HW 3 problems/solutions

HW 3 - Homework 3 1 Homework III and Solutions Problems III.1 Show that the rotation(in spin or isospin space U = exp(iI with I =(I1,I2,I3 = i/2

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Homework 3 1 11/1/2005 Homework III and Solutions Problems: III.1 Show that the rotation (in spin or isospin space) U ( θ ) = exp( i θ · I ) with I = ( I 1 , I 2 , I 3 ) = σ i /2 can be written as: () cos (/2 ) s in ) Ui θ =+ θσ θ 1 Solution: First, note that: ( ) 33 3 22 11 2 2 ,1 1 1 3 2 1 2 32 1 1 21 3 3 13 2 1 1 jk j k j j jk j j k jk j k l l ii θθσσ θ σ θθ ε σ θθσ θθσ θθσ == = ≠= ⋅= + + = = + + + = ∑∑ θ 1 θ 1 θ 1 Taylor expansion of the exponential gives: 1 00 0 2 2 1 !2 2 !2 2 1!2 1 1 2!2 2 2 1 1 1 2 1 kk k k k i U k i i k + ∞∞ = ⎛⎞ =⋅ = + ⎜⎟ + ⎝⎠ −⋅− + + −⋅ + = θθ σ θ σ θ σ 1 θ 1 ± ± 1 1 cos sin ! 2 2 2 k k + +⋅ = ⎟⎜ + ⎠⎝ θ 1 III.2 Show that the matrix U ( θ ) = exp( i θ · σ /2) is unitary and that its determinant is unity. In your proof, be wary of the fact that matrices do not necessarily commute! Solution: First, note that det( U ( θ )) = exp{Tr( i θ · σ /2)} = exp{0} = 1 (the proof of Pietch’s Theorem can be found in many books on linear algebra). Unitarity of U ( θ ) means UU = 1 : †† exp exp i U U e e ⋅− ⋅ =− = = 1
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Homework 3 2 11/1/2005 III.3 Take the deuteron as an analog of a quark-quark system. For the strong interactions, the deuteron is simply a nucleon-nucleon system. The nucleon comes in two flavors: the proton p = |½,+½ , and the neutron n = |½,-½ .
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This note was uploaded on 02/22/2009 for the course PHYSICS 557 taught by Professor Michaelrijssenbeek during the Fall '05 term at SUNY Stony Brook.

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HW 3 - Homework 3 1 Homework III and Solutions Problems III.1 Show that the rotation(in spin or isospin space U = exp(iI with I =(I1,I2,I3 = i/2

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