Dirac Contour Representation for Quantum Systems with Finite-Dimensional Hilbert Space in the Extended Complex Plane

The Dirac contour representation functions fk(z) and fb(z) are employed to represent theket states |f ⟩ and bra states ⟨f |, respectively, in quantum systems with a finite-dimensional Hilbertspace H_2j+1. The scalar product within these quantum systems is defined using a contour integral.Moreover, a numerical approach is utilized to examine the time evolution of both periodic and non-periodic systems, utilizing several Hamiltonian matrices. Furthermore, the stability of periodic systemsis investigated. In addition to these aspects, we study the most significant application of the Dirac con-tour representation, which is its capability to handle an extended Hilbert space, suitable for describingquantum physics at negative temperatures.