Chapter4_22 - PHY4604 R. D. Field Rotations in 3...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter4_22.doc University of Florida Rotations in 3 Dimensional Space (2) Rotations About the x-Axis, y-Axis, and z-Axis: It is easy to show that = ) cos( ) sin( 0 ) sin( ) cos( 0 0 0 1 ) ( x x x x x x R φ = ) cos( 0 ) sin( 0 1 0 ) sin( 0 ) cos( ) ( y y y y y y R = 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( ) ( z z z z z z R Arbitrary Rotation the 3-Space: The most general rotation in 3-space can be written as R( φ x , φ y , φ z ) = R x ( φ x )R y ( φ y )R z ( φ y ) ( i.e. three real parameters ). Infinitesimal Generators of O(3): Look at the infinitesimal transformation z z z z z z z z z z I I R δφ + = = = 1 0 0 0 1 0 1 1 0 0 0 ) cos( ) sin( 0 ) sin( ) cos( ) ( where I used 1 ) cos( 0  → ε and  → 0 ) sin( . Thus, r I I r R r z z z z r r r ) ( ) ( ' + = = and ) ( ) ( ) ( z z z z z z r I r r r r r = + . Hence r I d r d z z r r = and thus r e r z z I r r = ' where = 0 0 0 0 0 1 0 1 0 z I and = = 1 0 0 0 )
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

Ask a homework question - tutors are online