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European Journal of Physics

Semiclassical quantization of truncated potentials

Michael Berry, Kieron Burke

Published Aug 30, 2019• 7 citations
Publisher

Abstract

Abstract An infinite potential well, truncated at finite height, provides a simple model for studying the effect of nonanalyticity on semiclassical approximations. An exact quantization condition for the bound states separates the effects associated with the untruncated well from those of the truncation. Because the truncation occurs beyond the classical turning points, it has no effect to any finite order in powers of Planck’s constant. The truncation contribution is exponentially small and depends on the potential in the classically forbidden region. The contribution associated with the well, when consistently approximated beyond all semiclassical orders, also leads to a small exponential, depending on the potential in the classically allowed region. Both exponentially small contributions can be extracted by asymptotic analysis, with explicit results in the simple case of a linear well. This combination of several different semiclassical techniques could be pedagogically useful as an exercise in teaching physical asymptotics at the postgraduate level.

DOI
10.1088/1361-6404/ab4026

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Abstract An infinite potential well, truncated at finite height, provides a simple model for studying the effect of nonanalyticity on semiclassical approximations. An exact quantization condition for the bound states separates the effects associated with the untruncated well from those of the truncation. Because the truncation occurs beyond the classical turning points, it has no effect to any finite order in powers of Planck’s constant. The truncation contribution is exponentially small and depends on the potential in the classically forbidden region. The contribution associated with the well, when consistently approximated beyond all semiclassical orders, also leads to a small exponential, depending on the potential in the classically allowed region. Both exponentially small contributions can be extracted by asymptotic analysis, with explicit results in the simple case of a linear well. This combination of several different semiclassical techniques could be pedagogically useful as an exercise in teaching physical asymptotics at the postgraduate level.
Abstract Motivated by a quantum Zeno dynamics in a cavity quantum electrodynamics setting, we study the asymptotics of a family of symbols corresponding to a truncated momentum operator, in the semiclassical limit of vanishing Planck constant ℏ→0\hbar \rightarrow 0ℏ→0 ħ → 0 and large quantum number N→∞N\rightarrow \infty N→∞ N → ∞ , with ℏN\hbar NℏN ħ N kept fixed. In a suitable topology, the limit is the discontinuous symbol pχD(x,p)p\chi _D(x,p)pχD​(x,p) p χ D ( x , p ) where χD\chi _DχD​ χ D is the characteristic function of the classically permitted region D in phase space. A refined analysis shows that the symbol is asymptotically close to the function pχD(N)(x,p)p\chi _D^{(N)}(x,p)pχD(N)​(x,p) p χ D ( N ) ( x , p ) , where χD(N)\chi _D^{(N)}χD(N)​ χ D ( N ) is a smooth version of χD\chi _DχD​ χ D related to the integrated Airy function. We also discuss the limit from a dynamical point of view.
In the present investigation we assess the relevance of both classically forbidden phenomena and higher-order asymptotic contributions for the semiclassical energy quantization of a particle in the anharmonic oscillator potential V(x)=x 2 /2+ lambda x 4 . We propose an iterative method (Iq) for obtaining higher-order semiclassical corrections, which is similar to the WKB and phase-integral methods in the lowest order, but in higher orders the approximation differs significantly. A 'primitive' Bohr-Sommerfeld energy quantization is compared with a more complete semiclassical quantization, taking into account both complex trajectory contributions and higher-order (quantal) corrections.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
We study a class of PT-symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in \cite{BoMe14} that, when the potential is analytic, the eigenvalues stay real for a perturbation of size O(1). We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
We study a class of ‐symmetric semiclassical Schrödinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double‐well potential. In the simple well case, two of the authors have proved in that, when the potential is analytic, the eigenvalues stay real for a perturbation of size . We show here, in the double‐well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB‐analysis, leading to a fairly explicit quantization condition.
The semiclassical limit of the coherent state propagator $⟨{\mathbf{z}}^{\ensuremath{''}}\ensuremath{\mid}{e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}T∕\ensuremath{\hbar}}\ensuremath{\mid}{\mathbf{z}}^{\ensuremath{'}}⟩$ involves complex classical trajectories of the Hamiltonian $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{H}(\mathbf{u},\mathbf{v})=⟨\mathbf{v}\ensuremath{\mid}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\mid}\mathbf{u}⟩$ satisfying $\mathbf{u}(0)={\mathbf{z}}^{\ensuremath{'}}$ and $\mathbf{v}(T)={\mathbf{z}}^{\ensuremath{''}}*$. In this work we study mostly the case ${\mathbf{z}}^{\ensuremath{'}}={\mathbf{z}}^{\ensuremath{''}}$. The propagator is then the return probability amplitude of a wave packet. We show that a plot of the exact return probability brings out the quantal images of the classical periodic orbits. Then we compare the exact return probability with its semiclassical approximation for a soft chaotic system with two degrees of freedom. We find two situations where classical trajectories satisfying the correct boundary conditions must be excluded from the semiclassical formula. The first occurs when the contribution of the trajectory to the propagator becomes exponentially large as $\ensuremath{\hbar}$ goes to zero. The second occurs when the contributing trajectories undergo bifurcations. Close to the bifurcation the semiclassical formula diverges. More interestingly, in the example studied, after the bifurcation, where more than one trajectory satisfying the boundary conditions exist, only one of them in fact contributes to the semiclassical formula, a phenomenon closely related to Stokes lines. When the contributions of these trajectories are filtered out, the semiclassical results show excellent agreement with the exact calculations.