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 π 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 π / √ 2 π = e / √ 2 π , and the Euler formula verifies this: sines and cosines are the same if the angle changes by a whole multiple of 2 π . (For example, 2 π , 4 π , − 2 π , etcetera are physically all equivalent to a zero angle.) The derivative of e k i x / √ 2 π...
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- Fall '11
- mechanics, Eigenvalue, eigenvector and eigenspace, Orthogonal matrix, Eigenfunction, Inner product space, ekix elix