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Analytic approximation for eigenvalues in $\mathcal{PT}$ symmetric quantum mechanics

O. D. Skoromnik, И. Д. Феранчук

Published Aug 3, 2017• 0 citations
OpenAlex

Abstract

An analytical approximation for the eigenvalues of $\mathcal{PT}$ symmetric Hamiltonians is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for the whole spectrum of the system.

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Analytic approximation for eigenvalues in PT\mathcal{PT}PT symmetric quantum mechanics
An analytical approximation for the eigenvalues of PT\mathcal{PT}PT symmetric Hamiltonians is developed via simple basis sets of harmonic-oscillator wave functions with variable frequencies and equilibrium positions. We demonstrate that our approximation provides high accuracy for the whole spectrum of the system.
This paper proposes to broaden the canonical formulation of quantum mechanics.Ordinarily, one imposes the condition H † = H on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition.This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum.However, replacing this mathematical condition by the weaker and more physical requirement H ‡ = H, where ‡ represents combined parity reflection and time reversal PT , one obtains new classes of complex Hamiltonians whose spectra are still real and positive.This generalization of Hermiticity is investigated using a complex deformation H = p 2 + x 2 (ix) ǫ of the harmonic oscillator Hamiltonian, where ǫ is a real parameter.The system exhibits two phases: When ǫ ≥ 0, the energy spectrum of H is real and positive as a consequence of PT symmetry.However, when -1 < ǫ < 0, the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken.The phase transition that occurs at ǫ = 0 manifests itself in both the quantum-mechanical system and the underlying classical system.Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians H = p 2 + x 2N (ix) ǫ with N integer and ǫ > -N ; each of these complex Hamiltonians exhibits a phase transition at ǫ = 0.These PT -symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.
The harmonic oscillator is one of the most studied systems in Physics with a myriad of applications. One of the first problems solved in a Quantum Mechanics course is calculating the energy spectrum of the simple harmonic oscillator with analytic and algebraic approaches. In the algebraic solution, creation and annihilation operators are introduced to factorize the Hamiltonian. This work presents an algebraic solution for the simple harmonic oscillator in the context of Classical Mechanics, exploring the Hamiltonian formalism. In this solution, similarities between the canonical coordinates in a convenient basis for the classical problem and the corresponding operators in Quantum Mechanics are highlighted. Moreover, the presented algebraic solution provides a straightforward procedure for the quantization of the classical harmonic oscillator, motivating and justifying some operator definitions commonly used to solve the correspondent problem in Quantum Mechanics.
The following text develops a new method of solving the eigenvalue problem of the quantum mechanical harmonic oscillator. Thereby are introduced new notions, like that of a reduced basis or that of a huge algebra. Furthermore are presented formulas for iterated commutation relations. Commutation Relation and Hamilton Function Oscillatory properties of a diatomic molecule, a carbon-monoxide molecule for example, can be modelled by a linear elastic oscillator. The observables for describing the oscillatory states of the oscillator are taken to be the dis- �� location from the equilibrium state and the relative momentum of the two masses.