1.2. A Quantum System. Consider the system with hamiltonian H = H0 ...

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1.2. A Quantum System. Consider the system with hamiltonian H = H0 + H1 where H0 = hwBata+hwpbtb, H1 = hg (atb + abt) and the bosonic and fermionic creation annihilation operators satisfy [a, atj = 1, btb + bbt = 1, 62 =0=6t2 = = (a,b]==at,b=a,b=at,b]. = = = = 1.2.1. Show that N = ata + btb (the sum of boson and fermion numbers)commutes with H. So H, N have simultaneous eigenstates. 1.2.2. Show that the empty state for which NJ0 >= 0 is an eigenstate of H. Find the eigenvalue. 1.2.3. Find two eigenstates of H for which N = 1.

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