Abstract
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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References
E. Schrödinger, “Der stetige Übergang von der Mikro- zur Makromechanik,” Naturwissenschaften 14, 664–666 (1926).
J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932; Nauka, Moscow, 1964).
W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Weinheim, 2001; Fizmatlit, Moscow, 2005).
Coherent States: Applications in Physics and Mathematical Physics, Ed. by J. R. Klauder and B.-S. Skagerstam (World Sci., Singapore, 1985).
A. M. Perelomov, Generalized Coherent States and Their Applications (Springer, Berlin, 1986; Nauka, Moscow, 1987).
A. Weil, “Sur certains groupes d’opérateurs unitaires,” Acta Math. 111, 143–211 (1964).
D. Judge, “On the Uncertainty Relation for Angle Variables,” Nuovo Cimento 31(2), 332–340 (1964).
S. De Biévre and J. A. González, “Semiclassical Behaviour of Coherent States on the Circle,” in Quantization and Coherent States Methods (World Sci., Singapore, 1993), pp. 152–157.
K. Kowalski and J. Rembieliński, “On the Uncertainty Relations and Squeezed States for the Quantum Mechanics on a Circle,” J. Phys. A: Math. Gen. 35, 1405–1414 (2002).
J. A. González, M. A. del Olmo, and J. Tosiek, “Quantum Mechanics on the Cylinder,” arXiv: quant-ph/0306010.
K. Kowalski and J. Rembieliński, “Coherent States for the Quantum Mechanics on a Compact Manifold,” J. Phys. A: Math. Theor. 41(30), 304021 (2008).
K. E. Drexler, Nanosystems: Molecular Machinery, Manufacturing, and Computation (J. Wiley & Sons, New York, 1992).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis, Self-Adjointness (Academic, New York, 1975; Mir, Moscow, 1978).
M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].
P. Garbaczewski and W. Karwowski, “Impenetrable Barriers and Canonical Quantization,” Am. J. Phys. 72(7), 924–933 (2004).
S. P. Novikov, “1. Classical and Modern Topology. 2. Topological Phenomena in Real World Physics,” arXiv:math-ph/0004012.
A. S. Davydov, Quantum Mechanics (Nauka, Moscow, 1973) [in Russian].
S. M. Voronin and A. A. Karatsuba, The Riemann Zeta-Function (de Gruyter, Berlin, 1992; Fizmatlit, Moscow, 1994).
D. Mumford, Tata Lectures on Theta. I, II (Birkhäuser, Boston, 1983, 1984; Mir, Moscow, 1988).
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Original Russian Text © I.V. Volovich, A.S. Trushechkin, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 288–319.
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Volovich, I.V., Trushechkin, A.S. Squeezed quantum states on an interval and uncertainty relations for nanoscale systems. Proc. Steklov Inst. Math. 265, 276–306 (2009). https://doi.org/10.1134/S0081543809020254
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DOI: https://doi.org/10.1134/S0081543809020254