Article

# Pseudo-hermitian interaction between an oscillator and a spin half particle in the external magnetic field

Modern Physics Letters A (Impact Factor: 1.11). 01/2005; DOI: 10.1142/S0217732305016488

Source: arXiv

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**ABSTRACT:**Non-Hermitian but mathcal{P}_{\varphi }mathcal{T}_{\varphi } -symmetrized spherically-separable Dirac and Schrödinger Hamiltonians are considered. It is observed that the descendant Hamiltonians H r , H theta , and H phi play essential roles and offer some ``user-feriendly'' options as to which one (or ones) of them is (or are) non-Hermitian. Considering a mathcal{P}_{\varphi }mathcal{T}_{\varphi } -symmetrized H phi , we have shown that the conventional Dirac (relativistic) and Schrödinger (non-relativistic) energy eigenvalues are recoverable. We have also witnessed an unavoidable change in the azimuthal part of the general wavefunction. Moreover, setting a possible interaction V( theta)!=0 in the descendant Hamiltonian H theta would manifest a change in the angular theta-dependent part of the general solution too. Whilst some mathcal{P}_{\varphi }mathcal{T}_{\varphi } -symmetrized H phi Hamiltonians are considered, a recipe to keep the regular magnetic quantum number m, as defined in the regular traditional Hermitian settings, is suggested. Hamiltonians possess properties similar to the mathcal{PT} -symmetric ones (here the non-Hermitian mathcal{P}_{\varphi }mathcal{T}_{\varphi } -symmetric Hamiltonians) are nicknamed as pseudo- mathcal{PT} - symmetric.International Journal of Theoretical Physics 01/2009; 48(1):183-193. · 1.09 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This is a challenging paper that includes some reviews and new results. Since the non-commutative version of the classical system based on the compact group SU(2) has been constructed in (quant-ph/0502174) by making use of Jaynes–Commings model and so-called quantum diagonalization method in (quant-ph/0502147), we construct a non-commutative version of the classical system based on the non-compact group SU(1,1) by modifying the compact case. In this model the Hamiltonian is not hermite but pseudo hermite, which causes a big difference between the two models. For example, in the classical representation theory of SU(1,1), unitary representations are infinite dimensional from the starting point. Therefore, to develop a unitary theory of non-commutative system of SU(1,1) we need an infinite number of non-commutative systems, which means a kind of second non-commutativization. This is a very hard and interesting problem. We develop a corresponding theory though it is not always enough, and present some challenging problems concerning how classical properties can be extended to the non-commutative case. This paper is arranged for the convenience of readers as the first subsection is based on the standard model (SU(2) system) and the next one is based on the non-standard model (SU(1,1) system). This contrast may make the similarities and differences between the standard and non-standard models clearer.International Journal of Geometric Methods in Modern Physics 11/2011; 02(05). · 0.95 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different approaches to find such metric operators. We compare the different approaches of calculating pos- itive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations. Exceptional points and spontaneous symmetry breaking are also discussed in these models.Communications in Theoretical Physics 05/2013; 59(5). · 0.95 Impact Factor

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