masses of constituent particles on the masses of a free state are certain. When quarks are light, then the difference between current and valent masses of quarks is greater than valent masses of quarks, and when quarks are heavy, then the difference between these masses is insignificant. One of the alternative variants of taking nonlocality into account has been suggested for the definition of properties of hadrons at large distances. The dependence of the constituent masses of constituent particles on the radius of confinement is determined.     

A variational approach to bound states in quantum field theory

We consider here in a toy model an approach to bound state problem in a nonperturbative manner using equal time algebra for the interacting field operators. The potential is replaced by offshell bosonic quanta inside the bound state of nonrelativistic particles. The bosonic dressing is determined through energy minimisation, and mass renormalisation is carried out in a nonperturbative manner. Since the interaction is through a scalar field, it does not include spin effects. The model however nicely incorporates an intuitive picture of hadronic bound states in which the gluon fields dress the quarks providing the binding between them and also simulate the gluonic content of hadrons in deep inelastic collisions.     

Determination of the Glueball Mass Spectrum with the Spin-Orbit Interaction

Based on the investigation of the asymptotic behaviour of the polarization loop function for scalar charged particles in an external gauge field we determine the interaction Hamiltonian including the nonperturbative corrections of the relativistic character of motion and the large coupling constant. The mass spectrum of the bound state is analytically derived. The mechanism for arising of the constituent mass of the bound-state forming particles is explained. The change of the bound-state mass and of the constituent mass of particles is analyzed by varying the coupling constant. The mass spectrum of the two-gluon glueball is calculated taking into account spin-orbit and spin-spin interactions.     

Zur Quantenmechanik relativistischer Teilchen bei gegebenem Hilbert-Raum

The quantum theory of relativistic particles is developed in close analogy with that of nonrelativistic ones. In §1 we show how the Hilbert-space for the states of a physical system can be derived immediaty from experience. It turns out that both theories, the relativistic as well as the nonrelativistic one belong to the same Hilbert space. Lorentz invariance does not enter until we consider the transformations induced in the Hilbert space by the motion of the particles. These are formulated in §2. They form a representation of the inhomogeneous Lorentz-group or even of the Klein-group of conformai transformations in space-time. In §3 we develop the Hilbert-space from the so called “basic space”. This space is spanned by simultaneously observable quantities of specific single particles which we call ‘simple particles’. The characteristics of these single particles are: particle type (mass), momentum, spin, isotopic spin, etc. The configuration-space of a many particle system is taken as a set of all occupation numbers in the basic space. The Hilbert space is then the manifold of all square-integrable functions in the configuration space. In this paper the simple particles are considered to have zero rest-mass. Those having finite mass require the assumption of internal interactions and thus lead to complex structures. Mereover, particles of mass zero are distinguished by a higher degree of symmetry, i.e. by their invariance under transformations of the Klein-group. Taking into account further group-theoretical properties of the Klein-group, we assume that all aprticles are built up from neutrinos of two types. Only these we consider as simple particles. To begin with, we consider the simple case of a world with one kind of simple particles having no internal degrees of freedom. In §4 operators are given representing the inhomogeneous Lorentz-group and the Klein-group. For particles without any interaction these can easily be determined. In the case of particles with interaction it is possible to derive all operators of the inhomogeneous Lorentz-group from the interaction energy operator. This operator must satisfy certain conditions, one of which is quite involved and leads to a nonlinear operator equation. A special and rather simple approximate solution is given in §5. The resulting two-body-equation given in §6 leads to some interesting features concerning scattering and virtual and bound states. The cross-section and the wave function of the virtual state of the particle-pair, as well as the mass of a resonant state are finite. There is no bound state after a limiting process. The whole is strictly Lorentz-invariant, although we started from an expression only approximately Lorentz-invariant. Finally we state the corresponding field equations in §7. Our specific expression for the interaction energy leads to a strictly local coupling between the particles. However, the kinetic terms (which are bilinear in creation- and annihilation-operators) are of a non-local type, perhaps describing something like an interaction between the particles and the vacuum.     

Mesons in the Low-Energy Limit of QCDh

Quark-antiquark bound states are described within the Bethe-Salpeter approach for a class of quark models with instantaneous 4-quark interaction. Thereby decompositions of the Bethe-Salpeter vertex and wave functions according to their Lorentz structures and the particle content are used. Normalization conditions for the bound state functions are given. As an application of the general scheme, we determine the mass spectrum of low-lying mesons without expanding in energy. This calculation is performed for a special Nambu-Jona-Lasinio model which follows from the so-called QCD for hadrons, QCD_h, by a localization of the potential. Furthermore the pion and kaon decay constants are determined. We receive good agreement with the experimental data.     

抛物量子阱中束缚极化子的极化势和结合能

利用改进的Lee-Low-Pines(LLP)方法,用变分法计算了无限深抛物量子阱中同时考虑与体纵光学声子和界面纵光学声子相互作用的束缚极化子的极化势和结合能.数值计算得出:阱宽较大时极化势很小,阱宽较小时极化势较大,所以对于较窄的抛物阱必须考虑极化势.对于给定阱宽的抛物阱,随着远离阱中心极化势迅速减小,当到达阱的界面附近极化势又开始增大.阱宽较小时,束缚极化子的结合能随着阱宽L的增大而急剧减小;阱宽较大时,结合能减小的非常缓慢,最后接近体材料中的三维值.     

Non-Relativistic Treatment of a Generalized Inverse Quadratic Yukawa Potential

A bound state solution is a quantum state solution of a particle subjected to a potential such that the particle's energy is less than the potential at both negative and positive infinity. The particle's energy may also be negative as the potential approaches zero at infinity. It is characterized by the discretized eigenvalues and eigenfunctions,which contain all the necessary information regarding the quantum systems under consideration. The bound state problems need to be extended using a more precise method and approximation scheme. This study focuses on the non-relativistic bound state solutions to the generalized inverse quadratic Yukawa potential. The expression for the non-relativistic energy eigenvalues and radial eigenfunctions are derived using proper quantization rule and formula method, respectively. The results reveal that both the ground and first excited energy eigenvalues depend largely on the angular momentum numbers, screening parameters, reduced mass, and the potential depth. The energy eigenvalues,angular momentum numbers,screening parameters,reduced mass,and the potential depth or potential coupling strength determine the nature of bound state of quantum particles. The explored model is also suitable for explaining both the bound and continuum states of quantum systems.     

Bemerkungen zum Kleinschen Paradoxon

The problem of stationary energy eigenstates of relativistic spin 1/2-particles in external potentials exceeding twice the particles rest mass (Klein's paradox) is investigated, neglecting the interaction of the particles with the radiation field. This corresponds to theH ₀ of the “bound state interaction representation” in quantum electrodynamics. For simplicity, explicit calculations are based on a one dimensional square well potential. In potentials of this type which vanish at infinity and which do not become critically singular at the origin there always exists a stationary ground state of fixed total charge. Though this state, naturally, cannot be attributed to a sharp number of particles, no real paradox occurs.     

Determination of the ground state energies of the H 2 + , D 2 + and H 2 + molecular ions taking into account relativistic corrections

On the basis of determination of the asymptotic behavior of correlation functions of the corresponding field currents with the corresponding quantum numbers an analytic method for determination of the energy spectrum of three-body Coulomb system is suggested. Our results show that the constituent masses of particles, which we have defined as masses of particles in a bound state, differ from masses of particles in a free-state. The constituent mass to the free state mass relation for the electron is greater than the same mass relation for the proton, deuteron and triton. It was also found that this constituent electron mass has different values in each systems, i.e. in H ₂ ⁺ , D ₂ ⁺ and T ₂ ⁺ hydrogen molecular ions. The contributions of exchange and self-energy diagrams were taken into account in the determination of the energy spectrum of the three-body Coulomb system. Our results show that the self-energy diagram contribution is inversely proportional to the square of the constituent mass of particles. This contribution is sufficient for the electron and is negligible for the proton, deuteron and triton. When defining the energy and the wave function (WF), it is necessary to take into account the contributions of both the exchange and self-energy diagrams.     

Trajectory of virtual, bound and resonant Efimov states

The pole trajectory of Efimov states for a three-body ααβ system with αα unbound and αβ bound is calculated using a zero-range Dirac-δ potential. It is shown that a three-body bound state turns into a virtual one by increasing the αβ binding energy. This result is consistent with previous results for three equal mass particles. The present approach considers the n-n-¹⁸C halo nucleus. However, the results have good perspective to be tested and applied in ultracold atomic systems, where one can realize such three-body configuration with tunable two-body interaction.     

Study of the one-dimensional periodic polaron structures

One-dimensional and quasi one-dimensional electron structures are of applied interest. For example, in one-dimensional (nano-capillary) electroneutral metal–ammonia systems, exotic electron properties are observed, such as a drastic (by several orders of magnitude) drop of the electrical conductivity with decreasing temperature, which resembles the superconductivity transition. In this work, we studied the possibility of one-dimensional filamentary polaron nano-structure in insulating media. It was established that the interpolaron pair potential for large polarons offers attraction properties. It is known that attraction between the particles may alter the collective properties of a many-particle system. We demonstrated that the initially uniform distribution of the particles becomes unstable in one-dimensional systems and may change to the nonuniform structured state under specific conditions imposed on the temperature, particle concentration, and parameters of the pair interpolaron potential. The possibility of existence of a periodic one-dimensional structure of small-amplitude polarons that is imposed on the polaron uniform distribution is estimated in terms of temperature and concentration criteria. A dispersion relation between existence of the one-dimensional polaron structure and translational velocity of the polarons is found. The upper limit of the translational velocity when the periodic contribution to the distribution vanishes is determined. Periodic contribution disappears virtually stepwise as the velocity approaches its critical value. It is shown that this specific polaron–polaron interaction leads to results that are in principal different from those observed for classical Coulomb electron interaction.     

Permutation symmetry and condensation of identical particles consisting of fermions

For identical particles, consisting of fermions, the upper bound for the number of particles that can occupy a single state is determined. In the macroscopic case, it is proportional to the square root of the number of possible ways of formation of particles of a given composition of all fermions present in the mixture (the normalization constant of the respective density matrix). Particles capable of accumulating in macroscopic quantities in one state can consist only of an even number of fermions of different kinds. In the case of atoms in a trap, this bound can approach arbitrarily close to the total number of atoms. Since the state of the centers of mass of the atoms is described by a symmetric wave function, they, like elementary bosons, can form a condensate, the coherence properties of the components of which are characterized by an antisymmetric wave function of a single atom in relative coordinates.     

The effective interaction for unbound nucleons and the optical potential

The effective interaction for an unbound nucleon in a complex nucleus is studied with particular emphasis upon the relationship of the unbound state interaction with the interaction for bound states. A successful unified treatment of unbound and bound states is given, and the unbound state effective interaction is related to the standard “optical model”. The attractive force for an unbound nucleon is shown to be of an exchange type and is best represented by a non-local, energy-independent potential which can be calculated, at least for not too high incident nucleon energy, from the attractive interaction between bound nucleons in a nucleus. We relate the unbound state potential to the bound state effective force as calculated by Negele using Hartree-Fock theory.

A non-local potential used by Elton, Webb, and Barrett in a successful analysis of electron scattering is shown, with minor modification, to work remarkably well in an analysis of elastic scattering of protons. Excellent agreement with experiment is obtained for the differential cross section and the polarization of protons with an incident energy up to 60 MeV scattered from ⁴⁰Ca, ⁵⁸Ni, ¹²⁰Sn and ²⁰⁸Pb. The real potential parameters are shown to be independent of the incident energy and physically reasonable; the imaginary potential parameters agree with expectations from Fermi gas and collective models. Bound state energy levels are also calculated with no readjustment of the real potential parameters in order to check the consistency of the interaction with the bound states, and the results are in good agreement with Negele.

The interaction of Skyrme as modified by Vautherin and Brink is also used in an analysis of elastic proton scattering. Good results are obtained for the differential cross section and the polarization for 20 MeV protons scattered from ⁵⁸Ni. At higher incident proton energy, and for larger nuclei, the results are poorer.     

Localization of a pair of bound particles in a random potential

We study the localization length l_c of a pair of two attractively bound particles moving in a one-dimensional random potential. We show in which way it depends on the interaction potential between the constituents of this composite particle. For a pair with many bound states N the localization length is proportional to N, independently of the form of the two particle interaction. For the case of two bound states, we present an exact solution for the corresponding Fokker–Planck equation and demonstrate that l_c depends sensitively on the shape of the interaction potential and the symmetry of the bound state wave functions.     

Efimov effect in an analytically solvable model

The Efimov effect is demonstrated in a model consisting of two heavy particles and a light one when the light-heavy interaction leads to a zero-energy two-body bound state. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable S-wave potential of Yamaguchi form. It is demonstrated that in the case of a- two-body zero-energy bound state the binding energy of the light particle in the two-center potential exactly yields an effective $\text{1}\text{r}{}^2$ potential for the relative motion of the heavies. If the light-heavy mass ratio is made small enough, infinitely many bound states (the Efimov effect) are obtained. The approach to this limit is studied and the nature of the potential for large scattering length is obtained. An upper bound for the number of bound states is calculated using a result of Bargmann and Calogero and Efimov's $\text{ln}\text{(}\text{a}\text{r}{}_0\text{)}$ result is found. We note that the long-range effect arises from the large extent of the bound state, the pair wave function being essentially $\text{exp}\text{(?}\text{r}\text{a}\text{)}$ when the scattering length a is large.     

Semiclassical Method to Schroedinger Equation with Position-Dependent Effective Mass

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schroedinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schroedinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.     

Exact Solutions of the Klein–Gordon Equation with Position-Dependent Mass for Mixed Vector and Scalar Kink-Like Potentials

The relativistic problem of spinless particles with position-dependent mass subject to kink-like potentials (~tanh αx) is investigated. By using the basic concepts of the supersymmetric quantum mechanics formalism and the functional analysis method, we solve exactly the position-dependent effective mass Klein–Gordon equation with the vector and scalar kink-like potential coupling, and obtain the bound state solutions in the closed form. It is found that in the presence of position-dependent mass there exists the symmetry that the discrete positive energy spectra and negative energy spectra are symmetric about zero energy for the case of a mixed vector and scalar kink-like potential coupling, and in the presence of constant mass this symmetry only appears for the cases of a pure scalar kink-like potential coupling or massless particles.     

He-He-Ba三原子体系弱束缚态计算

本文利用含有绝热近似的超球坐标方法计算了碱土金属原子Ba和氦原子组成的弱束缚三原子分子体系He₂Ba的基态性质. 系统地研究了该系统的道函数和超球势曲线特征, 进而得到体系的束缚能. 研究结果显示, ¹³⁸Ba 与⁴He, ³He 的各种组合⁴He-⁴He-¹³⁸Ba, ⁴He-³He-¹³⁸Ba和³He-³He-¹³⁸Ba都分别只有一个束缚态.     

磁场和温度对束缚磁极化子有效质量的影响

研究强、弱耦合情形下,库仑场中束缚磁极化子的性质.采用改进的线性组合算符方法研究束缚磁极化子的振动频率和有效质量的温度依赖性,对RbCl晶体进行数值计算,结果表明:在强耦合情形下,束缚磁极化子的振动频率随温度的升高和磁场的增强而增加;有效质量随温度的增加而增加,但随磁场的增强而减少.