dorsal/arxiv
View SchemaFrom Classical to Quantum Mechanics: "How to translate physical ideas into mathematical language"
| Authors | H. Bergeron |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0102084 |
| URL | https://arxiv.org/abs/quant-ph/0102084 |
| DOI | 10.1063/1.1386410 |
Abstract
In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the question of a "true quantization" with "h" must be seen as an independent problem not directly related with quantum formalism. Moreover, this non-standard formulation of Classical Mechanics exhibits a new kind of operation with no classical counterpart: this operation is related to the "quantization process", and we show why quantization physically depends on group theory (Galileo group). This analytical procedure of quantization replaces the "correspondence principle" (or canonical quantization) and allows to map Classical Mechanics into Quantum Mechanics, giving all operators of Quantum Mechanics and Schrodinger equation. Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We find also that this approach gives a natural semi-classical formalism: some exact quantum results are obtained only using classical-like formula. So this procedure has the nice property of enlightening in a more comprehensible way both logical and analytical connection between classical and quantum pictures.
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"abstract": "In this paper, we investigate the connection between Classical and Quantum\nMechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics\n(a system is described by a state in a Hilbert space, observables are\nself-adjoint operators and so on) - Quantum Mechanics properly that specifies\nthe Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that\nGeneral Quantum Axiomatics (up to a supplementary \"axiom of classicity\") can be\nused as a non-standard mathematical ground to formulate all the ideas and\nequations of ordinary Classical Statistical Mechanics. So the question of a\n\"true quantization\" with \"h\" must be seen as an independent problem not\ndirectly related with quantum formalism. Moreover, this non-standard\nformulation of Classical Mechanics exhibits a new kind of operation with no\nclassical counterpart: this operation is related to the \"quantization process\",\nand we show why quantization physically depends on group theory (Galileo\ngroup). This analytical procedure of quantization replaces the \"correspondence\nprinciple\" (or canonical quantization) and allows to map Classical Mechanics\ninto Quantum Mechanics, giving all operators of Quantum Mechanics and\nSchrodinger equation. Moreover spins for particles are naturally generated,\nincluding an approximation of their interaction with magnetic fields. We find\nalso that this approach gives a natural semi-classical formalism: some exact\nquantum results are obtained only using classical-like formula. So this\nprocedure has the nice property of enlightening in a more comprehensible way\nboth logical and analytical connection between classical and quantum pictures.",
"arxiv_id": "quant-ph/0102084",
"authors": [
"H. Bergeron"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.1386410",
"title": "From Classical to Quantum Mechanics: \"How to translate physical ideas into mathematical language\"",
"url": "https://arxiv.org/abs/quant-ph/0102084"
},
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