On Consistency between Lagrange and Hamilton Formalisms in Quantum Mechanics

Abstract

gean L = iiJ.igii(fi- v(q) was investigated*) where gii= gii is a function of qli = lr-vn). It was shown that in the quantum mechanics, the correct Hamiltonian should be**) H = t {pi, qi} - L-Z (q) with Z expressed in terms of gii and its derivative, and this H satisfied the canonical equation of motion. The equation of motion for qi, however, could not be derived from the ordinary variation principle due to the fact that oqi was a q-number. The proposed formalism was examined for some examples (the free Lagrangean for the polar coordinate system, etc.). If there exists the canonical transformation from qi to Oa (a= lr-vn) for which the Lagrangean has standard form L = !Qa 2 - V(O), the consistent equation of motion is derived from the equation for Qa. It was indicated in I that the Euler-Lagrange equation should be modified and that the ordinary variational method was not valid. In this paper, it is, however, proved that if oqi and oqi are appropriately regarded as q-numbcrs, the variation principle yields the consistent equation of