Abstract
Magnetic fields in terrestrial experiments have only tiny effects on the ground-state properties of conventional atoms. The reason is that the natural atomic unit for magnetic field strength, B 0= m 2 e 3 c/ħ3 = 2.35 × 105 Tesla, is enormous compared with laboratory fields, which are seldom larger than 10 T. Here m denotes the electron mass, e the elementary charge, and ħ and c have their usual meaning. The unit B 0 is the field strength B at which the magnetic length ℓ B = (ħc/(eB))1/2 (~ cyclotron radius for an electron in the lowest Landau level) is equal to the Bohr radius a0 = ħ 2 /(me 2). Equivalently, at B = B 0 the Landau energy ħω B , with ω B = eB/(mc) the cyclotron frequency, becomes equal to the Rydberg energy e 2 /a 0. For B ≪ B 0 distortions of ground-state wave functions and energy level shifts due to the magnetic field will therefore be small, and their standard treatment by means of perturbation theory is completely adequate.
This is the text of a lecture given by J. Yngvason at the XIth International Congress of Mathematical Physics, Paris 1994. It appeared originally in the proceedings of the congress, ed. by D. Iagolnitzer, pp. 185–205, International Press 1995, but has been updated and slightly expanded for the present publication. It summarizes the joint work [2], [3] and [5] of the three authors and is presented here in lieu of [2] and [3] which are too long to be included in this volume.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
G. Chanmugam, Magnetic Fields of Degenerate Stars, Ann. Rev. Astron. Astrophys. 30, 143–184 (1992)
E.H. Lieb, J.P. Solovej and J. Yngvason, Asymptotics of Heavy Atoms in High Magnetic Fields: I. Lowest Landau Band Regions, Commun. Pure Appl. Math. 47, 513–591 (1994)
E.H. Lieb, J.P. Solovej and J. Yngvason, Asymptotics of Heavy Atoms in High Magnetic Fields: II. Semiclassical Regions, Commun. Math. Phys 161, 77–124 (1994)
M.A. Kastner, Artificial atoms, Phys. Today 46, 24–31 (1993)
E.H. Lieb, J.P. Solovej and J. Yngvason, The Ground States of Large Quantum Dots in Magnetic Fields, Phys. Rev. B 51, 10646–10665 (1995)
E.H. Lieb, J.P. Solovej and J. Yngvason, Heavy Atoms in the Strong Magnetic Field of a Neutron Star, Phys. Rev. Lett. 69, 749–752 (1992)
E.H. Lieb and J.P. Solovej, Atoms in the Magnetic Field of a Neutron Star, in: Differential Equations with Applications to Mathematical Physics, W.F. Ames, J.V. Herod and E.M. Harrell II, eds., pp. 221–237, Academic Press 1993
E.H. Lieb, J.P. Solovej and J. Yngvason, Quantum Dots, in: Proceedings of the Conference on Partial Differential Equations and Mathematical Physics, University of Alabama, Birmingham, 1994, I. Knowles, ed., pp. 157–172, International Press 1995
E.H. Lieb and B. Simon, The Thomas-Fermi Theory of Atoms, Molecules and Solids, Adv. in Math. 23, 22–116 (1977)
E.H. Lieb, Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53, 603–641 (1981);
E.H. Lieb, Erratum, Rev. Mod. Phys. 54, 311 (1982)
B.B. Kadomtsev, Heavy Atoms in an Ultrastrong Magnetic Field, Soviet Phys. JETP 31, 945–947 (1970)
B.B. Kadomtsev and V.S. Kudryavtsev, Atoms in a superstrong magnetic field, JETP Lett. 13, 42–44 (1971)
M. Ruderman, Matter in Superstrong Magnetic Fields: The Surface of a Neutron Star, Phys. Rev. Lett. 27, 1306–1308 (1971).
R.O. Mueller, A.R.P. Rau and L. Spruch, Statistical Model of Atoms in Intense Magnetic Fields, Phys. Rev. Lett. 26, 1136–1139 (1971)
Y. Tomishima and K. Yonei, Thomas-Fermi Theory for Atoms in a Strong Magnetic Field, Progr. Theor. Phys. 59, 683–696 (1978)
B. Banerjee, D.H. Constantinescu, and P. Rehák, Thomas-Fermi and Thomas-Fermi-Dirac calculations for atoms in a very strong magnetic field, Phys. Rev. D 10, 2384–2395 (1974)
J. Yngvason, Thomas-Fermi Theory for Matter in a Magnetic Field as a Limit of Quantum Mechanics, Lett. Math. Phys. 22, 107–117 (1991)
V. Ivrii, Semiclassical Microlocal Analysis and Spectral Asymptotics, Springer (to be published)
A. Sobolev, The quasi-classical asymptotics of local Riesz means for the Schrödinger operator in a strong homogeneous magnetic field, Duke Math. J. 74, 319–429 (1994)
I. Fushiki, E.H. Gudmundsson, C.J. Pethick, and J. Yngvason, Matter in a Magnetic Field in the Thomas-Fermi and Related Theories, Ann. Phys. 216, 29–72 (1992)
O.E. Rögnvaldsson, I. Fushiki, C.J. Pethick, E.H. Gudmundsson and J. Yngvason, Thomas-Fermi Calculations of Atoms and Matter in Magnetic Neutron Stars: Effects of Higher Landau Bands, Astrophys. J. 216, 276–290 (1993)
K. Johnsen, MS thesis, Univ. of Iceland, 1994. See also: K. Johnsen and J. Yngvason, Density Matrix Functional Calculations for Matter in Strong Magnetic Fields: Ground States of Heavy Atoms, Phys. Rev. A, in press (1996)
E.H. Lieb, A Variational Principle for Many-Fermion Systems, Phys. Rev. Lett. 46, 457–459;
E.H. Lieb, A Variational Principle for Many-Fermion Systems, Erratum 47, 69 (1981)
E.H. Lieb, J.P. Solovej, Quantum coherent operators: A generalization of coherent states, Lett. Math. Phys. 22, 145–154 (1991)
E.H. Lieb and W.E. Thirring, Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter, Phys. Rev. Lett. 35, 687–689 (1975)
L. Erdos, Magnetic Lieb-Thirring inequalities, Comm. Math. Phys. 170, 629–669 (1995)
E.H. Lieb and S. Oxford, An Improved Lower Bound on the Indirect Coulomb Energy, Int. J. Quant. Chem. 19, 427–439 (1981)
P.L. McEuen, E.B. Foxman, J. Kinaret, U. Meirav, M.A. Kastner, N.S. Wingreen and S.J. Wind, Self consistent addition spectrum of a Coulomb island in the quantum Hall regime, Phys. Rev. B 45, 11419–11422 (1992)
N.C. van der Vaart, M.P. de Ruyter van Steveninck, L.P. Kouwenhoven, A.T. Johnson, Y.V. Nazarov, and C.J.P.M. Harmans, Time-Resolved Tunneling of Single Electrons between Landau Levels in a Quantum Dot, Phys. Rev. Lett. 73, 320–323 (1994)
E.H. Lieb and H.-T. Yau, The stability and instability of relativistic matter, Commun. Math. Phys. 118, 177–213 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lieb, E.H., Solovej, J.P., Yngvason, J. (2001). Asymptotics of Natural and Artificial Atoms in Strong Magnetic Fields. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04360-8_13
Download citation
DOI: https://doi.org/10.1007/978-3-662-04360-8_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-04362-2
Online ISBN: 978-3-662-04360-8
eBook Packages: Springer Book Archive