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

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Unformatted text preview: 1 Phys6103: Homework set #2 1. Levi-Civ` ıta Practice Evaluate the following expressions which exploit the Einstein summation convention. (a) δ ii (b) δ ij ² ijk (c) ² ijk ² ‘jk (d) ² ijk ² 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 σ x , σ y , and σ z used in quantum mechanics satisfy σ i σ j = δ ij + i² ijk σ k . If a and b are ordinary vectors, prove that ( σ · a )( σ · b ) = a · b + i σ · ( 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 δ ( ax ) = 1 | a | δ ( x ). (b) Use the identity in part (a) to prove that δ [ g ( x )] = X m 1 | g ( x m ) | δ ( x- x m ) where g ( x m ) = 0 ....
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This note was uploaded on 08/30/2011 for the course PHYS 6103 taught by Professor Grigoriev during the Summer '11 term at Georgia Tech.

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

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