PROBABILITY DISTRIBUTIONS OF ENERGY OF ONE-DIMENSIONAL PARTICLES OF TWO KINDS IN THE ENERGY EIGENSTATES

Starting from Voe Neumann's axiomatic system of quantum mechanics, this paper calculates the probability distributions of the kinetic and potential energy of the one-dimensional harmonic oscillator and the particle in one-dimonsional infinitely deep square potential well. The results are compared with the classical cases, and it is shown that their relation conforms to Bohr's correspondence principle.Then we put forward a new viewpoint:it seems that the concept of the total energy of the particle in quantum mechanics should be deliberated again; the two basic hypotheses in quantum mechanics--"Bohr's probabilistic interpretation" and "the postulate that the particle's energy eguals the eigenvalue in the energy eigenstate" ars not logically harmonious. At the same time, the limitation of von Neumann's axiomatic system is pointed out.     

对冯·诺意曼“量子概率论”的两点质疑

本文对于公认的量子力学数学基础—冯·诺意曼概率论提出了两点质疑。从物理上讲,它将导致“粒子的势能以一定的概率超过总能量”这种违背能量守恒原理的结论。从数学上讲,它在某些方面还不如玻恩的概率解释(即常被物理学家称为“经典概率论”的数学意义的概率论)。因此,量子力学的数学基础似乎需要进一步发展和完善。     

PROBABILITY DISTRIBUTIONS OF ENERGY OF ONE-DIMENSIONAL PARTICLES OF TWO KINDS IN THE ENERGY EIGENSTATES

Starting from Voe Neumann's axiomatic system of quantum mechanics, this paper calculates the probability distributions of the kinetic and potential energy of the one-dimensional harmonic oscillator and the particle in one-dimensional infinitely deep square potential well. The results are compared with the classical cases, and it is shown that their relation conforms to Bohr's correspondence principle. Then we put forward a new viewpoint: it seems that the concept of the total energy of the particle in quantum mechanics should be deliberated again; the two basic hypotheses in quantum mechanics-"Bohr's probabilistic interpretation" and "the postulate that the particle's energy eguals the eigenvalue in the energy eigenstate" ars not logically harmonious. At the sametime, the limitation of von Neumann's axiomatic system is pointed out.