
-
Previous Article
On some aspects of the geometry of non integrable distributions and applications
- JGM Home
- This Issue
-
Next Article
On motions without falling of an inverted pendulum with dry friction
Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada |
In this paper we give the Bohr-Sommerfeld-Heisenberg quantization of the mathematical pendulum.
References:
[1] |
N. Bohr,
On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25.
|
[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. |
[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. |
[4] |
R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. |
[5] |
R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. |
[6] |
P. A. M. Dirac,
The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653.
|
[7] |
P. A. M. Dirac,
The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947. |
[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. |
[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.
|
[10] |
J. Śniatycki,
Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980. |
[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. |
show all references
References:
[1] |
N. Bohr,
On the constitution of atoms and molecules (part Ⅰ), Philosophical Magazine, 26 (1913), 1-25.
|
[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. |
[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. |
[4] |
R. Cushman and J. Śniatycki, Bohr-Sommerfeld Heisenberg quantization of the $2$-dimensional harmonic oscillator, arXiv: 1207.1477. |
[5] |
R. Cushman and J. Śniatycki, Shifting operators in geometric quantization, arXiv: 1808.04002. |
[6] |
P. A. M. Dirac,
The fundamental equations of quantum mechanics, Proc. Roy. Soc. London, 109 (1925), 642-653.
|
[7] |
P. A. M. Dirac,
The Principles of Quantum Mechanics, 3d ed. Oxford, at the Clarendon Press, 1947. |
[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. |
[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.
|
[10] |
J. Śniatycki,
Geometric Quantization and Quantum Mechanics, Applied Mathematical Series 30 Springer Verlag, New York, 1980. |
[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. |

[1] |
Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991 |
[2] |
Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. |
[3] |
Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829 |
[4] |
Fabiano Boaventura de Miranda, Cristiano Torezzan. A shape-gain approach for vector quantization based on flat tori. Advances in Mathematics of Communications, 2020, 14 (3) : 467-476. doi: 10.3934/amc.2020064 |
[5] |
Setsuro Fujiié, Jens Wittsten. Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3851-3873. doi: 10.3934/dcds.2018167 |
[6] |
Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli. Constrained systems, generalized Hamilton-Jacobi actions, and quantization. Journal of Geometric Mechanics, 2022, 14 (2) : 179-272. doi: 10.3934/jgm.2022010 |
[7] |
Mrinal Kanti Roychowdhury. Least upper bound of the exact formula for optimal quantization of some uniform Cantor distributions. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4555-4570. doi: 10.3934/dcds.2018199 |
[8] |
Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201 |
[9] |
Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, 2021, 29 (4) : 2673-2685. doi: 10.3934/era.2021008 |
[10] |
Dmitrii Rachinskii. On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3361-3386. doi: 10.3934/dcdsb.2018246 |
[11] |
Joachim Escher, Tony Lyons. Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach. Journal of Geometric Mechanics, 2015, 7 (3) : 281-293. doi: 10.3934/jgm.2015.7.281 |
[12] |
Stefano Marò. Relativistic pendulum and invariant curves. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139 |
[13] |
Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure and Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775 |
[14] |
Ravi Shanker Dubey, Pranay Goswami. Mathematical model of diabetes and its complication involving fractional operator without singular kernal. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2151-2161. doi: 10.3934/dcdss.2020144 |
[15] |
Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3387-3399. doi: 10.3934/dcdss.2021017 |
[16] |
Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373 |
[17] |
Mari Paz Calvo, Jesus M. Sanz-Serna. Carrying an inverted pendulum on a bumpy road. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 429-438. doi: 10.3934/dcdsb.2010.14.429 |
[18] |
Jean-Francois Bertazzon. Symbolic approach and induction in the Heisenberg group. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1209-1229. doi: 10.3934/dcds.2012.32.1209 |
[19] |
Jean-Marie Souriau. On Geometric Mechanics. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 595-607. doi: 10.3934/dcds.2007.19.595 |
[20] |
Chunpeng Wang, Jingxue Yin, Bibo Lu. Anti-shifting phenomenon of a convective nonlinear diffusion equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1211-1236. doi: 10.3934/dcdsb.2010.14.1211 |
2021 Impact Factor: 0.737
Tools
Metrics
Other articles
by authors
[Back to Top]