Communications in Theoretical Physics >
An indirect approach for quantum-mechanical eigenproblems: duality transforms
∗Author to whom any correspondence should be addressed.
Received date: 2021-12-20
Revised date: 2022-02-15
Accepted date: 2022-02-16
Online published: 2022-05-03
Supported by
National Natural Science Foundation of China(11575125)
National Natural Science Foundation of China(11675119)
National Natural Science Foundation of China(11947124)
Copyright
We suggest an indirect approach for solving eigenproblems in quantum mechanics. Unlike the usual method, this method is not a technique for solving differential equations. There exists a duality among potentials in quantum mechanics. The first example is the Newton–Hooke duality revealed by Newton in Principia. Potentials that are dual to each other form a duality family consisting of infinite numbers of family members. If one potential in a duality family is solved, the solutions of all other potentials in the family can be obtained by duality transforms. Instead of directly solving the eigenequation of a given potential, we turn to solve one of its dual potentials which is easier to solve. The solution of the given potential can then be obtained from the solution of this dual potential by a duality transform. The approach is as follows: first to construct the duality family of the given potential, then to find a dual potential which is easier to solve in the family and solve it, and finally to obtain the solution of the given potential by the duality transform. In this paper, as examples, we solve exact solutions for general polynomial potentials.
Key words: duality family; general polynomial potential; exact solution
Yu-Jie Chen , Shi-Lin Li , Wen-Du Li , Wu-Sheng Dai . An indirect approach for quantum-mechanical eigenproblems: duality transforms[J]. Communications in Theoretical Physics, 2022 , 74(5) : 055103 . DOI: 10.1088/1572-9494/ac5585
We are very indebted to Dr G Zeitrauman for his encouragement. This work is supported in part by the Special Funds for Theoretical Physics Research Program of the NSFC under Grant No. 11947124, and NSFC under Grant Nos. 11575125 and 11675119.
In this appendix, we provide an exact solution of the eigenproblem of the potential
The radial equation of the potential (
The choice of the boundary condition has been discussed in [
The regular solution is a solution satisfying the boundary condition at r = 0 [
The biconfluent Heun equation (
The biconfluent Heun function ${\rm{N}}\left(2l+1,0,\tfrac{-E}{\sqrt{\xi }},\tfrac{-2{\rm{i}}\mu }{{\xi }^{1/4}},z\right)$ has an expansion at z = 0 [
Only ${y}_{l}^{\left(1\right)}\left(z\right)$ satisfies the boundary condition for the regular solution at r = 0, so the radial eigenfunction reads
The irregular solution is a solution satisfying the boundary condition at r → ∞ [
The biconfluent Heun equation (
To construct the solution, we first express the regular solution (
The regular solution (
The boundary condition of bound states, ${\left.u\left(r\right)\right|}_{r\to \infty }\to 0$, requires that the coefficient of the second term must vanish since this term diverges when r → ∞ , i.e.,
Equation (
The eigenfunction, from equations (
In this appendix, we provide an exact solution of the eigenproblem of the potential
The radial equation reads
The choice of the boundary condition has been discussed in [
The regular solution is a solution satisfying the boundary condition at r = 0 [
The biconfluent Heun equation (
The biconfluent Heun function ${\rm{N}}\left(2l+1,\tfrac{{\rm{i}}\mu }{{\xi }^{3/4}},-\tfrac{E}{{\xi }^{1/2}}-\tfrac{{\mu }^{2}}{4{\xi }^{3/2}},0,z\right)$ has an expansion at z = 0 [
Only ${y}_{l}^{\left(1\right)}\left(z\right)$ satisfies the boundary condition for the regular solution at r = 0, so the radial eigenfunction reads
The irregular solution is a solution satisfying the boundary condition at r → ∞ [
The biconfluent Heun equation (
To construct the solution, we first express the regular solution (
The regular solution (
The boundary condition of bound states, ${\left.u\left(r\right)\right|}_{r\to \infty }\to 0$, requires that the coefficient of the second term must vanish since this term diverges when r → ∞ , i.e.,
Equation (
The eigenfunction, from equations (
In this appendix, we provide an exact solution of the eigenproblem of the potential
The radial equation reads
The choice of the boundary condition has been discussed in [
The regular solution is a solution satisfying the boundary condition at r = 0 [
The biconfluent Heun equation (
The biconfluent Heun function ${\rm{N}}\left(4l+2,\tfrac{{\rm{i}}\sqrt{2}\xi }{{\left(-E\right)}^{3/4}},-\tfrac{{\xi }^{2}}{2{\left(-E\right)}^{3/2}},\tfrac{-{\rm{i}}4\sqrt{2}\mu }{{\left(-E\right)}^{1/4}},z\right)$ has an expansion at z = 0 [
Only ${y}_{l}^{\left(1\right)}\left(z\right)$ satisfies the boundary condition for the regular solution at r = 0, so the radial eigenfunction reads
The irregular solution is a solution satisfying the boundary condition at r → ∞ [
The biconfluent Heun equation (
To construct the solution, we first express the regular solution (
The regular solution (
The boundary condition of bound states, ${\left.u\left(r\right)\right|}_{r\to \infty }\to 0$, requires that the coefficient of the second term must vanish since this term diverges when r → ∞ , i.e.,
Equation (
The eigenfunction, from equations (
For scattering states, E > 0, we introduce
The singularity of the S-matrix on the positive imaginary axis corresponds to the eigenvalues of bound states [
The scattering wave function can be constructed with the help of the S-matrix. The scattering wave function can be expressed as a linear combination of the radially ingoing wave ${u}_{\mathrm{in}}\left(r\right)$ and the radially outgoing wave ${u}_{\mathrm{out}}\left(r\right)$, which are conjugate to each other, i.e., [
In this appendix, we provide an exact solution of the eigenproblem of the potential
The radial equation reads
The choice of the boundary condition has been discussed in [
The regular solution is a solution satisfying the boundary condition at r = 0 [
The biconfluent Heun equation (
The biconfluent Heun function ${\rm{N}}\left(2l+1,\tfrac{{\rm{i}}\kappa }{{\xi }^{3/4}},-\tfrac{E}{{\xi }^{1/2}}-\tfrac{{\kappa }^{2}}{4{\xi }^{3/2}},\tfrac{-{\rm{i}}2\mu }{{\xi }^{1/4}},z\right)$ has an expansion at z = 0 [
Only ${y}_{l}^{\left(1\right)}\left(z\right)$ satisfies the boundary condition for the regular solution at r = 0, so the radial eigenfunction reads
The irregular solution is a solution satisfying the boundary condition at r → ∞ [
The biconfluent Heun equation (
To construct the solution, we first express the regular solution (
The regular solution (
The boundary condition of bound states, ${\left.u\left(r\right)\right|}_{r\to \infty }\to 0$, requires that the coefficient of the second term must vanish since this term diverges when r → ∞ , i.e.,
Equation (
The eigenfunction, from equations (
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