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arXiv:quant-ph/0411137 · Journal of Physics A Mathematical and General

-symmetric cubic anharmonic oscillator as a physical model

Ali Mostafazadeh

Published Jul 6, 2005• 134 citations
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Abstract

We perform a perturbative calculation of the physical observables, in particular, pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator and the classical Hamiltonian for the -symmetric cubic anharmonic oscillator, . Ignoring terms of order 4 and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass, and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum–classical correspondence for this -symmetric system, compute the associated conserved probability density and comment on the issue of factor ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.

DOI
10.1088/0305-4470/38/29/010

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We perform a perturbative calculation of the physical observables, in particular, pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator and the classical Hamiltonian for the -symmetric cubic anharmonic oscillator, . Ignoring terms of order 4 and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass, and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum–classical correspondence for this -symmetric system, compute the associated conserved probability density and comment on the issue of factor ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.
The PT-symmetric cubic oscillator with Hamiltonian H_3 = - (1/2) d_x^2 + (1/2) x^2 + i G x^3 is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with L^2(R) boundary conditions. Eigenfunctions of the stationary Schr\{o}dinger equation H_3 psi_n(x) = E_n psi_n(x) are manifestly complex, while the energy eigenvalues E_n are real and positive. Although H_3 does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re psi_n(x) and Im psi_n(x)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when G>0. Furthermore, we find that the eigenfunctions are concentrated in an allowed region where the energy E_n is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex gain and loss terms mutually compensate and lead to the real energy eigenvalues.
We take a qualitative comparative look at quantum and classical quartic anharmonic oscillators. It has been shown that the behavior of the quantum anharmonic oscillator mimics that of the classical anharmonic oscillators with the structurally same Hamiltonian in the coherent state basis of the harmonic oscillators. The associated equation of motion allows us to use Lindstet-Poincare perturbation method to compute the classical frequency of the oscillation order by order, by taking care of its dependence on amplitude and the quantum corrections. We also derive a bound for periodicity of such oscillations in both the classical and quantum cases.
We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. We show that the resulting operators are "quasi-Hermitian" relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, p_W, quantization yields a) continuous mutually unbiased bases (MUB) b) orthogonal bases (with Dirac delta normalization) c) biorthogonal bases (with Dirac delta normalization) d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.
We present a systematic perturbative construction of the most general metric operator (and positive-definite inner product) for quasi-Hermitian Hamiltonians of the standard form, , in one dimension. We show that this problem is equivalent to solving an infinite system of iteratively decoupled hyperbolic partial differential equations in (1 + 1)-dimensions. For the case that v ( x ) is purely imaginary, the latter have the form of a nonhomogeneous wave equation which admits an exact solution. We apply our general method to obtain the most general metric operator for the imaginary cubic potential, v ( x ) = i x 3 . This reveals an infinite class of previously unknown - as well as non- -inner products. We compute the physical observables of the corresponding unitary quantum system and determine the underlying classical system. Our results for the imaginary cubic potential show that, unlike the quantum system, the corresponding classical system is not sensitive to the choice of the metric operator. As another application of our method we give a complete characterization of the pseudo-Hermitian canonical quantization of a free particle moving in that is consistent with the usual choice for the quantum Hamiltonian. Finally, we discuss subtleties involved with higher dimensions and systems having a fixed length scale.