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Unformatted text preview: Physics 127B FINAL EXAM (due 10am, Tuesday, March 15) The final must be done on your own, with no discussions with others. You may consult your class notes, homeworks, the class textbooks and resources (listed on the course web page), but no other books or resources. Solve all 4 problems. Show all your work. If you use results from class notes, homework problems, or textbooks, quote your source. Please do the exam in a blue exam book or on clean note paper with the pages stapled together. Place your exam in my mailbox in West Bridge mailroom by 10am on Tuesday, March 15. No extensions. Problem 1: Phase separation instability in a variational study We are studying a challenging quantum many-body problem using variational approach. The system has N particles moving on a lattice with M sites, described by some Hamiltonian. We would like to find the lowest energy state (i.e., ground state) of the system for each N . For each N , we have a variational wavefunction that produces a uniform liquid state and we calculate the trial energy E N, trial . Figure below shows the trial energy per site trial ( ) E N, trial /M plotted vs particle density per site N/M . a) Devise an appropriate Maxwell construction to analyze the stability of such uniform liquid states to phase separation, providing all details of the proof (e.g., it is crucial that we plot energy per site vs density and not , say, energy per particle vs density).-0.7-0.6-0.5-0.4-0.3-0.2-0.1 0.1 0.2 0.3 0.4 0.5 Energy per site, = E/M sites density per site, = N/M sites b) Apply the Maxwell construction to the variational energies in the figure. In the phase-separated regime, what are the states that the system is separating into?...
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