Squeezed Quantum States on an Interval and Uncertainty Relations for Nanoscale Systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 288-319.

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We construct families of squeezed quantum states on an interval and analyze their asymptotic behavior. We study the localization properties of a kind of such states constructed on the basis of the theta function. For the coordinate and momentum dispersions of a quantum particle on an interval, we obtain estimates that apply, in particular, to nanoscale systems.
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I. V. Volovich; A. S. Trushechkin. Squeezed Quantum States on an Interval and Uncertainty Relations for Nanoscale Systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 288-319. http://geodesic.mathdoc.fr/item/TM_2009_265_a24/

[1] Schrödinger E., “Der stetige Übergang von der Mikro- zur Makromechanik”, Naturwissenschaften, 14 (1926), 664–666 | DOI | Zbl

[2] Fon Neiman I., Matematicheskie osnovy kvantovoi mekhaniki, Nauka, M., 1964, 367 pp. | MR

[3] Shlyaikh V. P., Kvantovaya optika v fazovom prostranstve, Fizmatlit, M., 2005, 760 pp.

[4] J.R. Klauder, B.-S. Skagerstam (eds.), Coherent states: Applications in physics and mathematical physics, World Sci., Singapore, 1985, 911 pp. | MR | Zbl

[5] Perelomov A. M., Obobschennye kogerentnye sostoyaniya i ikh primeneniya, Nauka, M., 1987, 270 pp. | MR

[6] Weil A., “Sur certains groupes d'opérateurs unitaires”, Acta math., 111 (1964), 143–211 | DOI | MR | Zbl

[7] Judge D., “On the uncertainty relation for angle variables”, Nuovo Cim., 31:2 (1964), 332–340 | DOI | MR | Zbl

[8] De Bièvre S., González J. A., “Semiclassical behaviour of coherent states on the circle”, Quantization and coherent states methods, World Sci., Singapore, 1993, 152–157 | DOI

[9] Kowalski K., Rembieliński J., “On the uncertainty relations and squeezed states for the quantum mechanics on a circle”, J. Phys. A: Math. and Gen., 35 (2002), 1405–1414 | DOI | MR | Zbl

[10] González J. A., del Olmo M. A., Tosiek J., Quantum mechanics on the cylinder, E-print , 2003 arXiv: quant-ph/0306010 | MR

[11] Kowalski K., Rembieliński J., “Coherent states for the quantum mechanics on a compact manifold”, J. Phys. A: Math. and Theor., 41:30 (2008), Pap. 304021 | DOI | MR

[12] Drexler K. E., Nanosystems: Molecular machinery, manufacturing, and computation, J. Wiley Sons, New York, 1992, 576 pp.

[13] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 2: Garmonicheskii analiz. Samosopryazhennost, Mir, M., 1978, 393 pp. | MR

[14] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969, 528 pp. | MR | Zbl

[15] Garbaczewski P., Karwowski W., “Impenetrable barriers and canonical quantization”, Amer. J. Phys., 72:7 (2004), 924–933 | DOI

[16] Novikov S. P., 1. Classical and modern topology. 2. Topological phenomena in real world physics, E-print , 2000 arXiv: math-ph/0004012 | MR

[17] Davydov A. S., Kvantovaya mekhanika, Nauka, M., 1973, 704 pp. | MR

[18] Voronin S. M., Karatsuba A. A., Dzeta-funktsiya Rimana, Fizmatlit, M., 1994, 376 pp. | MR | Zbl

[19] Mamford D., Lektsii o teta-funktsiyakh, Mir, M., 1988, 448 pp. | MR