homework2 - Homework 2 PHZ 5156 Due Tuesday September 12 1...

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Unformatted text preview: Homework 2 PHZ 5156 Due Tuesday, September 12 1. a) Show that the basis functions e n ( x ) = 1 √ 2 e inπx with n = 0 , ± 1 , ± 2 , ... forms an orthonormal set using the definition for the inner product as, h m | n i = Z 1- 1 e * m ( x ) e n ( x ) dx In other words, show that h m | n i = δ m,n . b) This orthonormal basis is complete . This means we can use the basis to rep- resent any single-valued function with a finite number of discontinuities. For the function f ( x ) = exp ( x ) defined from- 1 ≤ x ≤ 1, find the coefficients f n in the expansion f ( x ) = ∞ X n =-∞ f n e n ( x ) c) A basis can also be used to represent an operator ˆ Ω as a matrix. The representation in the orthonormal basis above is given by Ω mn = h m | ˆ Ω | n i Determine the matrix representation of the operator ˆ Ω = ∂ 2 ∂x 2 . What operator in quantum mechanics is this related to? 2. Consider the Helmholtz equation d 2 f ( x ) dx 2 + k 2 f ( x ) = 0 a) Find all solutions f n ( x ) to the Helmholtz equation that satisfy the boundary...
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homework2 - Homework 2 PHZ 5156 Due Tuesday September 12 1...

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