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Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. III
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1973
Year
Spectral TheoryQuantum DynamicHamiltonian TheoryEngineeringVelocity Dependent PotentialPhysicsVariational AnalysisPotential TheoryQuantum Mechanical PropertyVariational PrincipleLagrangian LcArbitrary Point TransformationHamiltonian SystemLagrangian MethodCalculus Of VariationVariational Inequalities
Schwinger’s variational principle is formulated for the multi-dimensional quantum system which corresponds to the classical system described by the Lagrangian Lc(\dotx,x)=(M/2)gij(x)\dotxi\dotxj-v(x). The c-number variations of coordinates and time are sufficient to give the laws of quantum mechanics. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relations are derived from this principle and there is no inconsistency. An appropriate choice of the Lagrangian operator is essential in our formulation. It is shown that an arbitrary point transformation is entitled to be called a canonical transformation. Ambiguities in the quantal Lagrangian are also discussed.