From Lagrangian to Quantum Mechanics with Symmetries

Details From Lagrangian to Quantum Mechanics with Symmetries From Lagrangian to Quantum Mechanics with Symmetries

2012-02-14

arxiv.org/abs/1202.3150v1

Physics

Mathematical Physics

To appear in: Proceedings of Symmetries in Science XV, Journal of Physics: Conference Series, (2012)

Scientific paper

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.

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