Find the first gap energy at ka = using (1.2.6) in the limit b 0 and U 0 b is small. You should

Find the first gap energy at ka = π using (1.2.6) in the limit b → 0 and U₀b is small. You should write approximations for sinKa and cos Ka near Ka = π, that is, Ka ≃ π + (ΔK)a, where ±K is small. You should find that you get an equation in terms of K that is factorizable into two terms that can equal zero. The difference between the energies E = h̄²K²/2m for these two solutions for K is the energy gap.

Do your zero-point energy and gap energy vanish in the limit U₀b → 0?

kb 2K sin(Ka) + cos(Ka) = cos (ka). (1.2.6)