Chapter4_13 - ) j m values Number of States (2j+1) 0 0 1 (a...

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PHY4604 R. D. Field Department of Physics Chapter4_13.doc University of Florida The Spectrum of States We see that <jm|(J - ) op (J + ) op |jm> = j(j+1)-m(m+1) = |c + | 2 0 <jm|(J + ) op (J - ) op |jm> = j(j+1)-m(m-1) = |c - | 2 0 and hence > ± ± + >= ± 1 | ) 1 ( ) 1 ( | ) ( jm m m j j jm J op . Also, 0 <jm|(J x ) 2 op |jm> + <jm|(J y ) 2 op |jm> = <jm|(J 2 ) op -(J z ) 2 op |jm> = j(j+1)-m 2 . and hence m 2 j(j+1) and j(j+1) 0. In addition, j(j+1)-m(m+1) 0 implies that m max = j j(j+1)-m(m-1) 0 implies that m min = -j We know that (J + ) op |jm max > = 0 and (J - ) op |jm min > = 0 so start at m min = -j and start applying the raising operator (J + ) op . It takes 2j+1 steps to get to m max = +j and each step is an integer ( i.e. +1) and hence 2j+1 = an integer . This means that j = 0, 1, 2,. .. ( positive integers ) or L , , , 2 5 2 3 2 1 = j ( positive half-integers
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Unformatted text preview: ) j m values Number of States (2j+1) 0 0 1 (a 1 of SU2)) 2 1 2 1 2 1 , + 2 (a 2 of SU2)) 1 -1,0,1 3 (a 3 of SU2)) 2 3 2 3 2 1 2 1 2 3 , , , + + 4 (a 4 of SU2)) 2 -2,-1,0,1,2 5 (a 5 of SU2)) Orbital Angular Momentum: Note that for the orbital angular momentum op op p r L r r r = , the allowed eigenvalues of the operator (L 2 ) op are integral, l = 0, 1, 2, . .. . The half-integral eigenvalues correspond to something else! m min =-j J + m = -j+1 J + m = -j+2 m max = +j m = j-1 J + 2j+1 steps Refer to the set of states with a given j by the number of states ( i.e. 2j+1)!...
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