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HW3 - and be so that $ ± ²3 ” • 1 2 is symmetric is...

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MA/PH 607 9/9/2011 Homework Handout III A. Read § 4.1 in the online notes, doing any in-notes exercises. Also skim/read §3.2 of A&W (Arfken & Weber), at the starting with Basic Definitions bottom of page 177. Skip everything in §3.2 before that (most of pages 176 & 177) but read about the stuff on “direct products” and “trace” even though we won’t do much with them). Starting on page 187 of A&W, “be sure you can do” # 1, 2 (note: is the c d A B AB BA ß œ ° “commutator” for ), 3, 7, (11), (13 ‘Pauli spin matrices’), (15), 20, 34, 36 (38 & 43 AB may also be interesting — look at them, at least). B. Compute where , and then find some nonzero matrices A A # œ # ‚ # α " # $ satisfying . (Compare with problem 3.2.6 on page 188 of A&W.) A 0 # œ C. Find the conjugate, the transpose, and the adjoint for each of the following matrices: A B C œ œ œ "3 #3 $ # ± $3 &3 $3 3 °#3 ) $ ° %3 # ± $3 % ° &3 ( ± )3 Ô × Õ Ø , , D. What should
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Unformatted text preview: and be so that + , # $ ± ²3 + , ” • 1. 2. is symmetric? is Hermitian (i.e., self adjoint)? E. Give an example of a anti-Hermitian matrix with no zero entries. # ‚ # F. Do NOT read §3.3 or §3.4 of A&W, but, starting on page 212, do 4, 5 and 6. G. Let and (assume the basis is orthonormal). Compute the l Ù œ l Ù œ + @ + @ + @ a v Ô × Ô × Õ Ø Õ Ø " " # # $ $ following (remember, scalars are complex) and, if possible, compare each with : Ø l Ù a v 1. 2. 3. trace Ø l Œ l Ù l Ù Œ Ø l Ø l Œ l Ù a v v a a v & ± Read § 4. and § 4.4 in the notes, doing any in-text or in-H. $ Elementary Matrix Theory lecture exercises. Also skim §3.1 of A&W. Starting on page 174 of A&W, do # 1 and 2 (rewrite the system in “matrix/vector form” first, and think about what the problem is really about). Then do #6b on page 188....
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