量子力学中升降算符的探讨论文 摘 要：升降算符这一概念在量子力学中有非常广泛的应用，因为在量子力学中研究力学量算符的本征值问题时，利用升降算符可以使计算更加简便，物理图景更加显著。升降算符分为谐振子哈密顿量、坐标与动量、磁量子数和角动量量子数升降算符。本文主要探究这四种升降算符，推导其相应的公式并对其性质作了阐述，同时计算出球谐函数的最低本征态，由最低本征态就可以推出任意的球谐函数，从而使求本征态的计算更加方便。对轨道角动量量子数算符中存在的问题，给出其完整形式，并进行了进一步的完善。
关键词：量子力学；升降算符；量子数
Discussion on the Raising and Lowering Operators in Quantum Mechanics
Abstract: In the quantum mechanics, the raising and lowering operators concept has a widely application. As researching the problem of mechanical quantity operator eigenvalue in quantum mechanics, the raising and lowering operators can make the calculation easier and physical picture more significant. The raising and lowering operators can be divided into four categories of operators: the harmonic oscillator Hamiltonian, the position and momentum, the magnetic quantum number and the angular momentum quantum number. This paper mainly discussed the four operators mentioned above, the corresponding formula is derived, the properties are discussed and the minimum state of spherical harmonic function is calculated. Starting from the lowest explore eigenstate we can calculate arbitrary spherical harmonic function, which makes the eigenstate calculation more convenient. According to the problem existed in the angular momentum quantum number operator, we give its complete form and made a further improvement.
Key Words: Quantum mechanics; Raising and lowering operators; Quantum number
目 录
摘要 1
引言 1
1.简谐振子的升降算符 2
1.1量子力学的哈密顿算符 2
2.坐标和动量的升降算符 4
2.1坐标和动量算符的讨论 4
3.磁量子数的升降算符 5
3.1角动量的定义 5
3.2磁量子数和轨道角动量量子数的关系 6
4.方向算符 7
4.1方向算符和轨道角动量关系 8
4.2方向算符对球谐函数的作用 8
5.轨道角动量升降算符 9
5.1轨道角动量升降算符的对易关系 9
5.3轨道角动量升降算符的讨论 11
5.4 、 、 和 在球谐函数下作用 12
5.5轨道角动量量子数的另外形式 14
6.总结 16
参考文献 16
致谢 17,3426