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Hamiltonian
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>由于力学量必须是可观测量，因此其本征值必须是实数的，从而相应的算符必须 >是厄密算符；反之，我们说，厄密算符可以用来表示量子力学中的力学量算符， >体系的哈密顿量只是其中的一种，为什么只是单单就指定哈密顿量呢？ >当然，可能把这种厄密算符只“处理”为哈密顿量而不失一般性，而且哈密顿量 >决定了体系的演化特征，地位非同一般，尽管将厄密算符只专门对应某个哈密顿 >量，这样做对于原来的数学问题不一定对应得最直接最简单。 ----------------------------------------------------------------------- I think H is the natural choice. first, we do not want anything that is classified by a compact symmetry group. in this case, everything will be determinded by group theory, such as angular momentum. then, we probably don't want anything depends on angle. a discrete angluar distribution is kind of strange since it seems to contain arbiturary partial waves. Therefore, things like p1.p2 appear to be less likely. then, it appears that H is a natural choice, although others might be possible. >不过，sage大哥说离散角分布是个奇怪的东西，我觉得不奇怪啊。角动量在给定 >方向上的投影量子化，意味着角动量方向与给定方向夹角的离散化分布，这也是 >通常所说的所谓“空间量子化”。 -------------------------------------------------------------------------- Angular momentum quantization does not mean that a observable will be a discrete function of angle. Imagine we do some scattering experiments, the angular momentum is quantized because (roughly ) SO(3) is compact. However, the scattering amplitude is always a continuous function of angle. This has nothing to do with quantization of space-time which presumably have something to do with the quantum gravity. my point is that if you measure the position of anything, it is a continuous function of angle. a real discretization of space-time will mean that we have some wave-function which is a delta-function in angle. It will contain all the partial waves and will be hard to prepare as a pure state. the eigen-value of angular momentum is discrete. However, all the spherical harmonics are continous function of angle. Probability amplitude is just a representation of state |psi>. <n|psi> qualifies as a probability amplitude with a variable n which can be discrete. Fundamentally, we don't discretize x, therefore, <x|psi> will always be continuous. |