dimensions are described by 3 × 3 orthogonal matrices with positive determinant, which is what SO(3) means. Conversely, our spin-1/2 rotation operators ˆ R φ,k are 2 × 2 unitary ma- trices with positive determinant, which is what SU(2) means. It is interesting that SU(2) being a two-fold cover of SO(3) (requiring two complete turns to get back to where you started) has nothing to do with quantum mechanics, but rather is built into the structure of rotations in three dimensions. See for a nice discussion of this. Mathematicians figured out all this stuff before Planck came up with his constant in 1900, setting in motion the discovery of quantum mechanics. 3
(b) Show that [ ˆ A, ˆ B ] = 0 and [ ˆ B, ˆ C ] = 0 but [ ˆ A, ˆ C ] 6 = 0. This means that A and B are compatible observables and also B and C are compatible observables, but A and C are incompatible observ- ables. Find a basis (two normalized and mutually orthogonal spinors) which is composed of eigenspinors of both ˆ B and ˆ C , and show that these basis vectors are not eigenspinors of ˆ A . (c) The basis vectors found in part 5b are states of definite B and C , but not states of definite A . Show this explicitly by computing the uncertainties Δ A , Δ B , Δ C in each of the two basis states. Use h A 2 i = h e k | ˆ A 2 | e k i , etc. 4
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