I am interested in understanding possible applications for the study of quantum systems with constant magnetic fields.

For definiteness, consider the Landau Hamiltonian $$H_{0} = \left(-i\frac{\partial}{\partial x} - \frac{b}{2}y\right)^{2} + \left(-i\frac{\partial}{\partial y} + \frac{b}{2}x\right)^{2}$$ describing a spinless, non-relativistic 2D charged particle subject to a constant magnetic field of intensity $b > 0$, or the analogous Hamiltonian in 3D.

What sort of applications/results motivate the study of "perturbations" of this system, such as obtained by introducing an electric field that decays at infinity, or a low intensity (non-constant) magnetic field, for example?

  • $\begingroup$ I think this kind of question would generally fall under "symmetry breaking". A more general question is the classification of all global and local symmetries of a given Hamiltonian and the study of all first and higher order symmetry breaking terms on the dynamic solution of that Hamiltonian and on its spectrum. This can also be combined with thermodynamics, in which case one ends up with the study of phase transitions. $\endgroup$
    – CuriousOne
    Commented Dec 13, 2014 at 0:57
  • $\begingroup$ Could you explain to me some more? I don't know what symmetry breaking means and how it relates to applications of the study of this operator. $\endgroup$
    – Geno Whirl
    Commented Dec 15, 2014 at 19:23


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