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A) Show that the operators σˆ x , σˆ y , σˆ z each has eigenvalues +1, −1. (b) Determine the normalised eigenvectors of each. Are |1 and |2 the eigenvectors of any of the matrices? 7 In the example on transition dipole moment between two energy states with = 0 we have chosen energy states ψ100 and ψ210 of atomic hydrogen. (a) What is the transition dipole moment between states ψ100 and ψ200 of atomic hydrogen? (b) How does the transition dipole moment depend on the parity of the energy states ψnlm ?

12) The expectation value of the square of the field amplitude is different from zero even when n = 0. This simply shows that a vacuum field not only has a non-zero energy but also has nonzero fluctuations. We shall see later that these vacuum fluctuations lead to many interesting effects in quantum optics. An important challenge in quantum optics and quantum information science is a preparation or excitation of the field into a particular photon number state [12, 13]. It is a great practical difficulty to realize a single photon number state.

19) Sx = 2 2i It follows from Eqs. 20) where ∈ mn is the Levi–Civita tensor defined as ⎧ ⎨ 1 lmn = x yz, yzx, zx y (even permutation of x yz) ∈ mn = −1 lmn = x zy, yx z, zyx (odd permutation of x yz) ⎩ 0 when two or more indices are equal. 21) On the basis of the states |1 and |2 , the Hermitian spin operators are represented by matrices 1 1 01 1 1 0 i Sx = σx = , Sy = σy = , 2 2 10 2 2 −i 0 1 1 −1 0 σz = , 0 1 2 2 where σx , σ y and σz are the familiar Pauli spin matrices. 3 Atomic Dipole Moment We have already shown that the operator Sz represents the energy of the electron.

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