【作者】 彭妮；

【导师】 刘全慧；

【作者基本信息】 湖南大学 ， 理论物理， 2004， 硕士

【摘要】 1926年，量子力学建立。迄今为止，量子力学取得了巨大的成功，还没有发现有与其相违背的实验，但不是量子力学所有的基础性问题都已经很好的解决。 本文研究了量子力学的基本性问题中的两个具体问题。一个是海森堡对应原理在半空间谐振子中的应用的问题。另一个问题是约束体系的量子哈密顿中涉及到的算符次序问题。 在本文中，我们首先对于半空间谐振子给出了包括位置、动量算符及其平方的矩阵元，定态上的不确定关系等等量子力学的基本结果。由于位置矩阵元的结果较复杂，我们借助海森堡对应原理对半空间谐振子的位置矩阵元及其平方的矩阵元给出了很好的近似表达式。 然后，我们讨论了约束体系中的算符次序问题。对于非约束体系，量子力学动能表达式为，其中p_i为笛卡尔动量算符，这一结论与坐标的选取无关。但是，对于约束体系这一结论不再成立。当我们将二维椭球面嵌入三维平直空间后，就可以在三维直角坐标系中描述在这个二维椭球面上的运动。动能的正确形式为，其中p_i是厄密的动量算符，f_i(x，y，z)为坐标x，y，z的非平凡函数。于是，我们在二维椭球约束体系中扁椭球和长椭球情况下得到了函数f_i(x，y，z)，给出了动能算符的明确形式，并讨论了相关问题。更多还原

【Abstract】 The quantum mechanics was founded in 1926. Since then, it has achieved huge success and no experiment we have found is disobedient with it. However, not every fundamental question is studied enough and adequately.In this paper two concrete problems concerned fundamental questions in quantum mechanics are discussed. One is the use of the Heisenberg correspondence for the harmonic oscillator in half space. The other is the operator ordering problem in quantum harniton of constrained systems.Firstly, the elementary quantum-mechanical results for the harmonic oscillator in half space are carried out. These results include expectation values for position, momentum and their square, the uncertainty relation in the eigenstates, etc. Since the result of expectation value for position of the harmonic oscillator in half space is quite complicated, the Heisenberg correspondence principle is used to give the approximate expressions for position and its square of the harmonic oscillator in half space, and the expressions prove to be very accurate by numerical calculations.Secondly, the operator ordering problem in quantum hamiton of constrained systems is discussed. For an unconstrained system, the quantum kinetic energy operator can be written in terms of where pi are Cartesian momentum, that isirrespective of the choosing in coordinate. But, the same result cannot be applied to the constrained system. Since the motion on an ellipsoid surface is representable in 3-dimensional Cartesian coordinate, the quantum kinetic operator turns to be where Cartesian momentum Pi are hermitian operators and functions fi(x,y,z) are now nontrivial in quantum mechanics. So we have the fractions fi(x,y,z) and the specific form of the quantum kinetic operator onthe oblate ellipsoid surface and the prolate ellipsoid surface, and we also discuss the interrelated problem.更多还原