Semiclassical quantization is exact only for the so called \emph{solvable} potentials, such as the harmonic oscillator. In the \emph{nonsolvable} case the semiclassical phase, given by a series in $\hbar$, yields more or less approximate results and eventually diverges due to the asymptotic nature of the expansion. A quantum phase is derived to bypass these shortcomings. It achieves exact quantization of nonsolvable potentials and allows to obtain the quantum wavefunction while locally approaching the best pre-divergent semiclassical expansion. An iterative procedure allowing to implement practical calculations with a modest computational cost is also given. The theory is illustrated on two examples for which the limitations of the semiclassical approach were recently highlighted: cold atomic collisions and anharmonic oscillators in the nonperturbative regime.
We demonstrate how a certain new form of the quantization condition proposed earlier can be used outside the class of potentials for which this form ensures exact spectra. Taking this form as a base we get an improved interpolating condition for the potential wells with coinciding an + and - infinities asymptotical values. Accuracy of the above conditions is discussed.
The semiclassical quantization rule is derived for a system with a spherically symmetric potential $V(r) \sim r^ν$ $(-2<ν<\infty)$ and an Aharonov-Bohm magnetic flux. Numerical results are presented and compared with known results for models with $ν= -1,0,2,\infty$. It is shown that the results provided by our method are in good agreement with previous results. One expects that the semiclassical quantization rule shown in this paper will provide a good approximation for all principle quantum number even the rule is derived in the large principal quantum number limit $n \gg 1$. We also discuss …
A semiclassical Bohr-Sommerfeld approximation is derived for an N-particle, two-mode Bose-Hubbard system modeling a Bose-Einstein condensate in a double-well potential. This semiclassical description is based on the `classical' dynamics of the mean-field Gross-Pitaevskii equation and is expected to be valid for large N. We demonstrate the possibility to reconstruct quantum properties of the N-particle system from the mean-field dynamics. The resulting semiclassical eigenvalues and eigenstates are found to be in very good agreement with the exact ones, even for small values of N, both for subcr…
We derive the semiclassical WKB quantization condition for obtaining the energy band edges of periodic potentials. The derivation is based on an approach which is much simpler than the usual method of interpolating with linear potentials in the regions of the classical turning points. The band structure of several periodic potentials is computed using our semiclassical quantization condition.
Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these methods are intriguingly exact rather than approximate, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.
We present a semiclassical quantization condition, i.e., quantum-classical correspondence, for steady states of nonadiabatic systems consisting of fast and slow degrees of freedom (DOFs) by extending Gutzwiller's trace formula to a nonadiabatic form. The quantum-classical correspondence indicates that a set of primitive hopping periodic orbits, which are invariant under time evolution in the phase space of the slow DOF, should be quantized. The semiclassical quantization is then applied to a simple nonadiabatic model and accurately reproduces exact quantum energy levels. In addition to the sem…
We propose a new method of quantization of a wide class of dynamical systems that originates directly from the equations of motion. The method is based on the correspondence between the classical and the quantum Poisson brackets, postulated by Dirac. This correspondence applied to open (non-Hamiltonian) systems allows one to point out the way of transition from the quantum description based on the Lindblad equation to the dynamical description of their classical analogs by the equations of motion and vice versa. As the examples of using of the method we describe the procedure of the quantizati…
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 the standard Langer modification where the factor l(l+1) is replaced by (l+1/2)^2, are avoided. The semiclassical energy levels are shown to be exact to first order in $\hbar$ with all higher order contributions vanishing. The wave functions and dipole matrix elements are also discussed.
A certain modification of the semiclassical quantization condition based on the summarization of the known power expansion in the squared Planck constant is proposed. Corresponding deviation from exact spectra arises only together with the deviation of the studied potential from the supersymmetric one so that on the base of the presented method a new kind of the perturbation theory not connected with the Planck constant may be developed
We investigate symmetric oscillators, and in particular their quantization, by employing semiclassical and quantum phase functions introduced in the context of Liouville-Green transformations of the Schrödinger equation. For anharmonic oscillators, first order semiclassical quantization is seldom accurate and the higher order expansions eventually break down given the asymptotic nature of the series. A quantum phase that allows in principle to retrieve the exact quantum mechanical quantization condition and wavefunctions is given along with an iterative scheme to compute it. The arbitrariness …
The exact semiclassical quantization condition represents a cumbersome series expansion, so that only the main term of it is usually taken into account. We propose a way to find next terms without new additional calculations. Results are exact for a wide class of actual potentials.
We study the canonical and the coherent state quantization of a particle moving in a magnetic field on a non-commutative plane. Starting from the so called θ-modified action, we perform the canonical quantization and analyze the gauge dependence of the obtained quantum theory. We construct the Malkin-Man'ko coherent states of the system in question, and the corresponding quantization. On this base, we study the relation between the coherent states and the "classical" trajectories predicted by the θ-modified action. In addition, we construct different semiclassical states, making use of special…
We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of section (SOS) and use a finite approximation for the transfer operator in coordinate space. Computation of the eigenvalues of this matrix combined with a quantization condition, determines a set of semiclassical eigenvalues which are compared with those obtained by solving the Schrödinger equation. The classical dynamics of this billiard system undergoes a smoot…
We derive a semiclassical approximation to the Husimi functions of stationary states of spin systems. We rederive the Bohr-Sommerfeld quantization for spin by locating the poles of the corresponding local Green function. The residues correspond to the Husimi functions, which are seen to agree very well with exact calculations.
A systematic method for calculating higher-order corrections of the relativistic semiclassical fixed-energy amplitude is given. The central scheme in computing corrections of all orders is related to a time ordering operation of an operator involving the Van Vleck determinant. This study provides us a new viewpoint for quantization.
We perform a systematic WKB expansion to all orders for a one-dimensional system with potential $V(x)=U_0/\cos^2{(αx)}$. We are able to sum the series to the exact energy spectrum. Then we show that at any finite order the error of the WKB approximation measured in the natural units of the mean energy level spacing does not go to zero when the quantum number goes to infinity. Therefore we make the general conclusion that the semiclassical approximations fail to predict the individual energy levels within a vanishing fraction of the mean energy level spacing.
We carry out an exact quantization of a PT symmetric (reversible) Liénard type one dimensional nonlinear oscillator both semiclassically and quantum mechanically. The associated time independent classical Hamiltonian is of non-standard type and is invariant under a combined coordinate reflection and time reversal transformation. We use von Roos symmetric ordering procedure to write down the appropriate quantum Hamiltonian. While the quantum problem cannot be tackled in coordinate space, we show how the problem can be successfully solved in momentum space by solving the underlying Schrödinger e…
The three-disk scatterer has served as a paradigm for semiclassical periodic orbit quantization of classical chaotic systems using Gutzwiller's trace formula. It represents an open quantum system, thus leading to spectra of complex eigenenergies. An interesting general feature of open quantum systems described by non-Hermitian operators is the possible existence of exceptional points where not only the complex eigenvalues but also their respective eigenvectors coincide. Using Gutzwiller's periodic orbit theory we show that exceptional points exist in a three-disk scatterer if the system's geom…
We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape invariant potentials. For these potentials, the ground state energies computed via the WKB method typically deviate from the exact results by about 10%, a recently suggested modification using nonintegral Maslov indices is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels.
A way of construction of semiclassical wave function (SWF) based on the Maslov - Fedoriuk approach is proposed which appears to be appropriate also for systems with chaotic classical limits. Some classical constructions called skeletons are considered. The skeletons are generalizations of Arnolds' tori able to gather chaotic dynamics. SWF's are continued by caustic singularities in the configuration space rather then in the phase space using complex time method. The skeleton formulation provides us with a new algorithm for the semiclassical approximation method which is applied to construct SW…
We study the spectrum of the hydrogen atom in Snyder space in a semiclassical approximation based on a generalization of the Born-Sommerfeld quantization rule. While the corrections to the standard quantum mechanical spectrum arise at first order in the Snyder parameter for the $l=0$ states, they are of second order for $l\neq 0$. This can be understood as due to the different topology of the regions of integration in phase space.
Stochastic mechanics (SM), as proposed by Edward Nelson and others in the 20th century, aims to reconstruct quantum mechanics (QM) from a more fundamental theory of classical point particles interacting with a classical-like ether, where said interaction causes the particles to undergo a diffusion process that conserves their average total energy. However, Timothy Wallstrom and others have emphasized that SM runs into the problem that it cannot recover the Schroedinger equation of QM unless an ad hoc quantization condition is assumed. In this thesis, I reformulate SM so that the quantization c…
The semiclassical quantization conditions for all partial waves are derived for bound states of two interacting anyons in the presence of a uniform background magnetic field. Singular Aharonov-Bohm-type interactions between the anyons are dealt with by the modified WKB method of Friedrich and Trost. For s-wave bound state problems in which the choice of the boundary condition at short distance gives rise to an additional ambiguity, a suitable generalization of the latter method is required to develop a consistent WKB approach. We here show how the related self-adjoint extension parameter affec…
The explicit semiclassical treatment of logarithmic perturbation theory for the nonrelativistic bound states problem is developed. Based upon $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions for the one-dimensional anharmonic oscillator is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and exited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the harmonic oscillator perturbed by $λx^{6}$ are considered…
We derive semiclassical quantization conditions for systems with spin. To this end one has to define the notion of integrability for the corresponding classical system which is given by a combination of the translational motion and classical spin precession. We determine the geometry of the invariant manifolds of this product dynamics which support semiclassical solutions of the wave equation. The semiclassical quantization conditions contain a new term, which is of the same order as the Maslov correction. This term is identified as a rotation angle for a classical spin vector. Applied to the …
Extracting the physical content of Liouvillian eigenmodes: Semiclassical quantization
School of Physical Sciences, National Institute of Science Education and Research（国立科学教育与研究中心物理科学学院）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Max-Planck-Institut für Physik Komplexer Systeme（马克斯·普朗克复杂系统物理研究所）；Technische Universität Dresden（德累斯顿工业大学）；Raman Research Institute（拉曼研究所）
The methods of the high energy semiclassical quantization in the rational polygon billiards used in our earlier papers are generalized to an arbitrary rational multi-connected polygon billiards i.e. to the billiards which is a rational polygon with other rational polygons inside them "rotated" with respect to the "mother" ones by rational angles. The respective procedure is described fully and its most important aspects are discussed. This generalization allows us to apply the method to arbitrary billiards with curved boundaries and with multi-connected areas where the respective semiclassical…
The exactness of the semiclassical method for three-dimensional problems in quantum mechanics is analyzed. The wave equation appropriate in the quasiclassical region is derived. It is shown that application of the standard leading-order WKB quantization condition to this equation reproduces exact energy eigenvalues for all solvable spherically symmetric potentials.
We study the Maslov correction to semiclassical states by using a Kähler regularized BKS pairing map from the energy representation to the Schrödinger representation. For general semiclassical states, the existence of this regularization is based on recently found families of Kähler polarizations degenerating to singular real polarizations and corresponding to special geodesic rays in the space of Kähler metrics. In the case of the one-dimensional harmonic oscillator, we show that the correct phases associated with caustic points of the projection of the Lagrangian curves to the configuration …
Mixed Quantum Classical (MQC)-IVR is a recently introduced semiclassical framework that allows for selective quantization of the modes of a complex system. In the quantum limit, MQC reproduces the semiclassical Double Herman-Kluk IVR results, accurately capturing nuclear quantum coherences and conserving zero-point energy. However, in the classical limit, while MQC mimics the Husimi-IVR for real-time correlation functions with linear operators, it is significantly less accurate for non-linear correlation functions with errors even at time zero. Here, we identify the origin of this discrepancy …
We introduce an improved semiclassical dynamics approach to quantum vibrational spectroscopy. In this method, a harmonic-based phase space sampling is preliminarily driven toward non-harmonic quantization by slowly switching on the actual potential. The new coordinates and momenta serve as initial conditions for the semiclassical dynamics calculation, leading to a substantial decrease in the number of chaotic trajectories to deal with. Applications are presented for model and molecular systems of increasing dimensionality characterized by moderate or high chaoticity. They include a bidimension…
The connection between Poincaré spheres for polariz-ation and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic 2-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and Generalized Hermite-Laguerre Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the 2-dimensional oscillator, which acquire an extra significance as fa…
We consider a spherical symmetric black hole in the Schwarzschild metric and apply Bohr-Sommerfeld quantization to determine the energy levels. The canonical partition function is then computed and we show that the entropy coincides with the Bekenstein-Hawking formula when the maximal number of states for the black hole is the same as computed in loop quantum gravity, proving in this case the existence of a semiclassical limit and obtaining an independent derivation of the Barbero-Immirzi parameter.
Despite the seminal connection between classical multiply-periodic motion and Heisenberg matrix mechanics and the massive amount of work done on the associated problem of semiclassical (EBK) quantization of bound states, we show that there are, nevertheless, a number of previously unexploited aspects of this relationship that bear on the quantum-classical correspondence. In particular, we emphasize a quantum variational principle that implies the classical variational principle for invariant tori. We also expose the more indirect connection between commutation relations and quantization of act…
