Quantization of the classical action and eigenvalue problem

Finite dimensional quantizations of the (q,p) plane : new space and momentum inequalities

We present a N-dimensional quantization a la Berezin-Klauder or frame quantization of the complex plane based on overcomplete families of states (coherent states) generated by the N first harmonic oscillator eigenstates. The spectra of position and momentum operators are finite and eigenvalues are equal, up to a factor, to the zeros of Hermite polynomials. From numerical and theoretical studies of the large $N$ behavior of the product $λ\_m(N) λ\_M(N)$ of non null smallest positive and largest eigenvalues, we infer the inequality $δ\_N(Q) Δ\_N(Q) = σ\_N \overset{<}{\underset{N \to \infty}{\to}…

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides the first explicit quantization of a classically chaotic system.

Hamilton-Jacobi Approach to the Quantization of Classically Non-Separable but Integrable Two-Dimensional Systems: the Role of the Classical Caustics

The quantization method based on the quantum Hamiltonian Jacobi equation, is extended to two-dimensional non-separable but integrable Hamiltonians. It is shown that each wave function for those systems corresponds to a well-defined family of classical trajectories, enveloped by a caustic. The energy eigenvalues and the values of the wave functions on the caustic are obtained by solving the 1-dim quantum Hamilton Jacobi equation of the caustic' arcs. Results are in good agreement with those obtained by usual methods.

Response to: [Comment on Quantization of the damped harmonic oscillator [Serhan et al, J. Math. Phys. 59, 082105 (2018)]]

This is a response to a recently reported comment [1] on paper [J. Math. Phys.59, 082105 (2018)] regarding the quantization of damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of [2] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be firstly achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can …

Numerical accuracy of Bogomolny's semiclassical quantization scheme in quantum billiards

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…

Geometric quantization of completely integrable Hamiltonian systems in the action-angle variables

We provide geometric quantization of a completely integrable Hamiltonian system in the action-angle variables around an invariant torus with respect to polarization spanned by almost-Hamiltonian vector fields of angle variables. The associated quantum algebra consists of functions affine in action coordinates. We obtain a set of its nonequivalent representations in the separable pre-Hilbert space of smooth complex functions on the torus where action operators and a Hamiltonian are diagonal and have countable spectra.

位置依赖非对易空间中的长度量子化

Quantization of Length in Spaces with Position-Dependent Noncommutativity

本研究针对位置依赖非对易空间的长度量子化问题，提出构建结合正则与李代数型非对易参数的梯级算子的新方法，实现了三方向长度量子化，建立了相关算子代数并推导了长度平方算子本征值。

Quantization with Action-Angle Coherent States

For a single degree of freedom confined mechanical system with given energy, we know that the motion is always periodic and action-angle variables are convenient choice as conjugate phase-space variables. We construct action-angle coherent states in view to provide a quantization scheme that yields precisely a given observed energy spectrum ${E_n}$ for such a system. This construction is based on a Bayesian approach: each family corresponds to a choice of probability distributions such that the classical energy averaged with respect to this probability distribution is precisely $E_n$ up to a c…

Formally exact quantization condition for nonrelativistic quantum systems

Based on the standard transfer matrix, a formally exact quantization condition for arbitrary potentials, which outflanks and unifies the historical approaches, is derived. It can be used to find the exact bound-state energy eigenvalues of the quantum system without solving an equation of motion for the system wave functions.

随机非厄米作用量理论在随机量子动力学中的应用：从正则量子化到路径积分量子化

Random non-Hermitian action theory for stochastic quantum dynamics: from canonical to path integral quantization

提出随机非厄米作用量理论，证明正则与路径积分量子化等价，揭示非厄米性导致波包局域化且随机性增大分布方差。

Geometric quantization of time-dependent completely integrable Hamiltonian systems

A time-dependent completely integrable Hamiltonian system is quantized with respect to time-dependent action-angle variables near an instantly compact regular invariant manifold. Its Hamiltonian depends only on action variables, and has a time-independent countable energy spectrum.

Quantization of almost-circular orbits in the Fokker action formalism. General scheme

General two-particle system is considered within the formalism of Fokker-type action integrals. It is assumed that the system is invariant with respect to the Aristotle group which is a common subgroup of the Galileo and Poincaré groups. It is shown that equations of motion of such system admit circular orbit solutions. The dynamics of perturbations of these solutions is studied. It is described by means of a linear homogeneous set of time-nonlocal equations and is analyzed in terms of eigenfrequencies and eigenmodes. The Hamiltonian description of the system is built in the almost circular or…

Enhanced quantization on the circle

We apply the quantization scheme introduced in [arXiv:1204.2870] to a particle on a circle. We find that the quantum action functional restricted to appropriate coherent states can be expressed as the classical action plus $\hbar$-corrections. This result extends the examples presented in the cited paper.

Canonical quantization of so-called non-Lagrangian systems

We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form which can always be treated as the Euler-Lagrange equations of a certain action. We construct such an action explicitly. It turns out that in the general case the hamiltonization and canonical quantization of such an ac…

Inflation of small true vacuum bubble by quantization of Einstein-Hilbert action

We study the quantization of the Einstein-Hilbert action for a small true vacuum bubble without matter or scalar field. The quantization of action induces an extra term of potential called quantum potential in Hamilton-Jacobi equation, which gives expanding solutions including the exponential expansion solutions of the scalar factor $a$ for the bubble. We show that exponential expansion of the bubble continues with a short period (about a Planck time $t_p$), no matter whether the bubble is closed, flat or open. The exponential expansion ends spontaneously when the bubble becomes large, i.e., t…

Group Theoretical Quantization and the Example of a Phase Space S^1 x R^+

The group theoretical quantization scheme is reconsidered by means of elementary systems. Already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions. A systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed, two-dimensional prototypes of which are T^*R^+ and S^1 x R^+ (interpreted as restrictions of T^*R and T^*S^1 to positive coordinate and momentum, respectively). In this framework (and under an additional, natural cond…

Any l-state solutions of the Woods-Saxon potential in arbitrary dimensions within the new improved quantization rule

The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulthén potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.

Examples of Enhanced Quantization: Bosons, Fermions, and Anyons

Enhanced quantization offers a different classical/quantum connection than that of canonical quantization in which $\hbar >0$ throughout. This result arises when the only allowed Hilbert space vectors allowed in the quantum action functional are coherent states, which leads to the classical action functional augmented by additional terms of order $\hbar$. Canonical coherent states are defined by unitary transformations of a fixed, fiducial vector. While Gaussian vectors are commonly used as fiducial vectors, they cannot be used for all systems. We focus on choosing fiducial vectors for several…

Inequivalent quantizations of the three-particle Calogero model constructed by separation of variables

We quantize the 1-dimensional 3-body problem with harmonic and inverse square pair potential by separating the Schrödinger equation following the classic work of Calogero, but allowing all possible self-adjoint boundary conditions for the angular and radial Hamiltonians. The inverse square coupling constant is taken to be $g=2ν(ν-1)$ with ${1/2} <ν< {3/2}$ and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral $D_6$ symmetry of its potential term $9 ν(ν-1)/\sin^2 3ϕ$. These are parametrized by a matrix $U\in U(2)$ sati…

Deformation Quantization of Geometric Quantum Mechanics

Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equati…