An Analytical Solution in Detuned Two Level Systems
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Finding the evolution of two level Hamiltonian is of great importance in quantum computation and quantum precision manipulation due to the requirement of quantum experiment control. However, the Schr\"odinger equation of an arbitrary time-dependent two level Hamiltonian is hardly solvable due to its non-commutativity Hamiltonian in different times. In this article, we expand and demonstrate an exact solution of Schr\"odinger equation respect to general two level systems with a few limitations. This analytical solution has lots of manipulative parameters and a few boundary restrictions, which could drive many applications. Furthermore, we show the adaptive capacity of our scheme, which demonstrated the widely use of our scheme, and make it suitable for most of experiment Hamiltonian directly.Keywords:
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Abstract The theory of vibrations of a composite particle when vibrational amplitudes are not constrained to be small according to the Eckart conditions is developed using the methods of differential topology. A global classical Hamiltonian appropriate for this system is given, and for the case of the molecular vibration–rotation problem, it is transformed into a global quantum Hamiltonian operator. It is shown that the zeroth‐order term in the global Hamiltonian operator is identical to the Wilson–Howard Hamiltonian; higher‐order terms are shown to give successively better approximations to the large amplitude problem. Generalized Eckart conditions are derived for the global classical Hamiltonian; the quantum equivalent of these conditions along with the quantum equivalent of the Eckart conditions are given. The spectrum of the global Hamiltonian operator is discussed and it is shown that the calculation of the vibration–rotation energy states of the system reduces to the same straight‐forward procedure, the solution of a secular determinant, as was carried out for the Wilson–Howard Hamiltonian at a later time by Nielsen.
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