Semiclassical Energy Research Articles

Minimal Length and the Quantum Bouncer: A Nonperturbative Study

We present the energy eigenvalues of a quantum bouncer in the framework of the Generalized (Gravitational) Uncertainty Principle (GUP) via quantum mechanical and semiclassical schemes. In this paper, we use two equivalent nonperturbative representations of a deformed commutation relation in the form [X,P]=i\hbar(1+\beta P^2) where \beta is the GUP parameter. The new representation is formally self-adjoint and preserves the ordinary nature of the position operator. We show that both representations result in the same modified semiclassical energy spectrum and agrees well with the quantum mechanical description.

Semiclassical Low Energy Scattering for One-Dimensional Schrödinger Operators with Exponentially Decaying Potentials

We consider semiclassical Schr\"odinger operators on the real line of the form $$H(\hbar)=-\hbar^2 \frac{d^2}{dx^2}+V(\cdot;\hbar)$$ with $\hbar>0$ small. The potential $V$ is assumed to be smooth, positive and exponentially decaying towards infinity. We establish semiclassical global representations of Jost solutions $f_\pm(\cdot,E;\hbar)$ with error terms that are uniformly controlled for small $E$ and $\hbar$, and construct the scattering matrix as well as the semiclassical spectral measure associated to $H(\hbar)$. This is crucial in order to obtain decay bounds for the corresponding wave and Schr\"odinger flows. As an application we consider the wave equation on a Schwarzschild background for large angular momenta $\ell$ where the role of the small parameter $\hbar$ is played by $\ell^{-1}$. It follows from the results in this paper and \cite{DSS2}, that the decay bounds obtained in \cite{DSS1}, \cite{DS} for individual angular momenta $\ell$ can be summed to yield the sharp $t^{-3}$ decay for data without symmetry assumptions.

Energy level structure and quantum phase transitions of spin systems with nonaxially symmetric Hamiltonians

A general spin system with a nonaxially symmetric Hamiltonian containing Jx, Jz-linear and Jz-quadratic terms, widely used in many-body fermionic and bosonic systems and in molecular magnetism, is considered for the variations of general parameters describing intensity interaction changes of each of its terms. For this model Hamiltonian, a semiclassical energy surface (ES) is obtained by means of the coherent-state formalism. An analysis of this ES function, based on catastrophe theory, determines the separatrix in the control parameter space of the system Hamiltonian: the loci of singularities representing semiclassical phase transitions. Here we show that distinct regions of qualitatively different spectrum structures, as well as a singular behavior of quantum states, are ruled by this separatrix: here we show that the separatrix not only describes ground-state singularities, which have been associated with quantum phase transitions, but also reveals the structure of the excited spectrum, distinguishing different quantum phases within the parameter space. Finally, we consider magnetic susceptibility and heat capacity of the system at finite temperature, in order to study thermal properties and thermodynamical phase transitions in the perspective of the separatrix of this Hamiltonian system.

H=xpModel Revisited and the Riemann Zeros

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, on average, as the nontrivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this Letter, we show that the Hamiltonian H = x(p + ℓ(p)²/p) contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L functions using different self-adjoint extensions of H. We discuss the relation of our work to Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.

Phase diagram of the fully frustrated transverse-field Ising model on the honeycomb lattice

Motivated by the current interest in the quantum dimer model on the triangular lattice, we investigate the phase diagram of the closely related fully frustrated transverse-field Ising model on the honeycomb lattice using classical and semiclassical approximations. We show that, in addition to the fully polarized phase at a large field, the classical model possesses a multitude of phases that break the translational symmetry which, in the dimer language, correspond to a plaquette phase and a columnar phase separated by an infinite cascade of mixed phases. The modification of the phase diagram by quantum fluctuations has been investigated in the context of linear spin-wave theory. The extrapolation of the semiclassical energies suggests that the plaquette phase extends down to zero field for spin 1/2, in agreement with the $\sqrt{12}\ifmmode\times\else\texttimes\fi{}\sqrt{12}$ phase of the quantum dimer model on the triangular lattice with only kinetic energy.

Application of Lie transform perturbation method for multidimensional non-Hermitian systems

Three-dimensional non-Hermitian systems are investigated using classical perturbation theory based on Lie transformations. Analytic expressions for total energy in terms of action variables are derived. Both real and complex semiclassical eigenvalues are obtained by quantizing the action variables. It was found that semiclassical energy eigenvalues calculated with the classical perturbation theory are in very good agreement with exact energies and for certain non-Hermitian systems second-order classical perturbation theory performed better than the second-order Rayleigh–Schroedinger perturbation theory.

Electron–positron pairs production in a macroscopic charged core

Classical and semi-classical energy states of relativistic electrons bounded by a massive and charged core with the charge-mass radio Q/M and macroscopic radius Rc are discussed. We show that the energies of semi-classical (bound) states can be much smaller than the negative electron mass-energy (−mc2), and energy-level crossing to negative energy continuum occurs. Electron–positron pair production takes place by quantum tunneling, if these bound states are not occupied. Electrons fill into these bound states and positrons go to infinity. We explicitly calculate the rate of pair-production, and compare it with the rates of electron–positron production by the Sauter–Euler–Heisenberg–Schwinger in a constant electric field. In addition, the pair-production rate for the electro-gravitational balance ratio Q/M=10−19 is much larger than the pair-production rate due to the Hawking processes.

Phase transition to Bose-Einstein condensation for a bosonic gas confined in a combined trap

We present a study of phase transition to macroscopic superfluidity for an ultracold bosonic gas confined in a combined trap formed by a one-dimensional optical lattice and a harmonic potential, focusing on the critical temperature of this system and the interference patterns of the Bose gas released from the combined trap. Based on a semiclassical energy spectrum, we develop an analytic approximation for the critical temperature $T_{c}$, and compare the analytic results with that obtained by numerical computations. For finite temperatures below $T_{c}$, we calculate the interference patterns for both the normal gas and the superfluid gas. The total interference pattern shows a feature of ``peak-on-a-peak". As a comparison, we also present the experimentally observed interference patterns of $^{87}$Rb atoms released from a one-dimensional optical lattice system in accord with our theoretical model. Our observations are consistent with the theoretical results.

Atoms and quantum dots with a large number of electrons: The ground-state energy

We compute the ground state energy of atoms and quantum dots with a large number N of electrons. Both systems are described by a non-relativistic Hamiltonian of electrons in a d-dimensional space. The electrons interact via the Coulomb potential. In the case of atoms (d=3), the electrons are attracted by the nucleus, via the Coulomb potential. In the case of quantum dots (d=2), the electrons are confined by an external potential, whose shape can be varied. We show that the dominant terms of the ground state energy are those given by a semiclassical Hartree-exchange energy, whose N to infinity limit corresponds to Thomas-Fermi theory. This semiclassical Hartree-exchange theory creates oscillations in the ground state energy as a function of N. These oscillations reflect the dynamics of a classical particle moving in the presence of the Thomas-Fermi potential. The dynamics is regular for atoms and some dots, but in general in the case of dots, the motion contains a chaotic component. We compute the correlation effects. They appear at the order N ln N for atoms, in agreement with available data. For dots, they appear at the order N.

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.

Calculation of semiclassical free energy differences along nonequilibrium classical trajectories

We have derived several relations, which allow the evaluation of the system free energy changes in the leading order in variant Planck's over 2pi(2) along classically generated trajectories. The results are formulated in terms of purely classical Hamiltonians and trajectories, so that semiclassical partition functions can be computed, e.g., via classical molecular dynamics simulations. The Hamiltonians, however, contain additional potential-energy terms, which are proportional to variant Planck's over 2pi(2) and are temperature-dependent. We discuss the influence of quantum interference on the nonequilibrium work and problems with unambiguous definition of the semiclassical work operator.

Semiclassical Ground-State Energies of Many-Electron Systems

A new semiclassical method is proposed to obtain accurate ground-state energies for many-electron systems. The method borrows its semiclassical character from Thomas-Fermi (TF) theory, but improves upon it by including exchange-correlation effects, at least approximately. We illustrate our method (correlated TF method) on simple models of 1D-interacting electrons, showing that it yields dramatic improvements over TF theory, particularly in the strongly correlated regime.

Semiclassical quantization of non-Hermitian 2-D systems: Classical (Lie transform) Perturbation theory

Both real and complex semiclassical eigen energies of two dimensional non-Hermitian Hamiltonian systems are obtained by classical (Lie transform) perturbation theory requiring the action variables I 1 and I 2 to satisfy the quantization condition I 1 =( n 1 +(1/2)) ℏ and I 2 =( n 2 +(1/2)) ℏ respectively where n 1 , n 2 are integers. Classical perturbation theory with Lie transform makes classical trajectories, which are non-periodic or non-quasi-periodic, periodic. It was observed that this method produces accurate eigen energies even when classical trajectories are not periodic or quasi-periodic. Eigen energies obtained by classical perturbation theory are compared with the same, determined by Rayleigh-Schroedinger perturbation theory

Semiclassical quantisation of finite-gap strings

We perform a first principle semiclassical quantisation of the general finite-gap solution to the equations of a string moving on × S3. The derivation is only formal as we do not regularise divergent sums over stability angles. Moreover, with regards to the AdS/CFT correspondence the result is incomplete as the fluctuations orthogonal to this subspace in AdS5 × S5 are not taken into account. Nevertheless, the calculation serves the purpose of understanding how the moduli of the algebraic curve gets quantised semiclassically, purely from the point of view of finite-gap integration and with no input from the gauge theory side. Our result is expressed in a very compact and simple formula which encodes the infinite sum over stability angles in a succinct way and reproduces exactly what one expects from knowledge of the dual gauge theory. Namely, at tree level the filling fractions of the algebraic curve get quantised in large integer multiples of ℏ = 1/(λ)1/2. At 1-loop order the filling fractions receive Maslov index corrections of ½ℏ and all the singular points of the spectral curve become filled with small half-integer multiples of ℏ. For the subsector in question this is in agreement with the previously obtained results for the semiclassical energy spectrum of the string using the method proposed in hep-th/0703191. Along the way we derive the complete hierarchy of commuting flows for the string in the × S3 subsector which are generated by the Taylor coefficients of the quasi-momentum p(x) through Hamilton's equation. Moreover, we also derive a very general and simple formula for the stability angles around a generic finite-gap solution which may be used in the study of stability properties of solutions in the × S3 subsector. We also stress the issue of quantum operator orderings and whether or not a given ordering preserves integrability since this problem already crops up at 1-loop in the form of the subprincipal symbol.

Statistical properties of many particle eigenfunctions

Wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy Green functions. Universal features of fixed energy (microcanonical) random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as N → ∞. This arises through a little known asymptotic limit of Bessel functions. Constraints due to symmetries, boundaries and collisions between particles can be included.

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.

Thomas–Fermi theory for atomic nuclei revisited

The recently developed semiclassical variational Wigner–Kirkwood (VWK) approach is applied to finite nuclei using external potentials and self-consistent mean fields derived from Skyrme interactions and from relativistic mean field theory. VWK consists of the Thomas–Fermi part plus a pure, perturbative ℏ 2 correction. In external potentials, VWK passes through the average of the quantal values of the accumulated level density and total energy as a function of the Fermi energy. However, there is a problem of overbinding when the energy per particle is displayed as a function of the particle number. The situation is analyzed comparing spherical and deformed harmonic oscillator potentials. In the self-consistent case, we show for Skyrme forces that VWK binding energies are very close to those obtained from extended Thomas–Fermi functionals of ℏ 4 order, pointing to the rapid convergence of the VWK theory. This satisfying result, however, does not cure the overbinding problem, i.e., the semiclassical energies show more binding than they should. This feature is more pronounced in the case of Skyrme forces than with the relativistic mean field approach. However, even in the latter case the shell correction energy for e.g., 208Pb turns out to be only ∼−6 MeV what is about a factor two or three off the generally accepted value. As an ad hoc remedy, increasing the kinetic energy by 2.5%, leads to shell correction energies well acceptable throughout the periodic table. The general importance of the present studies for other finite Fermi systems, self-bound or in external potentials, is pointed out.

Thouless energy as a unifying concept for Josephson junctions tuned through the Mott metal-insulator transition

The Thouless energy was introduced in the 1970s as a semiclassical energy for electrons diffusing through a finite-sized conductor. It turns out to be an important quantum-mechanical energy scale for many systems ranging from disordered metals to quantum chaos to quantum chromodynamics. In particular, it has been quite successful in describing the properties of Josephson junctions when the barrier is a diffusive normal-state metal. The Thouless energy concept can be generalized to insulating barriers by extracting an energy scale from the two-probe Kubo conductance of a strongly correlated electron system (metallic or insulating) via a generalized definition of the quantum-mechanical level spacing to many-body systems. This energy scale is known to determine the crossover from tunneling to Ohmic (thermally activated) transport in normal tunnel junctions. Here we use it to illustrate how the quasiclassical picture of transport in Josephson junctions is modified as the strongly correlated barrier passes through the Mott transition. Surprisingly, we find the quasiclassical form holds well beyond its putative realm of validity.

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.