一维有限深方势阱能量本征方程的数值解与近似解析解
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摘要
一维有限深方势阱的束缚态本征值问题无法精确求解.数值计算能量满足的超越方程,可以给出能谱和能量本征波函数.本文基于超越方程的泰勒级数展开,给出了一级近似下能谱及其波函数的解析解,发现束缚态个数由一个无量纲参数R确定,该参数正比于势阱宽度乘以势阱高度开方.除了最高能级波函数,能谱及其波函数的近似解析解与数值结果吻合.大R极限下,发现近似解析解退化为可精确求解的无限深势阱情况.
The bound states of one-dimensional finite square well cannot be solved exactly. Numerically,the energy spectrum and its wave function can be obtained from a transcendental equation. In this paper,we solve the transcendental equation with the Taylor series expansion,which to the first order gives analytical solutions of the energy spectrum and its wave function. We find that the number of bound states is determined by a dimensionless parameter R,which is proportional to the width of the well multiplied by the square of the potential height. Except the highest level wave function,the approximate analytical results show in well agreement with the numerical results. In the limit of large R,the approximate analytical results recover the exactly solvable problem of an infinite square well.
The bound states of one-dimensional finite square well cannot be solved exactly. Numerically,the energy spectrum and its wave function can be obtained from a transcendental equation. In this paper,we solve the transcendental equation with the Taylor series expansion,which to the first order gives analytical solutions of the energy spectrum and its wave function. We find that the number of bound states is determined by a dimensionless parameter R,which is proportional to the width of the well multiplied by the square of the potential height. Except the highest level wave function,the approximate analytical results show in well agreement with the numerical results. In the limit of large R,the approximate analytical results recover the exactly solvable problem of an infinite square well.
引文
[1]曾谨言.量子力学教程[M].3版.北京:科学出版社,2014:33-35.
[2]David J Griffiths.Introduction to Quantum Mechanics[M].Beijing:China Machine Press,2006:78-80.
[3]Barker B I,Rayborn G H,Ioup J W,et al.Approximating the finite square well with an infinite well:Energies and Eigenfunctions[J].Am J Phys,1991,59:1039-1041.
[4]王再军.用基函数展开法求解一维有限深方势阱模型[J].大学物理,2016,35(08):39-43.
[5]Venugopalan A,Agarwal G S.Superrevivals in the quantum dynamics of a particle confined in a finite squarewell potential[J].Phys Rev A 1999,59:1415-1421
[6]张涵,刘志伟,任政学,等.基于实稳定方法求解单粒子共振态的Wigner函数[J].大学物理,2018,37(01):17-24.
[2]David J Griffiths.Introduction to Quantum Mechanics[M].Beijing:China Machine Press,2006:78-80.
[3]Barker B I,Rayborn G H,Ioup J W,et al.Approximating the finite square well with an infinite well:Energies and Eigenfunctions[J].Am J Phys,1991,59:1039-1041.
[4]王再军.用基函数展开法求解一维有限深方势阱模型[J].大学物理,2016,35(08):39-43.
[5]Venugopalan A,Agarwal G S.Superrevivals in the quantum dynamics of a particle confined in a finite squarewell potential[J].Phys Rev A 1999,59:1415-1421
[6]张涵,刘志伟,任政学,等.基于实稳定方法求解单粒子共振态的Wigner函数[J].大学物理,2018,37(01):17-24.