Semiclassical Energy Levels Research Articles

Regular and chaotic motion domains in the channeling electron's phase space and mean level density for its transverse motion energy

The motion of charged particles in a crystal in the axial channeling regime can be both regular and chaotic. The chaos in quantum case manifests itself in the statistical properties of the energy levels set. These properties have been studied previously for the electrons channeling along [110] direction of the silicon crystal, in the case when the classical motion was completely chaotic. The case of channeling along [100] direction is of special interest because the classical motion here can be both regular and chaotic for the same energy depending on the initial conditions. The semiclassical energy level density (as well as its part that corresponds to the regular motion domains in the phase space) is computed for the 10 GeV channeling electrons and positrons. It is demonstrated that the level spacing distribution for both electrons and positrons can be better described by Berry-Robnik distribution than by both Wigner (completely chaotic case) or Poisson (completely regular case) distributions.

Magnetic breakdown investigated by means of the magnetic-field-containing relativistic tight-binding approximation method

We investigate the magnetic breakdown (MB) phenomena by means of the recently proposed magnetic-field-containing relativistic tight-binding approximation (MFRTB) method [Phys. Rev. B 91, 075122 (2015)]. In the MFRTB method, the MB phenomena can be described as the electron hopping between adjacent semiclassical orbits. It is shown that a set of magnetic energy bands is generated by the MB in the cluster that corresponds to the semiclassical energy level. It is also found that magnetic energy bands originating from the MB and those originating from the semiclassical orbit lying on the constant energy surface are hybridized to each other. Such hybridization leads to various subclusters that correspond to energy states of the so-called forbidden orbits. Magnetic oscillations related to the MB occur when the subcluster changes from the occupied state to unoccupied one and vice versa.

Applying Bogomolny's quantization method to generic classical systems.

The quantization method of Bogomolny [Nonlinearity 5, 805 (1992)] can potentially provide semiclassical estimates for energy levels of all bound states of arbitrary systems. This approach requires the formation of the transfer matrix TE as a function of energy E. Existing practical methods for calculating this matrix require a recalculation of many classical trajectories for each energy. This has hampered the application of Bogomolny's method to generic systems that do not possess special classical scaling properties. Generalizing earlier work [H. Barak and K. G. Kay, Phys. Rev. E 88, 062926 (2013)], we develop initial value representation formulas for TE that overcome this problem. These expressions are obtained from a generalized Herman-Kluk formula for the propagator that allows one to easily derive a family of semiclassical integral approximations for the Green's function that are, in turn, used to form the transfer matrix. Calculations for two-dimensional systems show that Bogomolny's method with the present expressions for TE produces accurate semiclassical energy levels from small transfer matrices.

Semiclassical excited-state signatures of quantum phase transitions in spin chains with variable-range interactions

We study the excitation spectrum of a family of transverse-field spin chain models with variable interaction range and arbitrary spin $S$, which in the case of $S=1/2$ interpolates between the Lipkin-Meshkov-Glick and the Ising model. For any finite number $N$ of spins, a semiclassical energy manifold is derived in the large-$S$ limit employing bosonization methods, and its geometry is shown to determine not only the leading-order term but also the higher-order quantum fluctuations. Based on a multi-configurational mean-field ansatz, we obtain the semiclassical backbone of the quantum spectrum through the extremal points of a series of one-dimensional energy landscapes -- each one exhibiting a bifurcation when the external magnetic field drops below a threshold value. The obtained spectra become exact in the limit of vanishing or very strong external, transverse magnetic fields. Further analysis of the higher-order corrections in $1/\sqrt{2S}$ enables us to analytically study the dispersion relations of spin-wave excitations around the semiclassical energy levels. Within the same model, we are able to investigate quantum bifurcations, which occur in the semiclassical ($S\gg 1$) limit, and quantum phase transitions, which are observed in the thermodynamic ($N\rightarrow\infty$) limit.

Semiclassical initial-value representation of the transfer operator

The ability of semiclassical initial-value representation (IVR) methods to determine approximate energy levels for bound systems is limited due to problems associated with long classical trajectories. These difficulties become especially severe for large or classically chaotic systems. This work attempts to overcome such problems by developing an IVR expression that is classically equivalent to Bogomolny's formula for the transfer matrix [E. B. Bogomolny, Nonlinearity 5, 805 (1992); Chaos 2, 5 (1992)] and can be used to determine semiclassical energy levels. The method is adapted to levels associated with states of desired symmetries and applied to two two-dimensional quartic oscillator systems, one integrable and one mostly chaotic. For both cases, the technique is found to resolve all energy levels in the ranges investigated. The IVR method does not require a search for special trajectories obeying boundary conditions on the Poincaré surface of section and leads to more rapid convergence of Monte Carlo phase space integrations than a previously developed IVR technique. It is found that semiclassical energies can be extracted from the eigenvalues of transfer matrices of dimension close to the theoretical minimum determined by Bogomolny's theory. The results support the assertion that the present IVR theory provides a different semiclassical approximation to the transfer matrix than that of Bogomolny for ℏ≠0. For the chaotic system investigated the IVR energies are found to be generally more accurate than those predicted by Bogomolny's theory.

Intra-Atomic Electric Field Radial Potentials in Step-Like Presentation

Within the frames of semiclassical approach, intra-atomic electric field potentials are parameterized in form of radial step-like functions. Corresponding parameters for 80 chemical elements are tabulated by fitting of the semiclassical energy levels of atomic electrons to their first principle values. In substance binding energy and electronic structure calculations, superposition of the semiclassically parameterized constituent-atomic potentials can serve as a good initial approximation of its inner potential: the estimated errors of the determined structural and energy parameters make up a few percent.

Semiclassical quantization of Bohr orbits in the helium atom

We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z2+e−e− configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons.

Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits

A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori and thereby allows one to characterize the periodic orbits by a set of winding numbers. With this knowledge, we construct the action variables as functions of the frequency ratios and carry out a semiclassical torus quantization. The semiclassical energy levels thus obtained agree well with exact quantum calculations.

Semiclassical description of a sixth-order quadrupole boson Hamiltonian

A sixth-order quadrupole boson Hamiltonian is treated through a time dependent variational principle approach choosing as trial function a coherent state with respect to zeroth b 0 † and second b 2 † + b −2 † components of the quadrupole bosons. The coefficients involved in the model Hamiltonian are chosen so that the classical effective potential energy term has two distinct minima. The classical system is described by two degrees of freedom and has two constants of motion. The equation of motion for the radial coordinate is analytically solved and the resulting trajectories are extensively studied. One distinguishes three energy regions exhibiting different types of trajectories. When one passes from the region characterized by two wells to the region of energies higher than the maximum value of the effective potential the trajectories period exhibits a singularity which reflects in fact a phase transition. The classical trajectories are quantized by a constraint of the integral action similar to the well known Bohr–Sommerfeld quantization condition. The semiclassical spectra corresponding to the two potential wells have specific properties. The tunneling process through the potential barrier is also studied. It is a remarkable fact that the transmission coefficients exhibit jumps in magnitude when the intrinsic momentum acquires certain values. It is an open question whether such discontinuities in the transmission coefficients show up also when the final states are not bound. Semiclassical energy levels are compared also with the exact eigenvalues of the initial Hamiltonian, obtained through a diagonalization procedure. The present approach suggests that the energy levels in the laboratory frame can be written as a sum of energy associated to the intrinsic degrees of freedom and a rotational term obtainable through a projection method. This is numerically confirmed for the ground band.

Kink scaling functions in 2D non-integrable quantum field theories

We determine the semiclassical energy levels for the ϕ 4 field theory in the broken symmetry phase on a 2D cylindrical geometry with antiperiodic boundary conditions by quantizing the appropriate finite-volume kink solutions. The analytic form of the kink scaling functions for arbitrary size of the system allows us to describe the flow between the twisted sector of c = 1 CFT in the UV region and the massive particles in the IR limit. Kink-creating operators are shown to correspond in the UV limit to disorder fields of the c = 1 CFT. The problem of the finite-volume spectrum for generic 2D Landau–Ginzburg models is also discussed.

Semiclassical energy levels of sine-Gordon model on a strip with Dirichlet boundary conditions

We derive analytic expressions of the semiclassical energy levels of sine-Gordon model in a strip geometry with Dirichlet boundary condition at both edges. They are obtained by initially selecting the classical backgrounds relative to the vacuum or to the kink sectors, and then solving the Schrödinger equations (of Lamè type) associated to the stability condition. Explicit formulas are presented for the classical solutions of both the vacuum and kink states and for the energy levels at arbitrary values of the size of the system. Their ultraviolet and infrared limits are also discussed.

Semiclassical Spectral Series of a Helium-like Atom in a Magnetic Field

We apply the complex WKB method (the Maslov complex germ theory) to the model of two electrons in a field with a fixed center. We construct semiclassical spectral series of the Pauli operator eigenvalues in the external magnetic field with the spin–orbital and spin–spin interactions and the quadrupole moment of the nucleus taken into account. These series correspond to a new type of closed phase trajectories, the relative equilibrium positions of the corresponding classical nonintegrable system. Explicit effective formulas are derived for the fine (Zeeman effect) and the hyperfine splitting of semiclassical energy levels of a helium-like ion with an arbitrary nucleus charge Z in the entire range of the magnetic field magnitude, including the extreme cases of weak and ultrastrong fields.

Centrifugal terms in the WKB approximation and semiclassical quantization of hydrogen

A systematic semiclassical expansion of the hydrogen problem about the classical Kepler problem is shown to yield remarkably accurate results. Ad hoc changes of the centrifugal term, such as in the WKB approximation and semiclassical quantization of hydrogen, where the factor $l(l+1)$ is replaced by ${(l+1/2)}^{2}$, are avoided. Expanding systematically in powers of \ensuremath{\Elzxh}, the semiclassical energy levels are shown to be exact to first order in $\ensuremath{\Elzxh}$ with all higher-order contributions vanishing. The wave functions and dipole matrix elements are also discussed.

Semiclassical energy levels of electrons in metals with band degeneracy lines

We show that in calculating the semiclassical energy levels of electrons in metals located in a magnetic field, one must determine whether or not the corresponding electron paths in the space of wave vectors k are attached to a band degeneracy line. Calculations in the two possible cases, i.e., with and without such attachment, differ by |e|ℏ/2m*c, where e is the electron charge and m* is the cyclotron mass of the electron. This shift in the energy levels is of a topological nature, and its existence depends neither on the specific form of the electron dispersion relation e(k) near the electron path nor on the shape or size of this path. The reason for this shift lies in the fact that the electron orbit is attached to the band degeneracy line, which is the line of singular points of the Bloch wave functions. In many respects this effect is similar to the Aharonov-Bohm effect if the band degeneracy line is considered an infinitely thin “solenoid.” This shift in energy levels should become apparent in studies of oscillation phenomena in metals. We give examples of metals in which the conditions for observing the shift is probably the most favorable.

Semiclassical initial value representation for rotational degrees of freedom: The tunneling dynamics of HCl dimer

The semiclassical initial value representation (SC-IVR) is emerging as a practical way of generalizing classical trajectory simulation methods to include (approximately) the effects of quantum mechanics (i.e., interference and tunneling). This paper describes the application of the SC-IVR approach to determine the low lying vibrational states of the HCl dimer on a realistic potential energy surface. Overall agreement of the semiclassical energy levels with accurate quantum values is very good, including a good description of the tunneling splitting in the ground state. Issues regarding the applications of the SC-IVR methodology to the angular variables related to rotational degrees of freedom are explicitly discussed.

Semiclassical wave functions and energy levels of Bose-condensed gases in spherically symmetric traps

The WKB-approximation for the Bogoliubov-equations of the quasi-particle excitations in Bose-gases with condensate is worked out in the case of spherically symmetric trap potentials on the basis of the resulting quantization rule. The excitation spectrum is calculated numerically and also analytically in certain limiting cases. It is found that the energy levels of a Bohr-Sommerfeld type quantization may be considerably shifted when the classical turning point gets close to the surface of the condensate.

WKB method for integrable but non-separable Hamiltonians

A recently proposed extension of the WKB method to integrable but non-separable Hamiltonian systems, based on the construction of global solutions of Dirichlet problems for the Hamilton - Jacobi equation, is applied to the construction of semiclassical wavefunctions for the Barbanis Hamiltonian. The wavefunctions obtained are compared with those generated for the same system, by means of the coherent Gaussian method, by Davis and Heller. It is shown that when the caustics are smooth curves, there is no energy degeneration, and the semiclassical wavefunctions obtained by the extended WKB method agree with those by Davis and Heller. But if internal caustics are present and to the same semiclassical energy level correspond, not simply to one as in the coherent Gaussian method, but several (four in the example investigated) WKB wavefunctions, they differ in the regions inside the internal caustics.

The Zeeman effect for the 'anisotropic hydrogen atoms' in the complex WKB approximation: II. Quantization of two-dimensional Lagrangian tori (with focal points) for the Pauli operator with spin-orbit interaction

For pt.I, see ibid, p.5799-5810. On the basis of the complex WKB method, new spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic field are constructed. These series correspond to conditionally periodic motions of a classical electron in the plane orthogonal to the field direction. The semiclassical wavefunctions are regular in the entire configuration space including the focal points and have the 'superscar' properties near the domain of classical motion, i.e. near the projection of stable two-dimensional Lagrangian tori on the configuration space. Both for Schrodinger and Pauli operators the analytical description of the semiclassical wavefunctions 'superscar' structure is obtained. This structure is non-uniform with respect to the space coordinates and has singularities with respect to the parameter h(cross),h(cross) to 0, near focal points. In the limit case of strong magnetic fields explicit analytical formulae for the semiclassical energy levels are obtained.

The Zeeman effect for the 'anistropic hydrogen atom' in the complex WKB approximation: I. Quantization of closed orbits for the Pauli operator with spin-orbit interaction

Using the complex WKB method, new semiclassical spectral series for perturbed states of an 'anisotropic' hydrogen atom, taking into account the electron's spin polarization in a homogeneous magnetic held (the anomalous Zeeman effect) are constructed. They are valid uniformly for all values of the field's magnitude. These series correspond to families (stable in the linear approximation) of periodic and conditionally periodic motions of a classical electron in the plane orthogonal to the direction of the field. The corresponding semiclassical wavefunctions have the property of quantum 'superscaring' in the coordinate space near these classical trajectories of motion. We obtain the limit cases of our semiclassical energy levels for strong and weak magnetic fields. We compare them with the only known approximations, i.e. for the hydrogen atom.

Semiclassical energy level statistics in the transition region between integrability and chaos: transition from Brody-like to Berry-Robnik behaviour

We study the energy level statistics of the generic Hamiltonian systems in the transition region between integrability and chaos and present the theoretical and numerical evidence that in the ultimate (far) semiclassical limit the Berry-Robnik (1984) approach is the asymptotically exact theory. However, before reaching that limit, one observes phenomenologically a quasi-universal behaviour characterized by the fractional power-law level repulsion and globally quite well described by the Brody (or Izrailev) distribution. We offer theoretical arguments explaining this extremely slow transition and demonstrate it numerically in improved statistics of the Robnik billiard and in the standard (Chirikov) map on a torus.