Prove the statements (10.2.7) and (10.2.8), and that the other terms vanish as stated, by explicitly evaluating
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Prove the statements (10.2.7) and (10.2.8), and that the other terms vanish as stated, by explicitly evaluating the matrix elements, using the band functions in (1.13.9) and (1.13.10), and substituting Φx = x, Φy = y, Φz = z, and Φs = 1.
Show also that the term αn is real for these states, because the factor i that appears in (10.2.9) cancels an i that occurs in the matrix elements.
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