The eigenvalues of the Hamiltonian are:
H∣ψ⟩=E∣ψ⟩for every eigenvector.
Write the Hamiltonian:
H=−2mℏ2dx2d2, substitute this to the first equation:
−2mℏ2dx2d2∣ψ⟩=E∣ψ⟩, the solution of this equation gives the set of eigenstates:
∣ψ⟩=C sin(aπnx). The corresponding energies (eigenvalues):
E=2man2π2ℏ2.
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