Uncertainty relation between angle and orbital angular momentum: interference effect in electron vortex beams
https://doi.org/10.17586/2220-8054-2015-6-2-205-212
Abstract
The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator. Instead of the angle variable operator, we introduce the complex position operator ˆZ = ˆx+iˆy and interpret the order parameter µ = ⟨ ˆZ⟩/⟨ ˆZ† ˆZ⟩ as a measure of certainty of the angle distribution. We prove the relation between the uncertainty of angular momentum and the angle order parameter. We also prove its generalizations and discuss experimental methods for testing these relations.
Supporting agencies: The author thanks Keisuke Watanabe, who discussed with me the tighter version of the uncertainty inequality, Eq. (36). He thanks Prof. Katsuhiro Nakamura and Prof. Davron Matrasulov for their warm hospitality for supporting his stays in Uzbekistan. This manuscript is written as a part of the proceedings of the workshop, Wave dynamics in low-dimensional branched structures, held during September 23–24, 2014 in Tashkent, Uzbekistan. This work is financially supported by the Grant-in-Aid for Scientific Research of Japan Society for the Promotion of Science, Grant No. 26400417.
About the Author
Sh. Tanimura
Nagoya University
Japan
Japan
Shogo Tanimura, Department of Complex Systems Science, Graduate School of Information Science
Nagoya 464-8601
References
1. Heisenberg W. ¨Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Physik, 43, P. 172 (1927).
2. Robertson H.P. The uncertainty principle. Phys. Rev., 34, P. 163 (1929).
3. Schiff L.I. Quantum Mechanics, 3rd edition, McGraw-Hill (1968).
4. Judge D. On the uncertainty relation for Lz and ϕ. Phys. Lett., 5, P. 189 (1963).
5. Kraus K. Remark on the uncertainty between angle and angular momentum. Z. Physik, 188, P. 374 (1965).
6. Carruthers P., Nieto M.M. Phase and angle variables in quantum mechanics. Rev. Mod. Phys., 40, P. 411 (1968).
7. Bernett S.M., Pegg D.T. Quantum theory of rotation angles. Phys. Rev. A, 41, P. 3427 (1990).
8. Ohnuki Y., Kitakado S. Fundamental algebra for quantum mechanics on SD and gauge potentials. J. Math. Phys., 34, P. 2827 (1993).
9. Tanimura S. Gauge field, parity and uncertainty relation of quantum mechanics on S1. Prog. Theor. Phys., 90, P. 271 (1993).
10. Schr¨odinger E. Zum Heisenbergschen Unsch¨arfeprinzip. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, 19, P. 296 (1930).
11. Franke-Arnold S., Barnett S.M., et al. Uncertainty principle for angular position and angular momentum. New J. Phys., 6, P. 103 (2004).
12. Pegg D.T., Barnett S.M., et al. Minimum uncertainty states of angular momentum and angular position. New J. Phys., 7, P. 62 (2005).
13. Uchida M. Tonomura A. Generation of electron beams carrying orbital angular momentum. Nature, 464, P. 737 (2010).
14. Verbeeck J., Tian H., Schattschneider P. Production and application of electron vortex beams. Nature, 467, P. 301 (2010).
15. McMorran B.J., Agrawal A., et al. Electron vortex beams with high quanta of orbital angular momentum. Science, 331, P. 192 (2011).
16. Hasegawa Y., Saitoh K., et al. Young’s interference experiment with electron beams carrying orbital angular momentum. J. Phys. Soc. Jap., 82, P. 033002 (2013).
17. Hasegawa Y., Saitoh K., Tanaka N., Uchida M. Propagation dynamics of electron vortex pairs. J. Phys. Soc. Jap., 82, P. 073402 (2013).
18. Tanimura S. The incompatibility relation between visibility of interference and distinguishability of paths. ArXiv quant-ph/0703118 (2007).
19. Ozawa M. Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A, 67, P. 042105 (2003).
20. Branciard C. Error-tradeoff and error-disturbance relations for incompatible quantum measurements. Proceedings of the National Academy of Science of the USA, 110 (17), P. 6742 (2013).
21. Hayashi M. Fourier analytic approach to quantum estimation of group action. ArXiv 1209.3463v2 (2012)
Review
For citations:
Tanimura Sh. Uncertainty relation between angle and orbital angular momentum: interference effect in electron vortex beams. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(2):205-212. https://doi.org/10.17586/2220-8054-2015-6-2-205-212
Views:
6
Email this article 