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Unformatted text preview: 18 CHAPTER 2. MATHEMATICAL PREREQUISITES Check that these functions are indeed periodic, orthonormal, and that they are eigenfunctions of id / d x with the real eigenvalues . . . , 3 , 2 , 1 , , 1 , 2 , 3 , . . . Completeness is a much more difficult thing to prove, but they are. The completeness proof in the notes covers this case. Answer: Any eigenfunction of the above list can be written in the generic form e k i x / 2 where k is a whole number, in other words where k is an integer, one of ..., 3, 2, 1, 0, 1, 2, 3, ... If you show that the stated properties are true for this generic form, it means that they are true for every eigenfunction. Now periodicity requires that e k i2 / 2 = e / 2 , and the Euler formula verifies this: sines and cosines are the same if the angle changes by a whole multiple of 2 . (For example, 2 , 4 , 2 , etcetera are physically all equivalent to a zero angle.) The derivative of e k i x / 2...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.
- Fall '11