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Progress of Theoretical Physics

On Consistency between Lagrange and Hamilton Formalisms in Quantum Mechanics

Reiji Sugano

Published Jul 1, 1971• 19 citations
Publisher

Abstract

Consistency between the Lagrangean and the Hamiltonian formalisms in the quantum mechanics is investigatedc for the type of Lagrangean L=½\dotqigij(q)\dotqj-v(q) as an extension of a previous paper. The variations δqi and δ\dotqi should be considered as q-numbers. When the Lagrangean can be transformed into the standard form L=½\dotQα2-V(Q), the commutation relations of δqi and δ\dotqi with qj and \dotqj are found with the help of the Q-coordinate system. It is shown that using the commutation relations, the variation principle leads to the same equation of motion as the canonical equation of motion which is obtained in the previous paper.

DOI
10.1143/ptp.46.297

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Consistency between the Lagrangean and the Hamiltonian formalisms in the quantum mechanics is investigatedc for the type of Lagrangean L=½\dotqigij(q)\dotqj-v(q) as an extension of a previous paper. The variations δqi and δ\dotqi should be considered as q-numbers. When the Lagrangean can be transformed into the standard form L=½\dotQα2-V(Q), the commutation relations of δqi and δ\dotqi with qj and \dotqj are found with the help of the Q-coordinate system. It is shown that using the commutation relations, the variation principle leads to the same equation of motion as the canonical equation of motion which is obtained in the previous paper.
Since the ordinary c-number variation principle does not lead, for a non-linear Lagrangian, to the result consistent between the Lagrangian and Hamiltonian formalisms, Schwinger's variation principle is reformulated for a type of Lagrangian L=½\dotqigij(q)\dotqj-u(q) by means of a q-number variation. The canonical momentum, the Hamiltonian, the canonical commutation relations and equation of motion are derived. Also the Euler-Lagrange equation is obtained, which is consistent with the canonical equation of motion. These consequences are exactly the same as those of previous papers, but different from the ordinary ones in the Euler-Lagrange equation and the Hamiltonian.
It is shown that the variations δqj and δQi cannot both be simultaneously zero at the limits of integration in application of Hamilton’s principle to the development of the theory of canonical transformations. This means that δF1 does not vanish. However, the derivation requires only that the δqj be zero at the end points. This lack of symmetry between the qj and Qi in the formulaion is noted with respect to the Lagrangian and the fundamental commutator relations of quantum mechanics. Further, what is meant by an independent variation of pj and qj (or Qi and Pi) in the formalism is discussed in the use of Hamilton’s principle in the derivation of Hamilton’s equations of motion.
A previous q-number variational method is extended to be applicable to any quantum system for which the classical Lagrangian is given by Lc=½gij\dotqi\dotqj+ui\dotqi-v. The q-number variation is necessary for the formulation to be form-invariant under a general space-time transformation. From the action principle, not only the Euler-Lagrange equation but also the commutation relations up to a constant factor are obtained. Futhermore the form of the quantum Lagrangian is, to some extent, decided in order for solutions to exist for the action principle. It is shown that the first Noether theorem holds and the quantization is consistent with the canonical formalism.
In a previous article in this series [E. M. Corson, J. Math. Phys. 4, 42(1963); herein referred to as I], we showed that a new variational principle, of Lagrangian structure, includes Hamilton's principle as a unique derived consequent and encompasses both the classical and quantal definitions of ``state'' and the corresponding Lagrangian and Schrödinger equations of motion, respectively. In this theory, the essential difference between classical and quantal domains was shown to arise solely from the superadded postulates of determinism vs indeterminism; other than this, the two basic postulates may be reduced to (a) Newton's equations and (b) the new variation principle. It was shown that in the quantal domain, Schrödinger's equations obtain for the transition amplitudes (propagators) between pure eigenstates of homologous variables in configuration space, at different times, thus clearly implying all the results of conventional Hamiltonian-operator quantum formalism. In the present article we show that our variational formulation, under the familiar classical canonical transformation to momentum space, also automatically leads to the corresponding Schrödinger equations for the transition amplitudes between pure eigenstates of homologous variables in momentum space, at different times, the unitarity of the formalism and the reciprocal Fourier integral relations between coordinate and momentum representations. Thus we bridge a long-standing gap in theory and derive, uniquely and apparently for the first time, the basic quantum commutators (P.B.'s) which are postulated in the more familiar Hamiltonian theory.