Question

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 1

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 )