关于半壁无限深势阱束缚态存在条件的讨论

DISCUSSION ON THE EXISTENCE CONDITION OF BOUND STATES IN HALF-WALL INFINITE POTENTIAL WELL

一维无限深势阱问题是量子力学中的经典问题,也是国内外量子力学初学者必学内容之一。一维半壁无限深势阱作为一维无限深势阱的变形,也是量子力学教学中常见的物理模型。然而关于半壁无限深势阱这一模型,许多教学和科研工作者甚至教材之中,都存在不严谨之处;尤其是对于束缚态存在条件的求解,许多教材与习题册没有对临界条件进行讨论验证。本文借助图解法对半壁无限深势阱束缚态存在条件进行了讨论并给出了严格解,纠正了长久以来人们对该问题的错误认识,对量子力学领域本科教学与实际科研工作有一定帮助。

One-dimensional infinite potential well is a classical problem in quantum mechanics,and it is one of the compulsory contents for beginners in quantum mechanics as well.As the deformation of the infinite potential well,half-wall infinite potential well is a common physical model in the teaching of quantum mechanics.However,much content of half-wall infinite potential well is not rigorous enough in teaching and research.Especially for the existence condition of bound states,many textbooks do not discuss and verify the critical conditions.In this paper,we discuss the existence condition of bound states in half-wall infinite potential well with graphic method and give a rigorous solution which is different from the textbooks.It corrects the long-standing cognitive bias of this problem,and is of great help to undergraduate teaching and practical scientific research in the field of quantum mechanics.