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American Journal of Physics

Quantum mechanical momentum operators and Hamilton’s variational principle

Kenneth R. Brownstein

Published Jul 1, 1976• 10 citations
Publisher

Abstract

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.

DOI
10.1119/1.10331

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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.
The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.
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.
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.
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 path-integral formulation of quantum mechanics is investigated which is closely related to that of Feynman. It differs from Feynman's formulation in that it involves the Hamiltonian function of the canonically conjugate coordinates and momenta. The classical limit yields the variational principle: $\ensuremath{\delta}ff(p\ifmmode\ddagger\else\textdaggerdbl\fi{}\ensuremath{-}N)dt=0$. A path-integral formula is also obtained for the energy eigenstate projection operator associated with the time-independent Schr\"odinger equation. The classical limit of the projection operator formula yields a modified form of the well-known variational principle for the phase-space orbit of given energy. Relativistically covariant Hamiltonian variational principles are analyzed and lead naturally to a relativistic scalar wave equation which involves a proper time variable which is canonically conjugate to the mass in the same manner as the ordinary time variable is conjugate to the energy in nonrelativistic quantum theory.
Abstract The transition from classical mechanics to quantum mechanics has been formulated in various ways. But all these ways have the common feature that they use as starting point mechanics not in its Newtonian form, but in the form which it was given by Lagrange and Hamilton. The essence of this form consists in the fact that the differential equations are represented as Euler equations of a variational principle for v functions, the generalized coordinates, of one variable, the time. In its purely mathematical aspect the quantization method may, therefore, be regarded as a mode of taking certain notions and certain relations arising from a variational principle for several dependent and one independent variable and of giving them a new meaning.