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Eigenfunction and Momentum

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I have attached a sheet with two problems. One on eigenfunctions and one on momentum for the ground state.

1. Show that the first two eigenfunctions for the particle in the box problem are orthogonal, i.e. that

2. Compute the average value of the momentum for the ground state (lowest energy) wave-function for a particle of mass in a 1D box of length .

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This solution provides assistance with the problems regarding eigenfunctions and momentum for the ground state.