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

ABSTRACT We consider a spin half particle in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction. We find that the energy eigenvalues for this system are real even though the interaction is not PT invariant.

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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.
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