Abstract
The Zeeman effect, which we discussed in some detail in Sect. 14.2, originates from the interaction between the magnetic moment of an electron bound to an atom and a uniform magnetic field. Experimentally this effect is often observed in atomic gases but can also manifest itself with bound electrons in semiconductors and dielectrics. What is important is that the quantum states of the electrons in all these cases belong to discrete energy eigenvalues. In metals and in the conduction band of semiconductors, on the other hand, the energy levels of electrons belong to the continuum spectrum, and in some instances, electrons can even be treated as free particles. The interaction between such unbound, almost free electrons and the uniform magnetic field results in some fascinating effects which had played and are still playing an important role in physics.
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- 1.
Störmer was born in Germany and got his Ph.D. in France, while Tsui was born in China to a family of farmers, did undergraduate work at Lutheran Augustana College in Illinois, and got his Ph.D. at the University of Chicago.
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These are not real “particles” of course and are usually called quasiparticle. They present a convenient theoretical model useful for the description of ground state properties of strongly interacting electrons.
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Sir Joseph Larmor (1857–1942) was a Northern Irish physicist famous for the discovery of Lorentz transformations 2 years before Lorentz and 8 years before Einstein. He also discovered the effects of time dilation and length contraction but believed that they are real material changes in length rather than pure kinematic effects. He believed in ether and did not believe in relativity, both special and general. Still, he held a post of Lucasian Professor of Mathematics at Cambridge University, which was established in 1663(!), and was held before him by Newton and after him by Dirac. At the time of writing, this post is held by Stephen Hawking.
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When considering the interaction of an atom with electromagnetic wave as in Sect. 15.3.1, I did not have to worry about the interaction between the magnetic field of the wave and the spin because normally such an interaction would be extremely weak. In the case considered in this chapter, the magnetic field can be strong enough to make this interaction relevant.
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This condition is obviously just an approximation because the wave function in the x direction has finite size, and requiring that its center remains inside the box does not make the probability of the particle described by the harmonic oscillator wave functions to be outside of the box to vanish automatically. The choice of xc to define the number of states is, in this sense, rather arbitrary (you could choose xc + △x, where △x is the uncertainty of the coordinate or something else for that matter). However, the error which I make here is of the order of △x/Lx and becomes negligibly small as Lx grows. The count of the number of states derived by this heuristic method is actually confirmed by a more rigorous (and immensely more complicated) mathematical approach to the problem.
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One can associate a delta-function δ(E − En) with each level and the identity expression ∑nNnδ(E − En) as a total density of states. This identification makes sense if you integrate it over any final energy interval to get the total number of states in it.
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Deych, L.I. (2018). Free Electrons in Uniform Magnetic Field: Landau Levels and Quantum Hall Effect. In: Advanced Undergraduate Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-71550-6_16
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