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Open Journal of Microphysics

Pseudo-Hermitian Matrix Exactly Solvable Hamiltonian

Ancilla Nininahazwe

Published Jan 1, 2019• 1 citation
Publisher

Abstract

The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian REF _Ref536606452 \r \h \* MERGEFORMAT [1] REF _Ref536606454 \r \h [2]. Like the direct approach to invariant vector spaces used in Refs. REF _Ref536606456 \r \h [3] REF _Ref536606457 \r \h [4], we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.

DOI
10.4236/ojm.2019.91001

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•Jan 17, 2005•15 citations
The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian REF _Ref536606452 \r \h * MERGEFORMAT [1] REF _Ref536606454 \r \h [2]. Like the direct approach to invariant vector spaces used in Refs. REF _Ref536606456 \r \h [3] REF _Ref536606457 \r \h [4], we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.
The non PT−P T-PT− symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field that couples to a harmonic oscillator via some pseudo-hermitian interaction is considered. We highlight all of the properties of both the original Mandal and Jaynes-Cummings Hamiltonians. We show that the Mandal Hamiltonian HMH_{M}HM​ is non hermitian and non invariant under the combined action of the parity operator P and the time-reversal operator T Even if the previous properties are not satisfied, it has been proved that the Mandal Hamiltonian HMH_{M}HM​ is pseudo-hermitian with respect to P and with respect to σ3\sigma_{3}σ3​ also [1,2]. Thus, we show that the original Jaynes-Cummings Hamiltonian is hermitian. We show the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator, similar to the direct approach to invariant vector spaces used in Refs [3,4].
It is shown that for a given Hermitian Hamiltonian possessing supersymmetry, there is always a non-Hermitian Jaynes–Cummings-type Hamiltonian (JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-Hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-Hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-Hermitian version of the standard Jaynes–Cummings (JC) model for which the parent supersymmetric Hamiltonian is the superoscillator. It is also shown that a non-Hermitian version of the several physically motivated generalizations of the JC model admits entirely real spectra. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time evolution is unitary.
Abstract In this work, we describe certain pseudo-Hermitian extensions of the harmonic and isotonic oscillators, both of which are exactly-solvable models in quantum mechanics. By coupling the dynamics of a particle moving in a one-dimensional potential to an imaginary-valued gauge field, it is possible to obtain certain pseudo-Hermitian extensions of the original (Hermitian) problem. In particular, it is pointed out that the Swanson oscillator arises as such an extension of the quantum harmonic oscillator. For the pseudo-Hermitian extensions of the harmonic and isotonic oscillators, we explicitly solve for the wavefunctions in the position representation and also explore their intertwining relations.
Non-Hermitian von Roos Hamiltonian's η\etaη-weak-pseudo-Hermiticity and exact solvability
A complexi…ed von Roos Hamiltonian is considered and a Hermitian …rst-order intertwining di¤erential operator is used to obtain the related position dependent mass -weak-pseudo-Hermitian Hamiltonians. Two user -friendly reference-target maps are introduced to serve for exactsolvability of some non-Hermitian -weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -symmetric Scarf II and a non-Hermitian periodic-type PT -symmetric Samsonov-Roy potentials are used as reference models in a user-friendly reference-target map and the corresponding isospectral Hamiltonians are obtained. It is observed that for each exactly-solvable reference Hamiltonian there is a corresponding set of exactly-solvable target Hamiltonians. PACS numbers: 03.65.Ge, 03.65.Fd, 03.65.Ca
It is shown that for a given Hermitian Hamiltonian possessing supersymmetry, there is alwayas a non-hermitian Jaynes-Cummings-type Hamiltonian(JCTH) admitting entirely real spectra. The parent supersymmetric Hamiltonian and the corresponding non-hermitian JCTH are simultaneously diagonalizable. The exact eigenstates of these non-hermitian Hamiltonians are constructed algebraically for certain shape-invariant potentials, including a non-hermitian version of the standard Jaynes-Cummings model for which the parent supersymmetric Hamiltonian is the superoscillator. The positive-definite metric operator in the Hilbert space is constructed explicitly along with the introduction of a new inner product structure, so that the eigenstates form a complete set of orthonormal vectors and the time-evolution is unitary.