量子力学における定常摂勅諭の再定式化III縮退を持つ系の摂動諭 : 2次元平面回転子を例として

概要

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In our previous papers we proposed a new reformulation for the stationary perturbation method in quantum mechanics, where the Schrodinger equation is translated into the Riccati equation through the transformation of logarithmic derivative type. It was shown that our method remarkably reduces cumbersome and lengthy calculations for wave functions and energy eigenvalues. In the present paper, we apply our method to the 2-dimensional rotator with a periodic potential, whose Schrodinger equation is called the Mathieu equation. It has been known that each of the excited states is doubly degenerate with respect to energy eigenvalue, and perturbation effect of higher orders yields the removal of degeneracy. By computing wave functions and eigenvalues to the 6-th order, or for some cases to the 8-th order, we show that degeneracy of the n-th excited states can be removed at the 2n-th order in perturbations. Our method leads not only to its simplicity and economy of calculations but also to the unified treatment of perturbations irrespective of degeneracy of states.
山口県立大学の論文
2005-03-25