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 quant...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.展开更多
In this paper, we present several expansions of the symbolic operator （1 ＋E）^x. Moreover, we derive some series transforms formulas and the Newton generating functions of {f（k）}.
基金supported in part by the Special Funds for Theoretical Physics Research Program of the NSFC under Grant No.11947124NSFC under Grant Nos.11575125 and 11675119。
文摘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.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘In this paper, we present several expansions of the symbolic operator （1 ＋E）^x. Moreover, we derive some series transforms formulas and the Newton generating functions of {f（k）}.
