Article

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

Modern Physics Letters A (Impact Factor: 1.34). 01/2005; DOI: 10.1142/S0217732305016488

Source: arXiv

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**ABSTRACT:**We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both cases we show that the energy spectrum is real and all the other features of nonrelativistic pseudo-Hermitian formulation are present. Using the spectral method, the positive definite metric operator (η) has been calculated explicitly for both the models to ensure positive definite norms for the state vectors.Modern Physics Letters A 11/2011; 25(20). · 1.34 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). · 0.62 Impact Factor - [Show abstract] [Hide abstract]

**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. · 1.19 Impact Factor

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