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Unformatted text preview: C/CS/Phy191 Problem Set 1 Solutions Out: September 06, 2007 1. Consider the normalized states cos θ 1 sin θ 1 and cos θ 2 sin θ 2 . Find the condition on θ 1 and θ 2 such that the superposition cos θ 1 sin θ 1 + cos θ 2 sin θ 2 is normalized to unity (“properly normalized”). ( Hint : Remember that cos ( θ 1 + θ 2 ) = cos θ 1 cos θ 2 sin θ 1 sin θ 2 ) Solution: From the condition that the vector is normalized, it follows that ( sin θ 1 + sin θ 2 ) 2 +( cos θ 1 + cos θ 2 ) 2 = 1 . Hence sin θ 1 sin θ 2 + cos θ 1 cos θ 2 = 1 / 2. Hence cos ( θ 1 θ 2 ) = 1 / 2. Therefore θ 1 θ 2 = 2 π / 3, or θ 1 θ 2 = 4 π / 3. 2. The kets h and v are states of horizontal and vertical photon polarization, respectively. Consider the states ψ 1 = 1 2 h + √ 3 v , ψ 2 = 1 2 h √ 3 v , ψ 3 = h , What are the relative orientations of the plane of polarization for these three states? Solution: Since h h = v v = 1 and v h = h v = 0, we have ψ 1 ψ 2 = 1 2 ψ 1 ψ 3 = 1 2 ψ 2 ψ 3 = 1 2 (1) The solution to cos α = 1 / 2 is given by α = 120 ◦ or α = 240 ◦ . Hence the three states correspond to states of linear polarization separated by 120 ◦ . 3. a) Find the eigenvectors, eigenvalues, and diagonal representations of the Pauli matrices I, X, Y, and Z, where I ≡ σ ≡ 1 0 0 1 X ≡ σ x ≡ 0 1 1 0 Y ≡ σ y ≡ i i Z ≡ σ z ≡ 1 1 Show that X 2 = Y 2 = Z 2 = I (so that calculating X n , Y n or Z n becomes really simple...) b) Find the points on the Bloch sphere that correspond to the normalized eigenvectors of the Pauli matrices. c) Find the action of the Z operator on a general qubit state Ψ = α + β 1 and describe this action on the Bloch sphere (i.e. how does the vector representing Ψ get rotated on the Bloch sphere?). C/CS/Phys C191, Fall 2007, 1 d) Form the matrix representation of the exponentiated operator e i γ Z / 2 and show how this exponen tiated operator acts on the Bloch sphere vector for Ψ . e) Similarly form the matrix representations of the exponentiated operators e i γ X / 2 and e i γ Y / 2 . Show explicitly how these act on the vector at the North pole of the Bloch sphere, i.e. on the qubit state Ψ = 0 , specifying the nature of the resulting rotation. Solution: a) Use standard linear algebra methods to find the eigenvalues and eigenvectors of the matrices as they are given (consult any standard textbook on linear algebra. If you are having trouble with this and need help, please let us know, since a good understanding of the concepts of eigenvalues and eigenvectors is essential to this course). Since the identity matrix σ ≡ I leaves all vectors unchanged, all vectors are eigenvectors of this matrix, and they all have eigenvalue 1....
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 Fall '07
 BIRGITTAWHALEY
 Linear Algebra, mechanics, Cos, North Pole, Bloch sphere

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