Notes on the quantum invariant and on the relation between it and quantum phase

Based on the construction theorem of quantum invariant, the results of Lewis and Riesenfeld are rededuced in a much simpler way and extended significantly. Meanwhile, it is pointed out that the Lewis-Riesenfeld phases are in general not of physical meaning unless the invariants take specific forms which have been worked out. As an example, the spin system in a magnetic field is discussed in detail.