J. M. Isidro
We show how to complement Feynman's exponential of the action so that it exhibits a Z_2 duality symmetry. The latter illustrates a relativity principle for the notion of quantum versus classical.
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A new approach to the problem of finding the asymptotical behaviour of large orders of semiclassical expansion is suggested. Asymptotics of high orders not only for eigenvalues, but also for eigenfunctions, are constructed. Thus, one can apply not only functional integral technique, which has been used up to now, but also method of direct analysis of the semiclassical expansion recursive relations.
The scattering properties of any complex scattering potential, v:R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system S_v. The application of the adiabatic approximation to S_v yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of S_v and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with larg…
We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form $V(x)=(ix)^{2N+1}+bix$ does not agree with an earlier WKB expression for $V(x)=-(ix)^{2N+1}$ the case $b=0$. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.
The Newton--Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton's characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling e…
We show that a strongly perturbed quantum system, being a semiclassical system characterized by the Wigner-Kirkwood expansion for the propagator, has the same expansion for the eigenvalues as for the WKB series. The perturbation series is rederived by the duality principle in perturbation theory.
We consider hamiltonian systems with spatially varying effective mass and slowly varying local potential in d dimensions. The Slater sum is defined as the diagonal element of the Bloch propagator. We derive a gradient expansion of the Slater sum up to the second order. We will show that the derived analytical expression is valid for d=1,2,3 and 4. A numerical example is shown to highlight the effect of the spatially varying effective mass.
In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)] (arXiv:1701.05583) developed a general theory of channel duality for classical-input quantum-output (CQ) channels. That result showed that a number of well-known duality results for linear codes on the binary erasure channel could be extended to general classical channels at the expense of using dual problems which are intrinsically quantum mechanical. One special case of this duality is a connection between coding for error correction (resp. wire-tap secrecy) on the quantum pure-state channel (PSC) and coding for wir…
We study the quantum mechanical motion in the $x^2y^2$ potentials with $n=2,3$, which arise in the spatially homogeneous limit of the Yang-Mills (YM) equations. These systems show strong stochasticity in the classical limit ($\hbar = 0$) and exhibit a quantum mechanical confinement feature. We calculate the partition function $Z(t)$ going beyond the Thomas-Fermi (TF) approximation by means of the semiclassical expansion using the Wigner-Kirkwood (WK) method. We derive a novel compact form of the differential equation for the WK function. After separating the motion in the channels of the equip…
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…
The necessary and sufficient conditions for the exactness of the semiclassical approximation for the solution of the Schrödinger and Klein-Gordon equations are obtained. It is shown that the existence of an exact semiclassical solution of the Schrödinger equation determines both the semiclassical wave function and the interaction potential uniquely up to the choice of the boundary conditions. This result also holds for the Klein-Gordon equation. Its implications for the solution of the Wheeler-DeWitt equation for the FRW scalar field minisuperspace models are discussed. In particular, exact se…
We construct a class of systems for which quantum dynamics can be expanded around a mean field approximation with essentially classical content. The modulus of the quantum overlap of mean field states naturally introduces a classical distance between classical phase points. Using this fact we analytically show that the time rate of change (trc) of two neighbouring classical trajectories is directly proportional to the trc of quantum correlations. Coherence loss and nonlocality effects appear as corrections to mean field dynamics and we show that they can be given in terms of classical trajecto…
Error bounds for the Floquet-Magnus expansion and their application to the semiclassical quantum Rabi model
本文提出一种非微扰方法推导周期驱动系统任意阶有效哈密顿量，并应用于半经典量子Rabi模型，提供误差界分析。
This is the second paper on semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding endpoints. The classical path, interpolating between this point and the classical vacuum, called "flucton", plus systematic one- and two-loop corrections, has been calculated in the first paper \cite{Escobar-Ruiz:2016aqv} for double-well potential and now extended for a number of quantum-mechanical problems (anharmonic oscillator, sine-Gordon potential). The method is based on systematic expansion in Feynman diagrams and thus can be extended to QFTs. We…
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 prac…
A class of spectral problems with a hidden Lie-algebraic structure is considered. We define a duality transformation which maps the spectrum of one quasi-exactly solvable (QES) periodic potential to that of another QES periodic potential. The self-dual point of this transformation corresponds to the energy-reflection symmetry found previously for certain QES systems. The duality transformation interchanges bands at the bottom (top) of the spectrum of one potential with gaps at the top (bottom) of the spectrum of the other, dual, potential. Thus, the duality transformation provides an exact map…
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 …
Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic…
The out-of-time-order correlator (OTOC), recently analyzed in several physical contexts, is studied for low-dimensional chaotic systems through semiclassical expansions and numerical simulations. The semiclassical expansion for the OTOC yields a leading-order contribution in $\hbar^2$ that is exponentially increasing with time within an intermediate, temperature-dependent, time-window. The growth-rate in such a regime is governed by the Lyapunov exponent of the underlying classical system and scales with the square-root of the temperature.
On connection between perturbation theory and semiclassical expansion in quantum mechanics
本文证明一维非简谐振子、径向振子和受扰库仑问题的微扰论与半经典展开在能量上一致，并通过匹配波函数微扰级数与半经典展开，提出高精度局部近似及微扰级数求和方法。
In quantum mechanics, systems can be described in phase space in terms of the Wigner function and the star-product operation. Quantum characteristics, which appear in the Heisenberg picture as the Weyl's symbols of operators of canonical coordinates and momenta, can be used to solve the evolution equations for symbols of other operators acting in the Hilbert space. To any fixed order in the Planck's constant, many-body potential scattering problem simplifies to a statistical-mechanical problem of computing an ensemble of quantum characteristics and their derivatives with respect to the initial…
For the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$ the Perturbation Theory (PT) in powers of coupling constant $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) is developed constructively in $r$-space and in $(gr)$ space, respectively. The Riccati-Bloch (RB) equation and Generalized Bloch (GB) equation are introduced as ones which govern dynamics in coordinate $r$-space and in $(gr)$-space, respectively, exploring the logarithmic derivative of wavefunction $y$. It is shown that PT in powers …
While a wealth of results has been obtained for chaos in single-particle quantum systems, much less is known about chaos in quantum many-body systems. We contribute to recent efforts to make a semiclassical analysis of such systems feasible, which is nontrivial due to the exponential proliferation of orbits with increasing particle number. Employing a recently discovered duality relation, we focus on the collective, coherent motion that together with the also present incoherent one typically leads to a mixture of regular and chaotic dynamics. We investigate a kicked spin chain as an example of…
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.
We present an analytic calculation of the semiclassical electron-positron pair creation rate by time-dependent electrical fields. We use two methods, first the imaginary time method in the WKB-approximation and second the world-line instanton approach. The analytic tools for both methods are generalized to time-dependent electric fields with more than one component. For the WKB method an expansion of the momentum spectrum of produced pairs around the canonical momentum $\vec{P}=0$ is presented which simplifies the computation of the pair creation rate. We argue that the world-line instanton me…
While plenty of results have been obtained for single-particle quantum systems with chaotic dynamics through a semiclassical theory, much less is known about quantum chaos in the many-body setting. We contribute to recent efforts to make a semiclassical analysis of many-body systems feasible. This is nontrivial due to both the enormous density of states and the exponential proliferation of periodic orbits with the number of particles. As a model system we study kicked interacting spin chains employing semiclassical methods supplemented by a newly developed duality approach. We show that for th…
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