Help please (concerning eigenfunctions and the Schrödinger equation)

[medofx]

Homework Statement

in the attachment

Relevant Equations

in the attachment

i have an exam in 2 days and in this question i don't know how should i proceed after that i simplified the wave function but i don't know how to confirm that it's an eigenfunction

 

[berkeman]

Welcome to the PF.

Your attachment with your work is very hard to read. Can you make a higher-quality scan of it and attach that? Better yet would be for you to type your work into the forum Edit window using LaTeX -- there is a tutorial on LaTeX at the Help pages (see INFO at the top of the page). Thanks

 

[medofx]

hopefully this is better

 

[kuruman]

hopefully this is better

Your attempt at a solution is not relevant to the question. To show that the given wavefunction is an eigenfunction, you need to operate on it with the Hamiltonian and see whether you get back the same wavefunction times a constant, in this case one of the energies $E_n$.