| 68 | 0 | 76 |
| 下载次数 | 被引频次 | 阅读次数 |
一维谐振子是量子力学中少数可以精确解析求解的模型之一，可以用于检验量子力学的理论与方法，具有丰富的内涵。利用傅里叶变换与厄米多项式的生成函数，实现了一维谐振子坐标表象中本征波函数到动量表象中本征波函数的变换，验证了两种表象中对一维谐振子描述的等价性与联系。利用代数解法分别得到一维谐振子在坐标表象与动量表象中本征能量与本征波函数，进一步讨论了两种表象的等价性。强调了物理学中“降维打击”思想，加深学生对量子力学的理解，激发学生的学习兴趣。
Abstract：The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of “solving physical problems in higher dimensional space” is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.
[1] 周世勋，陈灏，肖江.量子力学教程[M].3版.北京：高等教育出版社，2022.ZHOU S X,CHEN H,XIAO J.Introduction to quantum mechanics[M].3rd ed.Beijing:Higher Education Press,2022.(in Chinese)
[2] 钱伯初.量子力学[M].北京：高等教育出版社，2006.QIAN B C.Quantum mechanics[M].Beijing:Higher Education Press,2006.(in Chinese)
[3] 曾谨言.量子力学[M].5版.北京：科学出版社，2013.ZENG J Y.Quantum mechanics[M].5th ed.Beijing:Science Press,2013.(in Chinese)
[4] 蒋学华.求解谐振子能量本征值的几种方法[J].南阳师范学院学报(自然科学版),2002,1(6):33-35.JIANG X H.Several solutions to solving harmonic oscillator's energy eigenvalue[J].Journal of Nanyang Teachers College (Natural Science Edition),2002,1(6):33-35.(in Chinese)
[5] 王可，陶才德.一维线性谐振子量子束缚态能级的几种计算方法[J].绵阳师范学院学报，2005,24(5):57-61.WANG K,TAO C D.A discussion of the energy calculation means of linear harmonic oscillator[J].Journal of Mianyang Normal University,2005,24(5):57-61.(in Chinese)
[6] 韩东峰，康莉，孙孟乐.一维谐振子能量本征值两种解法探讨[J].洛阳工业高等专科学校学报，2005,15(2):21-22,51.HAN D F,KANG L,SUN M L.Two kinds of methods and discussion about eigenvalues of energy of one-dimension harmonic oscillator[J].Journal of Luoyang technology college,2005,15(2):21-22,51.(in Chinese)
[7] 姜荣.线性谐振子的算符理论[J].科技信息(科学教研),2008,25(9):257-259.JIANG R.The operator theory of linear resonance[J].Science & Technology Information,2008,25(9):257-259.(in Chinese)
[8] 刘海宽，张海丰，马佳.升降算符在一维谐振子能级讨论中的应用[J].商丘师范学院学报，2020,36(12):22-26.LIU H K,ZHANG H F,MA J.Application of the raising and lowering operators in the discussion of one-dimensional harmonic oscillator energy levels[J].Journal of Shangqiu Normal University,2020,36(12):22-26.(in Chinese)
[9] 王长荣.一维谐振子能级和波函数的代数解法[J].云南师范大学学报(自然科学版),2001,21(3):22-24.WANG C R.Algebraical solution to the energy level and wave function of one-dimensional harmonic oscillator[J].Journal of Yunnan Normal University,2001,21(3):22-24.(in Chinese)
[10] 刘晓丽，赵岩，戴宝印.一维谐振子能量本征值在各种表象中的求解[J].鞍山师范学院学报，2000,2(3):51-54.LIU X L,ZHAO Y,DAI B Y.The solutions of one-dimensional harmonic oscillator energy eigenvalues in different representation[J].Journal of Anshan Teachers College,2000,2(3):51-54.(in Chinese)
[11] 熊志华，周珏.三种表象中的一维谐振子性质研究[J].江西科技师范学院学报，2003,23(4):20-22.XIONG Z H,ZHOU J.A study of the nature of a harmonic oscillator with different representations[J].Journal of Jiangxi Science & Technology Teachers' College,2003,23(4):20-22.(in Chinese)
[12] 罗凌霄.一维谐振子的动量几率幅和等式型不确定关系[J].许昌学院学报，2004,23(5):21-24.LUO L X.The momentum probability amplitude and the equality form uncertain relation of one-dimensional harmonic oscillator[J].Journal of Xuchang University,2004,23(5):21-24.(in Chinese)
[13] 郑华，王新刚，朱励霖.物理学中的“降维打击”[J].物理与工程，2023,33(3):46-48,63.ZHENG H,WANG X G,ZHU L L.Solving physical problems in higher dimensional space[J].Physics and Engineering,2023,33(3):46-48,63.(in Chinese)
[14] 倪致祥.厄米本征值问题的探究[J].大学物理，2008,27(2):39-41.NI Z X.Explorating for the Hermite eigenvalue problem[J].College Physics,2008,27(2):39-41.(in Chinese)
[15] 胡先权，王帮美.也谈厄米多项式的递推关系[J].大学物理，2009,28(6):22-23.HU X Q,WANG B M.Discussing on the recurrence relation of the Hermite polynomial also[J].College Physics,2009,28(6):22-23.(in Chinese)
[16] 吴崇试.厄米多项式的定义及其他[J].大学物理，2010,29(3):14-15.WU C S.Two kinds of the Hermite polynomials[J].College Physics,2010,29(3):14-15.(in Chinese)
[17] 王蒙，陶俊琦，程剑剑，等.物理学中常用的高斯与类高斯型积分[J].大学物理，2021,40(5):13-19.WANG M,TAO J Q,CHENG J J,et al.Gaussian and Gaussian-like integrals in physics[J].College Physics,2021,40(5):13-19.(in Chinese)
基本信息:
DOI：
中图分类号:O413.1
引用信息:
[1]纪怀昱,张子骏,郑华,等.一维谐振子坐标表象与动量表象的等价性[J].物理与工程,2025,35(03):14-19.
基金信息:
国家自然科学基金(11905120)