dorsal/arxiv
View SchemaSchrodinger revisited: an algebraic approach
| Authors | M. R. Brown, B. J. Hiley |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0005026 |
| URL | https://arxiv.org/abs/quant-ph/0005026 |
Abstract
Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the commutator of the Hamiltonian with the density operator, and an equation for the time development of the phase operator that involves the anti-commutator of the Hamiltonian with this density operator. We show that this latter equation plays two important roles: (i) it expresses the conservation of energy in a system where energy is well defined and (ii) it provides a simple way to evaluate the gauge changes that occur in the Aharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these operator (i.e. purely algebraic) equations also allow us to re-examine the Bohm interpretation, showing that it is in fact possible to construct Bohm interpretations in representations other than the x-representation. We discuss the meaning of the Bohm interpretation in the light of these new results in terms of non-commutative structures and this enables us to clarify its relation to standard quantum mechanics.
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"abstract": "Starting with the quantum Liouville equation, we write the density operator\nas the product of elements respectively in the left and right ideals of an\noperator algebra and find that the Schrodinger picture may be expressed through\ntwo representation independent algebraic forms in terms of the density and\nphase operators. These forms are respectively the continuity equation, which\ninvolves the commutator of the Hamiltonian with the density operator, and an\nequation for the time development of the phase operator that involves the\nanti-commutator of the Hamiltonian with this density operator. We show that\nthis latter equation plays two important roles: (i) it expresses the\nconservation of energy in a system where energy is well defined and (ii) it\nprovides a simple way to evaluate the gauge changes that occur in the\nAharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these\noperator (i.e. purely algebraic) equations also allow us to re-examine the Bohm\ninterpretation, showing that it is in fact possible to construct Bohm\ninterpretations in representations other than the x-representation. We discuss\nthe meaning of the Bohm interpretation in the light of these new results in\nterms of non-commutative structures and this enables us to clarify its relation\nto standard quantum mechanics.",
"arxiv_id": "quant-ph/0005026",
"authors": [
"M. R. Brown",
"B. J. Hiley"
],
"categories": [
"quant-ph"
],
"title": "Schrodinger revisited: an algebraic approach",
"url": "https://arxiv.org/abs/quant-ph/0005026"
},
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