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2008 by Taylor & Francis Group, LLC 34 Introduction to Quantum Control and Dynamics • Assume the state of the system is |0 . In what states can we find the system after a measurement of the energy, and with what probability? • Write a differential equation which gives the evolution of the components of the column vector representing an arbitrary state |ψ (in the basis |0 , |1 ). Assume = 1. 35), to show that Tr(Pj ρPj ) is the probability of finding the result λj for a measurement on an ensemble ρ.

26) by identifying eigenvalues λj and corresponding projections Pj . 52). 25) are such that the product of two of them is equal to 0. , that the expectation value of A at the state |ψ , A ψ , is equal to ψ|A|ψ . Use the spectral theorem and the fact that the element of the spectral family corresponding to an interval [λ1 , λ2 ) of the spectrum is λ2 dPA . λ1 © 2008 by Taylor & Francis Group, LLC Chapter 2 Modeling of Quantum Control Systems; Examples Quantum systems whose dynamics depends on one or more control functions will be named quantum control systems.

4) of the vector field F , F (k). This is perpendicular to k for every vector k, as in Part B. Under appropriate boundary conditions (the vector fields go to zero at infinity faster than |r|1 2 ), we have8 R I 3 8 This VL (r) · VT (r) dr = 0. 25) fact is an immediate consequence of Parseval-Plancherel theorem R I 3 F ∗ (r)G(r)dr = R I 3 F ∗ (k) · G(k)dk, where F and G are the spatial Fourier transforms of the fields F and G, respectively (cf. 4). B). © 2008 by Taylor & Francis Group, LLC Modeling of Quantum Control Systems; Examples z z ✻ ✻ F (r) ✘ ✾ ✘ ............................

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