Why is the expectation value of the energy associated with the 1-D "particle-in-a-box" the same as the eigen value of the Hamiltonian associated with the 1-D "particle-in-a-box" wave function?

Answer: The average potential energy of the PIB is 0 irrespective of the wave function.

Explanation:

⟨H⟩=⟨KE⟩+⟨V⟩

the nn quantum number

⟨KE⟩ = (π^2 ℏ^2) / (2mL^2)

the average kinetic energy of the wavefunction is dependent on

⟨V⟩=∫sin (kx) 0sin (kx) dx=0

The average potential energy of the PIB is 0 irrespective of the wave function.

⟨H⟩=⟨KE⟩ = (π^2 ℏ^2) / (2mL^2)