Prog. Theor. Phys. Vol. 48 No. 6 (1972) pp. 2082-2092

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Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. I

— One Dimensional Case —

Toshiharu Kawai

Department of Physics, Dalhousie University, Halifax, Nova Scotia, Canada

(Received March 10, 1972)

Abstract:

Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian L_c(\dotx, x) = (M/2) f^-1(x)\dotx² - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.

URL : http://ptp.ipap.jp/link?PTP/48/2082/
DOI : 10.1143/PTP.48.2082

References:

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Citing Article(s) :

Progress of Theoretical Physics Vol. 49 No. 4 (1973) pp. 1352-1361 :
Schwinger's Variation Principle by Means of Q-Number Variation for Non-Linear Lagrangian

Reiji Sugano
Progress of Theoretical Physics Vol. 50 No. 1 (1973) pp. 277-289 :
Quantization for Non-Linear Lagrangian

Teruya Ohtani
Progress of Theoretical Physics Vol. 50 No. 2 (1973) pp. 680-690 :
Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. III

Hideki Kamo and Toshiharu Kawai
Progress of Theoretical Physics Vol. 51 No. 2 (1974) pp. 592-599 :
A Note on Covariant Derivatives in the Quantum Field Theory

Takao Okabayashi
Progress of Theoretical Physics Vol. 52 No. 5 (1974) pp. 1687-1701 :
A Quantum-Theoretical Lagrangian Formalism for Quasi-Linear Field Theories. I

Takao Okabayashi and Hiroyuki Kikugawa