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Eigenvalues and Hamiltonian H

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I am confused on how to approach this problem on eigenfunctions and eigenvalues with the Hamiltonian H. Would you please include the work and an explanation, please.

Please see the attachment for the full problem.

Consider the particle in a box problem for a box of length ...
Verify the substitution that the solutions ...
Are indeed eigenfunctions of the Hamiltonian ... with eigenvalues ...?

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The solution discusses questions regarding eigenvalues and Hamiltonian H.

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