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December  2018, 10(4): 419-443. doi: 10.3934/jgm.2018016

Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum

Richard Cushman , and  Jędrzej Śniatycki

Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada

* Corresponding author: Richard Cushman

Received  July 2017 Revised  September 2018 Published  November 2018

In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.

Keywords: Bohr-Sommerfeld-Heisenberg quantization, mathematical pendulum, shifting operator, geometric quantization.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Richard Cushman, Jędrzej Śniatycki. Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum. Journal of Geometric Mechanics, 2018, 10 (4) : 419-443. doi: 10.3934/jgm.2018016
References:
[1]

N. Bohr, On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25.   Google Scholar

[2]

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

[3]

R. Cushman and J. Śniatycki, Bohr-Sommerfeld-Heisenberg theory in geometric quantization, J. Fixed Point Theory Appl., 13 (2013), 3-24.  doi: 10.1007/s11784-013-0118-3.  Google Scholar

[4]

R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. Google Scholar

[5]

R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. Google Scholar

[6]

P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653.   Google Scholar

[7]

P. A. M. Dirac, The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947.  Google Scholar

[8]

H. Dullin, Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations, 254 (2013), 2942-2963.  doi: 10.1016/j.jde.2013.01.018.  Google Scholar

[9]

W. Heisenberg, Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, (German) [On the quantum theoretical meaning of kinematic and mechanical relationship], Z. Phys., 33 (1925), 879-893.   Google Scholar

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J. Śniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980.  Google Scholar

[11]

A. Sommerfeld, Zur Theorie der Balmerschen Serie, (German) [On the theory of the Balmer series], Sitzungberichte der Bayerischen Akademie der Wissenschaften (Mü nchen), mathematisch-physikalische Klasse, (1915), 425-458. Google Scholar

show all references

References:
[1]

N. Bohr, On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25.   Google Scholar

[2]

R. H. Cushman and L. M. Bates, Global Aspects of Classical Integrable Systems, second edition, Birkhäuser, Basel, 2015. doi: 10.1007/978-3-0348-0918-4.  Google Scholar

[3]

R. Cushman and J. Śniatycki, Bohr-Sommerfeld-Heisenberg theory in geometric quantization, J. Fixed Point Theory Appl., 13 (2013), 3-24.  doi: 10.1007/s11784-013-0118-3.  Google Scholar

[4]

R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. Google Scholar

[5]

R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. Google Scholar

[6]

P. A. M. Dirac, The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653.   Google Scholar

[7]

P. A. M. Dirac, The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947.  Google Scholar

[8]

H. Dullin, Semi-global symplectic invariants of the spherical pendulum, J. Differential Equations, 254 (2013), 2942-2963.  doi: 10.1016/j.jde.2013.01.018.  Google Scholar

[9]

W. Heisenberg, Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen, (German) [On the quantum theoretical meaning of kinematic and mechanical relationship], Z. Phys., 33 (1925), 879-893.   Google Scholar

[10]

J. Śniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980.  Google Scholar

[11]

A. Sommerfeld, Zur Theorie der Balmerschen Serie, (German) [On the theory of the Balmer series], Sitzungberichte der Bayerischen Akademie der Wissenschaften (Mü nchen), mathematisch-physikalische Klasse, (1915), 425-458. Google Scholar

Figure 1.  The graph of $H\left( {x,y} \right) = \frac{1}{2}{y^2} - \cos x + 1$ with $\left( {x,y} \right) \in \left[ { - \mathit{\boldsymbol{\pi }},\mathit{\boldsymbol{\pi }}} \right] \times {\mathbb{R}}$
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Figure 2.  The ${\mathbb{Z}}_2$-reduced space in ${\mathbb{R}}^3$
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