dorsal/arxiv
View SchemaPath integrals from classical momentum paths
| Authors | John Hegseth |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0403005 |
| URL | https://arxiv.org/abs/quant-ph/0403005 |
Abstract
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.
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"abstract": "The path integral formulation of quantum mechanics constructs the propagator\nby evaluating the action S for all classical paths in coordinate space. A\ncorresponding momentum path integral may also be defined through Fourier\ntransforms in the endpoints. Although these momentum path integrals are\nespecially simple for several special cases, no one has, to my knowledge, ever\nformally constructed them from all classical paths in momentum space. I show\nthat this is possible because there exists another classical mechanics based on\nan alternate classical action R. Hamilton\u0027s Canonical equations result from a\nvariational principle in both S and R. S uses fixed beginning and ending\nspatial points while R uses fixed beginning and ending momentum points. This\nalternative action\u0027s classical mechanics also includes a Hamilton-Jacobi\nequation. I also present some important points concerning the beginning and\nending conditions on the action necessary to apply a Canonical transformation.\nThese properties explain the failure of the Canonical transformation in the\nphase space path integral. It follows that a path integral may be constructed\nfrom classical position paths using S in the coordinate representation or from\nclassical momentum paths using R in the momentum representation. Several\nexample calculations are presented that illustrate the simplifications and\npractical advantages made possible by this broader view of the path integral.\nIn particular, the normalized amplitude for a free particle is found without\nusing the Schrodinger equation, the internal spin degree of freedom is simply\nand naturally derived, and the simple harmonic oscillator is calculated.",
"arxiv_id": "quant-ph/0403005",
"authors": [
"John Hegseth"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP",
"physics.class-ph"
],
"title": "Path integrals from classical momentum paths",
"url": "https://arxiv.org/abs/quant-ph/0403005"
},
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