A NEW VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS
Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. It is shown that in a q-number theory such a limitation on the number of variations is necessary, since a completely arbitrary q-number variation implies an infinite number of conditions to be satisfied. This implication arises because, in a q-number theory, the system variables are matrices which, in most cases, have an infinite number of elements, each of which is independently varied in a completely arbitrary q-number variation. A review of Schwinger's quantum field theory formalism shows that the limitation to c- number variations, while accomplishing the desired limitation on the number of independent variations, is not representation-invariant, i.e., under a change of representation, c-number variations applied to the original system variables transform to q-number variations of the new system variables. In the Hamiltonian formulation of classical mechanics, invariant infinitesimal canonical transformations are generated by constants of the motion. It is also possible to generate infinitesimal invariant transformations from constants of the motion in the classical Lagrangian formulation. This latter possibility is exploited to obtain a q-number theory. It is demanded that the variations of the system variables in the action principle be the invariant infinitesimal transformations generated by constants of the motion. This requirement accomplishes the desired limitation of the number of independent variations in a representation-invariant manner. The resulting theory is developed in detail for a certain class of Lagrangian functions. A set of commutation relations is derived and a dynamical principle for determining the temporal development of the system is postulated. A discussion of the results, including consideration of the self-consistency of the formalism developed, is presented. Finally, a specific problem is worked through and it is shown that the results obtained correspond with those of the canonical method of quantization.
- Research Organization:
- Brooklyn Polytechnic Inst.
- NSA Number:
- NSA-16-028326
- OSTI ID:
- 4783183
- Journal Information:
- Dissertation Abstr., Vol. Vol: 22; Other Information: Orig. Receipt Date: 31-DEC-62
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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