dorsal/arxiv
View SchemaThe Uncertainty Relation in "Which-Way" Experiments: How to Observe Directly the Momentum Transfer using Weak Values
| Authors | J. L. Garretson, H. M. Wiseman, D. T. Pope, D. T. Pegg |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0310081 |
| URL | https://arxiv.org/abs/quant-ph/0310081 |
| DOI | 10.1088/1464-4266/6/6/008 |
| Journal | J. Opt. B: Quantum Semiclass. Opt. 6, s506 (2004) |
Abstract
A which-way measurement destroys the twin-slit interference pattern. Bohr argued that distinguishing between two slits a distance s apart gives the particle a random momentum transfer \wp of order h/s. This was accepted for more than 60 years, until Scully, Englert and Walther (SEW) proposed a which-way scheme that, they claimed, entailed no momentum transfer. Storey, Tan, Collett and Walls (STCW) in turn proved a theorem that, they claimed, showed that Bohr was right. This work reviews and extends a recent proposal [Wiseman, Phys. Lett. A 311, 285 (2003)] to resolve the issue using a weak-valued probability distribution for momentum transfer, P_wv(\wp). We show that P_wv(\wp) must be wider than h/6s. However, its moments can still be zero because P_wv(\wp) is not necessarily positive definite. Nevertheless, it is measurable in a way understandable to a classical physicist. We introduce a new measure of spread for P_wv(\wp): half of the unit-confidence interval, and conjecture that it is never less than h/4s. For an idealized example with infinitely narrow slits, the moments of P_wv(\wp) and of the momentum distributions are undefined unless a process of apodization is used. We show that by considering successively smoother initial wave functions, successively more moments of both P_wv(\wp) and the momentum distributions become defined. For this example the moments of P_wv(\wp) are zero, and these are equal to the changes in the moments of the momentum distribution. We prove that this relation holds for schemes in which the moments of P_wv(\wp) are non-zero, but only for the first two moments. We also compare these moments to those of two other momentum-transfer distributions and \hat{p}_f-\hat{p}_i. We find agreement between all of these, but again only for the first two moments.
{
"annotation_id": "1a767eda-4659-4b23-9ad3-f24fdaa2f112",
"date_created": "2026-03-02T18:02:02.807000Z",
"date_modified": "2026-03-02T18:02:02.807000Z",
"file_hash": "69b282535350d0b24098e326f0209585c201321d3318811b8bacdd3cbc8d90e4",
"private": false,
"record": {
"abstract": "A which-way measurement destroys the twin-slit interference pattern. Bohr\nargued that distinguishing between two slits a distance s apart gives the\nparticle a random momentum transfer \\wp of order h/s. This was accepted for\nmore than 60 years, until Scully, Englert and Walther (SEW) proposed a\nwhich-way scheme that, they claimed, entailed no momentum transfer. Storey,\nTan, Collett and Walls (STCW) in turn proved a theorem that, they claimed,\nshowed that Bohr was right. This work reviews and extends a recent proposal\n[Wiseman, Phys. Lett. A 311, 285 (2003)] to resolve the issue using a\nweak-valued probability distribution for momentum transfer, P_wv(\\wp). We show\nthat P_wv(\\wp) must be wider than h/6s. However, its moments can still be zero\nbecause P_wv(\\wp) is not necessarily positive definite. Nevertheless, it is\nmeasurable in a way understandable to a classical physicist. We introduce a new\nmeasure of spread for P_wv(\\wp): half of the unit-confidence interval, and\nconjecture that it is never less than h/4s. For an idealized example with\ninfinitely narrow slits, the moments of P_wv(\\wp) and of the momentum\ndistributions are undefined unless a process of apodization is used. We show\nthat by considering successively smoother initial wave functions, successively\nmore moments of both P_wv(\\wp) and the momentum distributions become defined.\nFor this example the moments of P_wv(\\wp) are zero, and these are equal to the\nchanges in the moments of the momentum distribution. We prove that this\nrelation holds for schemes in which the moments of P_wv(\\wp) are non-zero, but\nonly for the first two moments. We also compare these moments to those of two\nother momentum-transfer distributions and \\hat{p}_f-\\hat{p}_i. We find\nagreement between all of these, but again only for the first two moments.",
"arxiv_id": "quant-ph/0310081",
"authors": [
"J. L. Garretson",
"H. M. Wiseman",
"D. T. Pope",
"D. T. Pegg"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/1464-4266/6/6/008",
"journal_ref": "J. Opt. B: Quantum Semiclass. Opt. 6, s506 (2004)",
"title": "The Uncertainty Relation in \"Which-Way\" Experiments: How to Observe Directly the Momentum Transfer using Weak Values",
"url": "https://arxiv.org/abs/quant-ph/0310081"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "005b9be9-03b6-47a6-a9d8-7534e2e58a6e",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}