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solutions 06

# Introductory Nuclear Physics

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PHY431 Homework Set 6 Reading: Lecture Notes Homework: See below: Due date: Wednesday March 8 Hints and Solutions Problem VI.1 Using the isospin-lowering operator I - , and starting at the "fully stretched" state ++ = | uuu , derive the representation of the three other states in terms of quark states. Hints: no hints Solution: I - ++ = I - | uuu = N {| duu +| udu +| uud } = + , with N a normalization constant. Clearly N = 1/ 3 here. I - + = N {| ddu +| dud +| udd } = 0 , with again N = 1/ 3 . I - 0 = N {| ddd } = - , with N = 1 . Problem VI.2 Calculate the Isospin-3 ( I 3 ) eigenvalues (of the I 3 operator) of the bottom and the top quarks. Hints: no hints Solution: I 3 ( b ) = Q - 1 / 2 (B+S+C+B+T) = - 1 / 3 - 1 / 2 ( 1 / 3 +0+0-1+0) = 0 I 3 ( t ) = Q - 1 / 2 (B+S+C+B+T) = + 2 / 3 - 1 / 2 ( 1 / 3 +0+0+0+1) = 0, as expected for any quark not a u or d quark. Problem VI.3 Show that the states | π 0 and | η are othonormal, assuming that the |

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