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hw 9 solutions

# hw 9 solutions - 1 ECE 495N Fall09 GRIS 280 MWF 1130A 1220P...

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12/3/09 1 ECE 495N, Fall’09 GRIS 280, MWF 1130A – 1220P HW#9: Due Wednesday Dec.9 in class. This is the last HW for the semester. Pauli spin matrices: (2x2) Identity matrix: . x = 0 1 1 0 ± ² ³ ´ µ , y = 0 ± i + i 0 ² ³ ´ µ · , z = 1 0 0 ± 1 ² ³ ´ µ · I = 1 0 0 1 . ± ² ³ ´ µ 1. What are the eigenvalues of the (2x2) matrix . . ˆ n ± x sin ² cos ³ + y sin sin + z cos Show that the corresponding eigenvectors can be written as c s . ± ² ³ ´ µ , . s * c * ± ² ³ ´ µ , where c . cos ± 2 e ² i /2 , s . sin 2 e + i / 2 2. Consider a device with two spin-degenerate levels described by [ H ] = 0 0 ± ² ³ ´ µ . It is connected to four magnetic contacts with one pointing along + ˆ z and one along . ˆ z described by [ . 1 ] = ± i 2 0 0 ´ µ · ¸ ¹ , [ . 2 ] = ± i 2 0 0 ´ µ · ¸ ¹ where and are real numbers with say, > . The other two are identical contacts but one points along + ˆ n and one along . ˆ n where ˆ n = ˆ z cos + ˆ x sin . a. Show that . 1 and . 2 can be written in the form . i ( aI + b z ) /2 and . i ( aI . b z ) /2 respectively and obtain a and b.

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