We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi ( $HJ$ ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the $HJ$ theory on the tangent bundle of a configuration manifold, the quantum $HJ$ theory, $HJ$ problems for general differential operators and the $HJ$ problem for Lie groups.We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi ($HJ$) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the $HJ$ theory on the tangent bundle of a configuration manifold, the quantum $HJ$ theory, $HJ$ problems for general differential operators and the $HJ$ problem for Lie groups.

arXiv:0907.0964The Journal of Geometric MechanicsJan 1, 200918 citations

Tunnelling Times in Quantum Mechanics

James Puleston

In this essay I survey recent developments in the theory of tunnelling times in quantum mechanics. The study of this concept is polarised depending on one’s interpretation of quantum mechanics, which require different approaches and yield starkly different results. I survey this topic through the lenses of the orthodox (Copenhagen) and de Broglie-Bohm (pilot wave) interpretations, presenting the requisite theory and definitions and expounding the literature of the original authors through proofs and figures. I do this in sections 2 and 3 respectively. Before embarking on tunnelling times as a topic, I survey the conceptual aspects of how time is treated in classical and quantum mechanics. This topic has many subtle aspects which have hindered progress in understanding tunnelling times in the past. In this essay I hope to demonstrate the success that has been achieved in recent years, but also that further theoretical and experimental work is required, if we are to arrive at a universally accepted definition of tunnelling times in quantum mechanics - if indeed such a definition exists.In this essay I survey recent developments in the theory of tunnelling times in quantum mechanics. The study of this concept is polarised depending on one’s interpretation of quantum mechanics, which require different approaches and yield starkly different results. I survey this topic through the lenses of the orthodox (Copenhagen) and de Broglie-Bohm (pilot wave) interpretations, presenting the requisite theory and definitions and expounding the literature of the original authors through proofs and figures. I do this in sections 2 and 3 respectively. Before embarking on tunnelling times as a topic, I survey the conceptual aspects of how time is treated in classical and quantum mechanics. This topic has many subtle aspects which have hindered progress in understanding tunnelling times in the past. In this essay I hope to demonstrate the success that has been achieved in recent years, but also that further theoretical and experimental work is required, if we are to arrive at a universally accepted definition of tunnelling times in quantum mechanics - if indeed such a definition exists.

May 14, 20200 citations

We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (

HJ

) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the

HJ

theory on the tangent bundle of a configuration manifold, the quantum

HJ

theory,

HJ

problems for general differential operators and the

HJ

problem for Lie groups.