infinite square potential well,stationary state,Schr?dinger equation,eigenvalues problem,standing wave ,"/> 无限深方势阱本征值和本征态的三种求解方法

大学物理 ›› 2022, Vol. 41 ›› Issue (2): 26-.doi: 10.16854 / j.cnki.1000-0712.210230

• 教学讨论 • 上一篇    下一篇

无限深方势阱本征值和本征态的三种求解方法

李海凤,陈康康   

  1. 西安工业大学 基础学院物理系,陕西 西安 710021
  • 收稿日期:2021-05-08 修回日期:2021-09-08 出版日期:2022-02-26 发布日期:2022-03-03
  • 作者简介:李海凤( 1986—),内蒙古呼伦贝尔市人,西安工业大学基础学院物理系讲师,博士,主要从事理论物理研究和教学工作.     
  • 基金资助:
    国家自然科学基金( 21703166);陕西省教育厅自然科学专项( 17JK0374) 资助

Three kinds of methods for solving eigenvalues and eigenfunctions for the infinite square potential well

LI Hai⁃feng,CHEN Kang⁃kang   

  1. School of Science,Xi'an Technological University,Xi'an,Shaanxi 710021,China
  • Received:2021-05-08 Revised:2021-09-08 Online:2022-02-26 Published:2022-03-03

摘要: 一维无限深方势阱模型是量子力学理想模型,经典教材中势阱的边界一般取得比较特殊.或关于坐标原点具有对称性,或势阱左边界位于坐标原点.本文首先展示了如何利用3种方法求解一维任意边界无限深方势阱能量本征值和对应的本征态,不同方法得到的结果彼此之间等价,讨论分析了这3种方法的推导结果,然后得到关于一维任意边界无限深方势阱能量本征值和本征态的通式,从中比较容易看出这两个物理量均与阱宽有关,并且本征波函数与边界值有关,最后将一维结果拓展到二维和三维任意边界无限深方势阱情况.


关键词: 无限深方势阱, 定态薛定谔方程, 本征值问题, 驻波

Abstract:

Moving particle in the one-dimensional infinite square potential well is a quantum ideal model.In the traditional textbooks,the boundary condition of the potential well is particularly taken.Either it is symmetric about the coordinate origin point,or the left boundary value of the well is at the origin of the coordinate.First,it is presented how to solve the energy eigenvalues and eigenfunctions for this model with the arbitrary boundary condition by virtue of the three kinds of approaches.Moreover,the results obtained by the different methods are equivalent to each other.Then the derived results are discussed and analyzed.Furthermore,the general formulas for solving the energy eigenvalues and eigenstates for the one-dimensional infinite square potential well with a arbitrary boundary are acquired.It is easy to see that the two physical quantities are dependent on the well width,and the eigenfunctions are related to the boundary value.Finally,the one-dimensional results are extended to two and threedimensional infinite square potential wells with the arbitrary boundaries.

Key words: infinite square potential well')"> infinite square potential well, stationary state')">stationary state, Schr?dinger equation')">Schr?dinger equation, eigenvalues problem')">eigenvalues problem, standing wave ')">standing wave

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  • 引用本文

    李海凤, 陈康康. 无限深方势阱本征值和本征态的三种求解方法[J]. 大学物理, 2022, 41(2): 26-.

    LI Hai⁃feng, CHEN Kang⁃kang. Three kinds of methods for solving eigenvalues and eigenfunctions for the infinite square potential well [J]. College Physics, 2022, 41(2): 26-.