Hermitik olmayan lineer operatörler ile tutarlı bir quantum mekaniği sistemi kurmak

Conventionally, observables of quantum mechanical systems are represented byHermitian linear operators acting in a separable Hilbert space. This is mainly be-cause Hermitian operators have real eigenvalues and eigenvalues of an observablerepresent experimental results which are also real numbers. Recently, non-Hermtianoperators with real eigenvalues have been employed in the description of quantummechanical systems. The following key observations makes this possible: 1) A di-agonalizable linear operator H with a discrete spectrum has real eigenvalues if andonly if H is η-pseudo-Hertmitian, i.e.,H †= ηHη −1 , for a bounded, Hermitian, invert-ible and positive-definite η; 2) If we define a new inner product, namely ξ|ρ =ηξ|η|ρ ∀|ξ , |ρ in Hilbert space, then H is Hermitian with respect to this new innerproduct. η is not unique, but can be uniquely determined up to an arbitrary multi-plicative constant provided that one chooses an irreducible set of operators with realspectra and require them to be η-pseudo-Hertmitian. We give a detailed discussionof the related mathematical concepts and a proof of the uniqueness theorem for η.Furthermore we show, by explicit calculations, how this theorem applies for quantumsystems with a two-dimensional Hilbert space. In particular, for the latter systems weobtain the general form of a diagonalizable operator H with a real spectrum, calcu-late the most general η that makes it η-pseudo-Hertmitian, determine the form of anyother diagonalizable operator O with a real spectrum that is η-pseudo-Hertmitian,and finally demonstrate how choosing a particular O that together with H form anirreducible set fix η up to a multiplicative constant.1