Hamiltonian and commutation relations

Giuseppe Marmo

Juan Cesar Flores

In a first order approximation, the Bose-Hubbard Hamiltonian with on site interaction is obtained from the free Hamiltonian (U=0) and generalized commutation relation for the annihilation-creation operators. Similar generalized commutation relations were used for the first time in high energy physics. The spectrum of the system can be found formally by using the algebraic properties of the generalized operators.

G. N. Brodimas, A. D. Jannussis, D. Sourlas +1 more

Raoelina Andriambololona

A method to obtain all the operators diagonalizing a given Hamiltonian in second-quan-tization theory is presented. The importance not only of the dynamical commutation relations but also of the kinematical and mutual commutation relations is pointed out. Different classes of Hamiltonians (with examples), programs, degree of a program, order of calculation are introduced. Finally, the practical utilization of the method is given.

Eugene Paul Wigner

The commutator of the Hamiltonian with the operator corresponding to any physical quantity gives the operator which corresponds to the time derivative of that quantity. One can ask, hence, whether the postulate, that the quantum mechanical operators obey the classical equations of motion, uniquely determines the commutation relations. The answer is found to depend on the form of the Hamiltonian and is in the negative for a free particle and for the harmonic oscillator.