2. Two-level systems and non-Hermitian Quantum Mechanics Consider the following two-level system ди at 1u+i(JI +9) ai, u

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2. Two-level systems and non-Hermitian Quantum Mechanics Consider the following two-level system ди at 1u+i(JI +9) ai, ueca, where EO A 91 0 H= G Λ ΕΟ 0 92 where Eo, A, 40, 91, 92 are positive real numbers, and where H and HI+G are real symmetric matrices. ( -( (a) Discuss very briefly the following question: Assuming that one has available the most detailed possible physical de scription, is a non-Hermitian version of Quantum Mechanics necessary? In what circumstances might such a version of Quantum Mechanics be appropriate or convenient? [4 marks] (b) Take the following trial solution for Equation (2): u(t) = uoc-iwt. Hence, show that the eigenvalue problem breaks down into two cases: = wp = E + V4A2 - (91-92), W; = Ho - }(91 +92), 412 > (91-92), Case 1, and W = = Eo, wi = Ho - (91 +92) 3V (91-92)2 – 4A2, 41² < (91-92), Case 2. dt [6 marks) (c) For both cases 1 and 2, show that the norm ||1||3 = (u, u) :=u*?u satisfies d ||| ||= {u, (20I +G) u). [6 marks] (d) Derive the following bound on the growth of the L? norm: f | ||||< [140-min(91, 92)] || | ||2. [4 marks]