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# HW5 (2) - University of Colorado Department of Physics...

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University of Colorado, Department of Physics PHYS3220, Fall 09, HW#5 due Wed, Sep 23, 2PM at start of class 1. A few more properties of Hermitian operators (total: 10pts) a) Show that the sum of two Hermitian operators is a Hermitian operator. b) Suppose that ˆ A is a Hermitian operator and α is a number. Under what condition on α is α ˆ A a Hermitian operator? 2. A few useful properties of the Dirac delta function (total: 10pts) a) By multipying both sides of the following equations by a differentiable function f ( x ), and integrating over x , verify the following equations: δ ( x ) = δ ( - x ) (1) d dx δ ( x ) = - d dx δ ( - x ) (2) ( x ) = 0 (3) x d dx δ ( x ) = - δ ( x ) (4) b) Prove the following relations Z -∞ δ ( a - x ) δ ( x - b ) dx = δ ( a - b ) (5) f ( x ) δ ( x - a ) = f ( a ) δ ( x - a ) (6) 3. A few useful properties of Fourier transforms (total: 10pts) a) Proove Parseval’s theorem Z -∞ | f ( x ) | 2 dx = Z -∞ | g ( k ) | 2 dk, (7) for any regular function f ( x ) = 1 2 π Z -∞ g ( k ) exp( ikx ) dx with g ( k ) = 1 2 π Z -∞ f ( x ) exp( - ikx ) dx b) The convolution of two functions f 1 and f 2

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HW5 (2) - University of Colorado Department of Physics...

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