Three Spins
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 62
22-8-2016
1245
Three Spins
Consider three particles of spin 1/2 which have no motion. The raising (s+ = sx + isy) and lowering (s- = sx - isy) operators of the individual spins have the property
(i)
(ii)
where the arrows indicate the spin orientation with regard to the z-direction.
a) Write explicit wave functions for the four J = 3/2 states: (M = 3/2, 1/2, -1/2, -3/2).
b) Using the definition that 
construct the 4 × 4 matrices which represent the J+ and J- operators.
c) Construct the 4 × 4 matrices which represent Jx and Jy.
d) Construct from Jx, Jy, Jz the value of the matrix J2.
SOLUTION
a) We use the notation that the state with three spins up is |↑1↑2↑3⟩. This is the state with M = 3/2. We operate on this with the lowering operator J-, which shows that the states |3/2, M⟩ with lower values of M are
(1)
(2)
(3)
(4)
b) From the definition of J+ we deduce that
(5)
(6)
(7)
(8)
The matrix J- is the Hermitian conjugate of J+:
(9)
c) Because Jx = (J+ + J-)/2 and Jy = -i (J+ - J-)/2 we can construct
(10)
(11)
(12)
d) To find the matrix J2 = J2x + J2y + J2z, we square each of the three matrices and add them. This gives J2 = 15Ĩ/4, where Ĩ is the 4 × 4 unit matrix. This is what one expects, since the eigenvalue of J2 is J(J + 1), which is 15/4 when J = 3/2.
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