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Problem 9: Consider the Linear Systems df and A is given by: In each of the four cases: (a) Determine the type of the system, i.e., sink (node), source, saddle, spiral source, spiral sink, center. (b) Draw the phase portrait of the system. If the eigenvalues are real you need to compute the eigenvectors and indicate them clearly on the phase portrait.

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The free energy of fermions in a magnetic field. Consider the same system as in HW1: a lattice with N sites, where each site can accommodate zero, one, or two electrons. Electrons have a spin of 1/2, so each electron can have a spin of +1/2 or -1/2 relative to some fixed direction. Electrons are fermions and satisfy the Pauli exclusion principle. This means that if two electrons occupy the same lattice site, their spins have to be antiparallel. Since electrons are indistinguishable particles, there is only one state of two electrons with antiparallel spins. The system is in an external magnetic field B. The mass of an electron is m, so adding an electron to the system costs energy equal to mc^2. The interaction energy with the magnetic field is as usual es = -sB, where s = 1/2 and μ is the magnetic moment. There is also a strong repulsion between two electrons when they occupy the same lattice site, so the state with two electrons has energy 2 = 2mc^2 + U, where U > B. The system is in thermal equilibrium with the heat reservoir at temperature T. The reservoir can also freely supply electrons, so the number of electrons is not fixed but is determined solely by the condition of thermal equilibrium.
  1. Calculate the free energy of the system.
  2. By minimizing the free energy, calculate the average number of electrons in the system.
  3. Calculate the average energy and specific heat.
  4. In all of the above, consider and discuss separately the limits of weak B < mc^2 and strong B > mc^2 magnetic field.
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Question Text
Problem 9: Consider the Linear Systems df and A is given by: In each of the four cases: (a) Determine the type of the system, i.e., sink (node), source, saddle, spiral source, spiral sink, center. (b) Draw the phase portrait of the system. If the eigenvalues are real you need to compute the eigenvectors and indicate them clearly on the phase portrait.
TopicAll topics
SubjectPhysics
ClassClass 12