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22 lines
434 B
Groff
22 lines
434 B
Groff
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3.8
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a)
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check that the eigenvalues of the hermitian operator in example 3.1 are real. show that the eigenfunctions are orthogonal.
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Qf = if'
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the eigenvalues are 0,+- 1, etc., which are obviously real.
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pick two arbitrary eigenfunctions:
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f = A exp(-i q phi)
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g = A exp(-i q' phi)
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<f|g> = A*A int[exp(i q phi) exp(-i q' phi)] dphi[0,2pi]
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= A*A int[exp(i (q - q') phi)] dphi[0,2pi]
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= A*A [i(q-q')]^-1 [exp(i(q-q') phi)]|[0,2pi]
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