Homework 3

# Homework 3 -...

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Unformatted text preview: Physics 601 Homework 3­­­Due Friday September 24      Goldstein problems  8.1,  8.6, 8.9, 8.26, 8.23    In addition:    1. For a particle in 3 dimensions the angular momentum operator is given by  L = x × p  where  x  and  p  satisfy are canonical (i.e. they satisfy the canonical  Poisson bracket relations.  a. Show that  [ Li , L j ]PB = εijk Lk  where i,j,k  take on the values x,y,z.  (Note  that this is isomorphic to the  commutators for the angular  € € € momentum in quantum mechanics.  For those with mathematically  inclinations, this is the Lie algebra SO(3).)   € b. Show that  [ L, L2 ]PB = 0   (that is that [ Li , L2 ]PB = 0  for all i) where  L2 = L ⋅ L  .  c. Show that  [ L, f ( r)]PB = 0   where f is an arbitrary function and  r€ x ⋅ x .  = € Note that parts b. &c. reflect a deeper result  [ L, s]PB = 0  for any scalar s.  This  € reflects the fact that the angular momentum is the generator of rotations.     €  € 2. Label our canonical variables by a phase‐space vector     €   € q1 q2 ... qn η=  which satisfies canonical Poisson brackets  [ηi ,η j ]PB = J ij   p1 p2 ... € pn1   now suppose that  there is a one parameter continuous family of   canonical transformations depending on one parameter ε :  ξ (η;ε)   . a.  Show that  ξ (η;ε)   satisfies  [ξ i ,ξ j ]PB = J ij  if it satisfies the conditions   dξ i = [ξ i , g]PB with ξ (η;0) = η  for some  g.  Note that this is the same  dε € form as usual Hamiltonian time  with t replaced by ε and H replaced  € € by g.  The function g is called the generator of the transformation.       € € b. The time evolution of the phase‐space position under Hamiltonian  gives a transformation from an initial point in phase space to a      subsequent one.  That is  η( t ) is really a function the time and the  initial conditions:  η(η0 , t ) .  Note that the initial conditions are a set of  phase space variable which are canonical.  Now let me define a  canonical transformation  ξ (η, T )  which has the same functional  € relation as   η(η0 , t ) .  That is  ξ (η, T ) corresponds to the value the point  € in phase space that any point in phase space evolves into from  η  a  time T later.  Using part a) show that for any T ,  η(ξ , T )   satisfies the  € canonical Poission bracket relation with H acting as the generator of  € € time translations.  € €         ...
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## This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.

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