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Annals of Physics

Constraints in quantum mechanics via lagrangians, Hamiltonians, and canonical equations of motion

Pierre Amiot, James J. Griffin

Published Dec 1, 1975• 5 citations
Publisher
DOI
10.1016/0003-4916(75)90059-7

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Progress of Theoretical Physics•Apr 1, 1973•1 citation
The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.
A quantum mechanical model with one bosonic degree of freedom is discussed in detail. Conventionally, when a quantum mechanical model is constructed, one must know the corresponding classical model. And by applying the correspondence between the classical Poisson brackets and the canonical commutator, the canonical quantization condition can be obtained. In the quantum model, study of the corresponding classical model is needed first. In this model, the Lagrangian is an operator gauge invariant. After localization, in order to keep gauge invariance, the operator gauge potential must be introduced. The Eular-Lagrange equation of motion of the dynamical argument gives the usual operator equation of motion. And the operator gauge potential just gjves a constraint. This constraint is just the usual canonical quantization condition.
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.