Matrix D is the canonical form of A-a diagonal matrix with A's eigenvalues on the main diagonal. Produces matrices of eigenvalues ( D) and eigenvectors ( V) of matrix A, so that A*V = V*D. To request eigenvectors, and in all other cases, use eigs to find the eigenvalues or eigenvectors of sparse matrices. If S is sparse and symmetric, you can use d = eig(S) to returns the eigenvalues of S. Returns a vector containing the generalized eigenvalues, if A and B are square matrices. Returns a vector of the eigenvalues of matrix A. ![]() 2.Eig (MATLAB Functions) MATLAB Function Reference Using this method we obtain a relationship between the energies E and the perturbation parameter C see equations ( 5.49)-( 5.50) for the ℓ = 0 case, and equation ( 5.57) for the ℓ ≠ 0 case. įinally, in section 5 we apply an extended Nikiforov-Uvarov method to calculate the eigen-energies E n, ℓ for the radial Schrödinger equation ( 5.39) with potential V P. As such, there appears to be a critical screening parameter for which bound states cease to exist at some point this is what is seen with the Yukawa potential. In particular, for certain values of ℓ and C we see that an effective potential which contains V P can become repulsive. We discuss the ℓ = 0 and ℓ ≠ 0 bound states, where ℓ denotes the azimuthal quantum number. However, these bounds (including the famous Bargmann bound ) do not apply with V P as these require the finiteness of a certain integral, which for V P does not converge. ![]() These bounds can be translated into conditions on the screening parameter this is indeed the case for the Yukawa potential. For a given radial potential, there are quite a few upper and lower bounds on the number of bound states see for a review of some such bounds. Next, in section 4, we move to the analysis of bound states. Additionally, by using a small argument approximation for the modified Bessel function of the second kind, we obtain an estimate for the ground state energy assuming certain parameters are sufficiently small see ( 3.30). However, using a sharp estimate for the modified Bessel function from, we can provide an upper bound for the ground state energy see ( 3.24). It does not appear to be possible to minimize the Hamiltonian directly as it involves a modified Bessel function of the second kind see ( 3.22). In section 3, we appeal to the Hypervirial theorem and the variational principle to obtain an upper bound for the ground state energy of a system with potential V P. It turns out to be necessary to calculate coefficients of this series for negative values of j this is related to the so-called Kramers-Pasternack inverse relation. This method is based on expanding the expectation values 〈 r j〉 for any integer j into power series in the parameter C. More precisely, in section 2, we use the Hypervirial and Hellmann-Feynman theorems to calculate the energy levels for a Hydrogen atom with potential V P. Such problems continue to be of interest, especially if explicit or almost explicit solutions can be obtained. The purpose of this paper is to further study properties of solutions, including the number of bound states, of the Schrödinger equation with potential given by ( 1.1). Essentially, the applicability of this method reduces to saying that the potential should preserve the domain of a particular Hamiltonian (see, theorem 2.2). As further motivation for a different screened potential, it was seen in that the classical Runge-Gross method does not apply with the Yukawa potential, but will apply for V P. In particular, all derivatives of V P have a removable discontinuity at the origin, but the derivatives of the Yukawa potential are not bounded. There are some subtle differences between the potential V P given by ( 1.1) and the Yukawa potential, especially related to their behavior at the origin. ![]() The dilatation analytic potentials also contain the classical Coulomb potential as well as the Yukawa potential V Y. This is particularly useful in understanding so-called quasi-static energies and potential energy surfaces. This means, in particular, that the spectrum of the associated Hamiltonian can be completely characterized. Figure 1. Plot of V p( r) given by ( 1.1) for various values of the parameter C.īasic properties of this potential are discussed in, but let us mention that the potential belongs to the class of so-called dilatation analytic potentials.
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