Quantum Mechanics And Classical Correspondence

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.