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Il Nuovo Cimento B

Harmonic oscillator with complex mass and frequency and the violation of PT symmetry

A. D. Jannussis, V. Papatheou, H. N. Ioannidou

Published Apr 1, 2002• 0 citations
OpenAlex

Abstract

We show that the Schrodinger Hamiltonian of the harmonic oscillator with complex mass and frequency can have real orthogonal eigenfunctions with corresponding discrete complex eigenvalues for certain values of the parameters, and we study this case.

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In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.
We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.
We consider the nature of the wave function using the example of a harmonic oscillator. We show that the eigenfunctions ψn=znψ_n{=}z^nψn​=zn of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation with z∈Cz\in\mathbb Cz∈C are the coordinates of a classical oscillator with energy En=ℏωnE_n=\hbarωnEn​=ℏωn, n=0,1,2,... n=0,1,2,...\,n=0,1,2,.... They are defined on conical spaces C/Zn{\mathbb C}/{\mathbb Z}_nC/Zn​ with cone angles 2π/n2π/n2π/n, which are embedded as subspaces in the phase space C\mathbb CC of the classical oscillator. Here Zn{\mathbb Z}_nZn​ is the finite cyclic group of rotations of the space C\mathbb CC by an angle 2π/n2π/n2π/n. The superposition ψ=∑ncnψnψ=\sum_n c_nψ_nψ=∑n​cn​ψn​ of the eigenfunctions ψnψ_nψn​ arises only with incomplete knowledge of the initial data for solving the Schrödinger equation, when the conditions of invariance with respect to the discrete groups Zn{\mathbb Z}_nZn​ are not imposed and the general solution takes into account all possible initial data parametrized by the numbers n∈Nn\in\mathbb Nn∈N.
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.
Abstract In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.