Abstract
Quantum systems have a number of peculiar features that can all be traced back to one common origin, namely the universal “diabolic” shapes of energy surfaces in the vicinity of a degeneracy. The first such feature is the avoided-crossing effect, when energy levels do not cross but seem to repel each other when plotted in dependence of an external parameter. The next feature is the occurrence of geometric phases, also known as Berry phases, which accumulate during excursions in parameter space. We discuss experimental examples of such phases, both in the space of magnetic fields and in what we call the space of shapes. The Aharonov–Bohm phase is closely related to such geometric phases. Finally, in a short section we show that the hallmarks of quantum chaos can be understood in terms of locally defined 2×2 matrices and their avoided level crossings.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aharonov, Y., Anandan, J.: Phase change during a cyclic quantum evolution. Phys. Rev. Lett. 58, 1593–1596 (1987)
Aharonov, Y., Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)
Bayh, W.: Messung der kontinuierlichen Phasenschiebung von Elektronenwellen im kraftfeldfreien Raum durch das magnetische Vektorpotential einer Wolfram-Wendel. Z. Phys. 169, 492–510 (1962)
Berry, M.V.: Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392, 45–57 (1984)
Bitter, T., Dubbers, D.: Manifestation of Berry’s topological phase in neutron spin rotation. Phys. Rev. Lett. 59, 251–254 (1987)
Bohigas, O., Giannoni, M.J., Schmit, C.: Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984)
Gutzwiller, M.C.: Phase-integral approximation in momentum space and the bound states of an atom. J. Math. Phys. 8, 1979–2000 (1967)
Haake, F.: Quantum Signatures of Chaos, 3rd edn. Springer, Berlin (2010)
Lauber, H.-M., Weidenhammer, P., Dubbers, D.: Geometric phases and hidden symmetries in simple resonators. Phys. Rev. Lett. 72, 1004–1007 (1994)
Manini, N., Pistolesi, F.: Off-diagonal Berry phases. Phys. Rev. Lett. 85, 3067–3071 (2000)
Nakamura, K., Thomas, H.: Quantum billiard in a magnetic field: chaos and diamagnetism. Phys. Rev. Lett. 61, 247–250 (1988)
Simon, B.: Holonomy, the quantum adiabatic theorem, and Berry’s phase. Phys. Rev. Lett. 51, 2167–2170 (1983)
Stöckmann, H.-J.: Quantum Chaos—An Introduction. University Press, Cambridge (1999)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Dubbers, D., Stöckmann, HJ. (2013). Diabolic Points, Geometric Phases, and Quantum Chaos. In: Quantum Physics: The Bottom-Up Approach. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31060-7_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-31060-7_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31059-1
Online ISBN: 978-3-642-31060-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)