1. (a) Consider a particle on a one dimensional ring of perimeter L. (i) Calculate the transition

amplitude K(at, r't') = (rt; r't') using the path integral formalism. (ii) Verify that the

transition amplitude obeys the Schrodinger equation. (iii)Find the energy spectrum by taking

the Fourier transform of K(at, r't' = 0)|72, with respect to time.

Hint: Here you will do the integration over the paths for -co < I < co.i =2,..., N -1, where the

initial point is a fired point x1 = c', ti = t'. The particle can reach the final point with a winding

number 0, 41, 4:2, ..., thus the final position can take the values on = xx _ L,c + 2L, ... Here

winding number is the number of turns in counterclockwise direction minus the number of turns in

clockwise direction around the circle.

(b) Find the energy spectrum and the propagator K(rt. r't ) using the Schrodinger picture.

. Compare your result with part (a).

(c) Assume that the particle is initially localized in one half of the ring v(0 < z < L/2) =

(2/L) and v(L/2 < < < L) = 0. Find the probability to find the particle in 0 < r < L/2

interval at some later time t.