We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation in...We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.展开更多
We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And t...We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function(CHF),we find two linearly dependent solutions corresponding to the same eigenstate,from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant.After that,we are able to localize the positions of the eigenvalues on the real axis or on the complex plane whenτis a real number,a pure imaginary number,and a complex number,respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+n,n=0,1,2….The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple.The features of the angular probability distribution(APD)and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed.We find that for a real numberτ,the eigenvalue is a real number and the eigenfunction is a real function,and the eigenfunction system is an orthogonal complete system,and the APD is asymmetric in the northern and southern hemispheres.For a pure imaginary numberτ,the eigenvalue is still a real number and the eigenfunction is a complex function,but the APD is symmetric in the northern and southern hemispheres.Whenτis a complex number,the eigenvalue is a complex number,the eigenfunction is still a complex function,and the APD in the northern and southern hemispheres is also asymmetric.Finally,an approximate expression of complex eigenvalues is obtained when n is greater than∣s∣.展开更多
The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studyi...The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential.We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor(1-x^2)^(-p-1) as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction.The calculations are obtained systematically using some properties of the generalized hypergeometric series.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11975196)partially by SIP,Instituto Politecnico Nacional(IPN),Mexico(Grant No.20210414)。
文摘We propose a new scheme to study the exact solutions of a class of hyperbolic potential well.We first apply different forms of function transformation and variable substitution to transform the Schrodinger equation into a confluent Heun differential equation and then construct a Wronskian determinant by finding two linearly dependent solutions for the same eigenstate.And then in terms of the energy spectrum equation which is obtained from the Wronskian determinant,we are able to graphically decide the quantum number with respect to each eigenstate and the total number of bound states for a given potential well.Such a procedure allows us to calculate the eigenvalues for different quantum states via Maple and then substitute them into the wave function to obtain the expected analytical eigenfunction expressed by the confluent Heun function.The linearly dependent relation between two eigenfunctions is also studied.
基金supported by the National Natural Science Foundation of China(Grant No.11975196)partially by 20220355-SIP,IPN。
文摘We first convert the angular Teukolsky equation under the special condition ofτ≠0,s≠0,m=0 into a confluent Heun differential equation(CHDE)by taking different function transformation and variable substitution.And then according to the characteristics of both CHDE and its analytical solution expressed by a confluent Heun function(CHF),we find two linearly dependent solutions corresponding to the same eigenstate,from which we obtain a precise energy spectrum equation by constructing a Wronskian determinant.After that,we are able to localize the positions of the eigenvalues on the real axis or on the complex plane whenτis a real number,a pure imaginary number,and a complex number,respectively and we notice that the relation between the quantum number l and the spin weight quantum number s satisfies the relation l=∣s∣+n,n=0,1,2….The exact eigenvalues and the corresponding normalized eigenfunctions given by the CHF are obtained with the aid of Maple.The features of the angular probability distribution(APD)and the linearly dependent characteristics of two eigenfunctions corresponding to the same eigenstate are discussed.We find that for a real numberτ,the eigenvalue is a real number and the eigenfunction is a real function,and the eigenfunction system is an orthogonal complete system,and the APD is asymmetric in the northern and southern hemispheres.For a pure imaginary numberτ,the eigenvalue is still a real number and the eigenfunction is a complex function,but the APD is symmetric in the northern and southern hemispheres.Whenτis a complex number,the eigenvalue is a complex number,the eigenfunction is still a complex function,and the APD in the northern and southern hemispheres is also asymmetric.Finally,an approximate expression of complex eigenvalues is obtained when n is greater than∣s∣.
基金Supported by the National Natural Science Foundation of China under Grant No.11275165Partially by 20160978-SIP-IPN,Mexico
文摘The associated Legendre polynomials play an important role in the central fields,but in the case of′the non-central field we have to introduce the universal associated Legendre polynomials P^m'l_′(x) when studying the modified Pschl-Teller potential and the single ring-shaped potential.We present the evaluations of the integrals involving the universal associated Legendre polynomials and the factor(1-x^2)^(-p-1) as well as some important byproducts of this integral which are useful in deriving the matrix elements in spin-orbit interaction.The calculations are obtained systematically using some properties of the generalized hypergeometric series.
