HW2 - Quantum Mechanics I, Physics 3143: Assignment 2, Fall...

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Quantum Mechanics I, Physics 3143: Assignment 2, Fall 2010 1. To justify the energy eigenfunction expansion for the wave function of a particle in a box x [0 ,a ] it is useful to know that any square integrable function defined on the range x [ - a,a ] can be written as a Fourier series f ( x ) = X k = -∞ C k e iπkx/a 2 a | {z } e k ( x ) where ( e k ( x )) for k integer, is an orthonormal (Hilbert) basis for the square integrable functions. The coefficients are given by C k = Z a - a dx e - iπkx/a 2 a f ( x ) . By restricting attention to functions that are odd, f ( x ) = - f ( - x ) , show that the Fourier series reduces to f ( x ) = X n =1 A n r 2 a sin ( nπx/a ) | {z } u n ( x ) , for 0 < x < a , where A n = Z a 0 u * n ( x ) f ( x ) ≡ h u n | f i . Note that f ( x ) satisfies the boundary conditions for the particle in a box f (0) = f ( a ) = 0 . Hint : Use e ± = cosθ ± isinθ to rewrite the complex Fourier series in terms of cosines and sines, and then use the inversion symmetry
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HW2 - Quantum Mechanics I, Physics 3143: Assignment 2, Fall...

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