We discuss a methodology problem which is crucially important for solving the Sch?dinger equation in terms of the variational method. We present a complete analysis on the application of the hypervirial theorem for ju...We discuss a methodology problem which is crucially important for solving the Sch?dinger equation in terms of the variational method. We present a complete analysis on the application of the hypervirial theorem for judging the quality of the trial wavefunction without invoking the precise solutions.展开更多
Based on a Hamfltonian identity, we study one-dimensional generalized hypervirial theorem, Blanchardlike （non-diagonal case） and Kramers＇ （diagonal case） recurrence relations for arbitrary x^k which is independen...Based on a Hamfltonian identity, we study one-dimensional generalized hypervirial theorem, Blanchardlike （non-diagonal case） and Kramers＇ （diagonal case） recurrence relations for arbitrary x^k which is independent of the central potential V（x）. Some significant results in diagonal case are obtained for special k in xk （k ≥2）. In particular, we find the orthogonal relation 〈n1｜n2〉 = δh1,n2 （k = 0）, 〈n1[V＇（x）｜n2〉 = （En1-En2）^2〈n1｜x｜n2〉 （k = 1）, En = （n｜V＇（x）x/2｜n〉 ＋ （n｜V（x）｜n〉 （k = 2） and -4En（n｜x｜n） ~ 〈n｜V＇（x）x^2｜n〉 ＋ 4〈n｜V（x）x｜n〉 =0 （k=3）. The latter two formulas can be used directly to calculate the energy levels. We present useYul explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.展开更多
文摘We discuss a methodology problem which is crucially important for solving the Sch?dinger equation in terms of the variational method. We present a complete analysis on the application of the hypervirial theorem for judging the quality of the trial wavefunction without invoking the precise solutions.
基金Supported in part by Project 20150964-SIP-IPN,COFAA-IPN,Mexico
文摘Based on a Hamfltonian identity, we study one-dimensional generalized hypervirial theorem, Blanchardlike （non-diagonal case） and Kramers＇ （diagonal case） recurrence relations for arbitrary x^k which is independent of the central potential V（x）. Some significant results in diagonal case are obtained for special k in xk （k ≥2）. In particular, we find the orthogonal relation 〈n1｜n2〉 = δh1,n2 （k = 0）, 〈n1[V＇（x）｜n2〉 = （En1-En2）^2〈n1｜x｜n2〉 （k = 1）, En = （n｜V＇（x）x/2｜n〉 ＋ （n｜V（x）｜n〉 （k = 2） and -4En（n｜x｜n） ~ 〈n｜V＇（x）x^2｜n〉 ＋ 4〈n｜V（x）x｜n〉 =0 （k=3）. The latter two formulas can be used directly to calculate the energy levels. We present useYul explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.
