Answer Happy

A quantum particle is confined to move along a ring of circumference a. We fix a point on the ring and take r to be the distance to the particle as we travel clock- wise around the ring from that point. Note that 0 <r<a. We take the particle to be otherwise free, so that its Hamiltonian is h2 22 Ĥ 2m 8.r2 a) From Fourier analysis, we know that every function y(t) on the ring can be written as 4(x) = aoUo+ Ě (, U.(x) + b, V.(z)), where a, and the an, bn are complex numbers, and 1 U.(2) va 2 U (0) nao = = 2ппах, COS a a 2 2пnt, Vn() = sin a Show that the functions U.(I), U. (1), Vn() give the energy modes and de- termined the corresponding energy eigenvalues. Identify the ground state energy and comment on any degeneracy of the energy eigenvalues. b) The particle may propagate as a plane wave y(x) = Aeikt around the ring. Determine the allowed values of k and for each such value write the plane wave as a linear superposition of the energy modes. Conversely, give a description of the positive energy modes in terms of propagating waves.