# 量子力学中升降算符的探讨论文

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

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

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