hw2 - 1 Phys6103: Homework set #2 1. Levi-Civ` ta Practice...

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Unformatted text preview: 1 Phys6103: Homework set #2 1. Levi-Civ` ta Practice Evaluate the following expressions which exploit the Einstein summation convention. (a) ii (b) ij ijk (c) ijk jk (d) ijk ijk 2. Vector Identities Use the Levi-Civit`a symbol to prove that (a) ( A B ) ( C D ) = ( A C )( B D )- ( A D )( B C ) (b) ( f g ) = g ( f )- f ( g ) (c) ( A B ) ( C D ) = ( A C D ) B- ( B C D ) A (d) The 2 2 Pauli matrices x , y , and z used in quantum mechanics satisfy i j = ij + i ijk k . If a and b are ordinary vectors, prove that ( a )( b ) = a b + i ( a b ) . 3. Delta Function Identities A test function as part of the integrand is required to prove any delta function identity. With this in mind, (a) Prove that ( ax ) = 1 | a | ( x ). (b) Use the identity in part (a) to prove that [ g ( x )] = X m 1 | g ( x m ) | ( x- x m ) where g ( x m ) = 0 ....
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hw2 - 1 Phys6103: Homework set #2 1. Levi-Civ` ta Practice...

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