Abstract
We present some basic features of pseudo-hermitian quantum mechanics and illustrate the use of pseudo-hermitian Hamiltonians in a description of physical systems.
Highlights
The description of the system is completely different in quantum theory
We are dealing with a infinite dimensional Hilbert space in this case, which is a vector space complete with respect to the norm induced by the associated scalar product
The evolution equations of classical mechanics are replaced by the Schrödinger equation ih ¶ y = Hy ¶t and the initial conditions are replaced by the requirement that y lies in the domain of the Hamiltonian
Summary
The system is described by its coordinates in N-dimensional phase (configuration) space. [PT , H ] = 0 , where P is space reflection and T is time reversion These can be covered by a broad class of pseudo-hermitian operators which satisfy the following operator equation. We encounter a serious problem that the time evolution generated by these Hamiltonians is non-unitary with respect to the scalar product (1). The Hamiltonian H is self-adjoint with respect to the new scalar product and may serve as a generator of time translations. We will provide an example of pseudo-hermitian systems which appear in relativistic quantum mechanics It illustrates the construction of a proper positive-definite scalar-product and discusses consequent restriction of its ambiguity by the additional physical requirement of relativistic invariance
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