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(a) Show that on a Hilbert space H, any unitary operator Û : H H H preserves norms, i.e. V|4) E H the relation |Û [4) || = || |_) || holds. Moreover, show that all eigenvalues of û lie on the complex unit circle. [2] (b) Consider the so-called A-system illustrated in the figure on the right (for simplicity all displayed quantities can be assumed to be li> E; real-valued). Write the Hamiltonian H of this system both as an П. abstract Hilbert-space operator involving the sum of dyadic pro- Ω, les ducts and as a marix with respect to the basis {\g), (i), (e)}. [4P] Hint: Horizontal lines indicate (field-free) energy levels, e.g., (g|Hg) = Eg. Connecting arrows indicate couplings between these igs levels, e.g., (g\Î\i) = N1. (c) Consider the Hamiltonian î = a (g) (g| +Ble) (el (a,B E R) on a Hilbert space H with orthonormal basis states {\g), le)}. Write an explicit expression for the time-evolution operator û = e-] Ħt as a sum of dyadic products on H. [2] (d) The Heisenberg equation of motion allows to find the time evolution of expectation values via an ordinary first-order differential equation. Specifically, given a Hamiltonian H and an operator A without explicit time dependence, the expectation value (Â)|y(t)) = (V (t) |Â|(t)) behaves as d (A) perce) (L) = (, Â) wres) dt (t ħ (t Which statements can you make if the commutator (î, Â] vanishes? How does the solution fo

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