8.Odev Kuantum - i times a Hermitian operator(c With what...

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Quantum Mechanics I Homework Assignment # 8, due December 1 [1] Consider four Hermitian 2 × 2 matrices I , σ 1 , σ 2 , and σ 3 , where I is the unit matrix, and the others satisfy σ i σ j + σ j σ i = 2 δ ij . You must prove the following without using a specific representation or form for the ma- trices. (a) Prove that Tr ( σ i ) = 0. (You may need to use the cyclic property of trace: Tr ( XY Z ) = Tr ( Y ZX ) = Tr ( ZXY ) .) (b) Show that the eigenvalues of σ i are ± 1 and that det ( σ i ) = - 1. (c) Show that the four matrices are linearly independent and therefore that any 2 × 2 matrix can be expanded in terms of them. (d) From (c) we know that M = m 0 I + 3 X i =1 m i σ i , where M is any 2 × 2 matrix. Derive an expression for m i ( i = 0 , 1 , 2 , 3). [2] (a) Any 2 × 2 operator is a linear combination of I and three σ ’s. Consider U = u 0 I + i -→ u · -→ σ , where the number u 0 is real. Under what restrictions on u 0 and the numerical vector -→ u is U unitary? (b) Satisfy the unitarity conditions on u 0 and -→ u in such a way that U is seen to be an exponential function of
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Unformatted text preview: i times a Hermitian operator. (c) With what restriction on the number λ is 1 + iλ-→ σ ·-→ n 1-iλ-→ σ ·-→ n , (where-→ n * =-→ n ,-→ n ·-→ n = 1) , a unitary operator? Write this in the form of part (a) and in the form of part (b) . [3] Find the value of Tr ( 1 + ( σ x + σ y ) / √ 2 2 1 + (-σ x + σ y ) / √ 2 2 ) . What physical meaning can you give to this number? 1 [4] A spin-1/2 particle interacts with a magnetic field-→ B = B ˆ z through the Pauli interaction H = μ-→ σ ·-→ B where μ is the magnetic moment and-→ σ = ( σ x ,σ y ,σ z ) are the Pauli spin matrices. At t = 0 a measurement determines that the spin is pointing along the positive x-axis . What is the probability that it will be pointing along the negative y-axis at a later time t ? 2...
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