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Register a new userSemiclassical treatment of logarithmic perturbation theory
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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 $lambda x^{6}$ are considered.
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It is shown that for one-dimensional anharmonic oscillator with potential $V(x)= a x^2+ldots=frac{1}{g^2},hat{V}(gx)$ (and for perturbed Coulomb problem $V(r)=frac{alpha}{r} + ldots = g,tilde{V}(gr)$) the Perturbation Theory in powers of coupling constant $g$ (weak coupling regime) and semiclassical expansion in powers of $hbar^{1/2}$ for energies coincide. %The same is true for strong coupling regime expansion in inverse fractional powers in $g$ of energy. It is related to the fact that the dynamics developed in two spaces: $x (r)$-space and in $gx (gr)$ space, leads to the same energy spectra. The equations which govern dynamics in these two spaces, the Riccati-Bloch equation and the Generalized Bloch(GB) equation, respectively, are presented. It is shown that perturbation theory for logarithmic derivative of wave function in $gx (gr)$ space leads to true semiclassical expansion in powers of $hbar^{1/2}$ and corresponds to flucton calculus for density matrix in path integral formalism in Euclidean (imaginary) time.
The explicit semiclassical treatment of the logarithmic perturbation theory for the bound-state problem for the spherical anharmonic oscillator is developed. Based upon the $hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the quartic anharmonic oscillator are considered.
The explicit semiclassical treatment of logarithmic perturbation theory for the bound-state problem within the framework of the radial Klein-Gordon equation with attractive real-analytic screened Coulomb potentials, contained time-component of a Lorentz four-vector and a Lorentz-scalar term, is developed. Based upon $hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues for the Hulthen potential containing the vector part as well as the scalar component are considered.
In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= frac{1}{g^2},hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variational calculations it has led to variational energies of unprecedented accuracy for cubic anharmonic oscillator. In this paper the formalism is applied to both quartic and sextic, spherically-symmetric radial anharmonic oscillators with two term potentials $V(r)= r^2 + g^{2(m-1)}, r^{2m}, m=2,3$, respectively. It is shown that a two-parametric Approximant for quartic oscillator and a five-parametric one for sextic oscillator for the first four eigenstates used to calculate the variational energy are accurate in 8-12 figures for any $D=1,2,3ldots $ and $g geq 0$, while the relative deviation of the Approximant from the exact eigenfunction is less than $10^{-6}$ for any $r geq 0$.
We analyse the dynamics leading to radiative cooling of an atomic ensemble confined inside an optical cavity when the atomic dipolar transitions are incoherently pumped and can synchronize. Our study is performed in the semiclassical regime and assumes that cavity decay is the largest rate in the system dynamics. We identify three regimes characterising the cooling. At first hot atoms are individually cooled by the cavity friction forces. After this stage, the atoms center-of-mass motion is further cooled by the coupling to the internal degrees of freedom while the dipoles synchronize. In the latest stage dipole-dipole correlations are stationary and the center-of-mass motion is determined by the interplay between friction and dispersive forces due to the coupling with the collective dipole. We analyse this asymptotic regime by means of a mean-field model and show that the width of the momentum distribution can be of the order of the photon recoil. Furthermore, the internal excitations oscillate spatially with the cavity standing wave forming an antiferromagnetic-like order.
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