Consider a free particle constrained to move over the rectangular region 0≤ x ≤a, 0≤ y ≤ b.

The energy eigenfunctions, i.e. wavefunctions, of this system are:

The Hamiltonian for the system is:

a. Show that if the system is in one of its eigenstates, then ⟨E² ⟩-⟨E⟩² = 0

b. The energy equation for this system is

Construct an energy level diagram (include the first five energy levels) if the particle is constrained to a square (a=b). Don't forget degeneracy.

1/2 sin 4 sin b ; where n, = 1,2,3,... and ny = 1,2,3,... %3D a