rev5 - and B are both Hermitian, then show that i( AB-BA )...

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Physics (PHZ) 3113 Florida Atlantic University Mathematical Physics Fall, 2009 Review Problems V Recommended Reading: Chow, pp. 100–121. 1. (Chow 3.4) Given the matrices A := 0 1 0 1 0 1 0 1 0 , B := 1 0 0 0 1 0 0 0 1 and C := 1 0 0 0 0 0 0 0 - 1 , show that A and B commute, that B and C commute, but that A and C do not. 2. (Chow 3.7) Show that the matrix A := 1 4 0 2 5 0 3 6 0 is not invertible. 3. (Chow 3.9) Find the inverse of the matrix A := 1 2 3 2 5 3 1 0 8 and check your result by direct calculation. 4. (based on Chow 3.15) Let A and B be an arbitrary matrices whose product AB exists. a. Show that the products AA and A A both exist and are Hermitian. Are these products equal to one another? b. Show that the product B A exists and is equal to ( AB ) . c. If A and B are both Hermitian, then show that AB + BA is Hermitian as well. d. If A
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Unformatted text preview: and B are both Hermitian, then show that i( AB-BA ) is Hermitian as well. 5. (based on Chow 3.19) The Pauli spin matrices 1 := 0 1 1 0 , 2 := -i i and 3 := 1-1 play an important role in quantum mechanics. a. Show that each of these matrices is both Hermitian and unitary. Calculate the inverse of each. b. Show that the product of two Pauli matrices is i j = ij I + i X k ijk k , where ij and ijk are the Kronecker and Levi-Civita symbols, respectively. c. Calculate the commutator [ i , j ] of two Pauli matrices. 6. (Chow 3.21) Let A , B and C be square matrices of the same dimension. Show that tr AB = tr BA and tr ABC = tr BCA = tr CAB . Is tr ACB = tr ABC ?...
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This note was uploaded on 10/21/2011 for the course PHZ 3113 taught by Professor Staff during the Fall '10 term at FAU.

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rev5 - and B are both Hermitian, then show that i( AB-BA )...

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