Quantum mechanical momentum operators and Hamilton’s variational principle

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The Euler–Lagrange equations in classical field theory are not covariant with respect to a transformation of independent variables. A modified covariant version of these equations is presented. The modification consists of replacing the partial derivative ∂/∂qi by the Hermitian quantum mechanical momentum operator pi conjugate to the coordinate qi. This momentum operator is given by pi=−ih/[∂/∂qi+(1/2) ∂/∂qi(lnJ)], where J is the Jacobian of the transformation.