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Hard broglie wavelgnth and gas problem

Show that the de Broglie wavelength, λ, of a particle of mass m moving at a velocity of v where v<<c is related to the KE of the particle by: lambda=h/(2mEk)^0.5 [3] (ii) Consider the particle in a small rectangular box with sides of length a, b and c. The particle is moving at right angles to the “b-c” plane. Find an expression for the smallest possible energy of the particle. [3] (Think about the amplitude of the wave at the wall of the box) (iii) The box now contains many particles, and one “b-c” plane of the box is replaced by a piston. Show that as the length, a, is slowly decreased the resulting change in wavelength ensures that Boyle’s Law is obeyed