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We apply the method of State-dependent Diagonalization to study the eigenstates of the relativistic quantum harmonic oscillator in the low relativistic limit. The relativistic corrections of the energy eigenvalues of the quantum harmonic oscillator are evaluated for different values of the relativistic parameter α ≡ ħω0 / m0c2. Unlike the conventional exact diagonalization, this new method is shown to be very efficient for evaluating the energy eigenvalues and eigenfunctions. We have also found that for non-zero α the eigenfunctions of the system become more localized in space and that the ground state of the SHO (i.e., the α = 0 case) turns into a squeezed state. Furthermore, since our system is a special case of the quantum harmonic oscillator with a velocity-dependent anharmonic potential, this new approach should be very useful for investigating the cases with more complicated velocity-dependent anharmonic potentials.

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The thermopower S of a structurally disordered Hubbard model with a quasi-exponential transfer integral is studied. Two cases are considered: (i) the infinite-U case near half-filling and (ii) the finite-U case with various electronic concentrations. Approximations for the required correlation functions are constructed based on their short-time expansions which are carried out to higher orders in the electronic hopping. For both cases, S is found to be in accord with the generalized Heikes formula at high temperatures. In the temperature range where the thermal energy is comparable to the characteristic hopping energy, the thermopower of case (ii) varies rapidly with decreasing temperature.

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