A particle of mass M moves in a periodic potential with the form, V Vo [15 2 16 2πx 1 - cos + 16 : (1 6πx - COS where Vo is the depth of the lattice and d is a constant with units of length. The potential, represented in Figure 1, has minima at x = nd where n is an integer. V 0 d 2d 3d 4d 5d 6d x Figure 1: For use with Q11 In this question, you may find the following approximation useful: cos(0) 102/2, if is small. (a) Show that close to x = 0 this potential can be approximated by that of a harmonic oscillator. Derive its angular frequency w in terms of Vo, d and M and use this expresssion to determine the energy eigenfunction of the ground state for this potential. (b) Use the tight-binding approximation to write down energy eigenfunctions for a particle moving in this potential. Use Bloch's theorem to show that the probability density associated with these eigenfunctions has the periodicity of the lattice.

A particle of mass M moves in a periodic potential with the form, V Vo [15 2 16 2πx 1 - cos + 16 : (1 6πx - COS where Vo is the depth of the lattice and d is a constant with units of length. The potential, represented in Figure 1, has minima at x = nd where n is an integer. V 0 d 2d 3d 4d 5d 6d x Figure 1: For use with Q11 In this question, you may find the following approximation useful: cos(0) 102/2, if is small. (a) Show that close to x = 0 this potential can be approximated by that of a harmonic oscillator. Derive its angular frequency w in terms of Vo, d and M and use this expresssion to determine the energy eigenfunction of the ground state for this potential. (b) Use the tight-binding approximation to write down energy eigenfunctions for a particle moving in this potential. Use Bloch's theorem to show that the probability density associated with these eigenfunctions has the periodicity of the lattice.