1. Consider a one-particle with mass m, one-dimensional system with length l and potential energy V=V0for41lx43l,V=0for0x41land43lxl and V= elsewhere, where V0=ml22 as shown in Figure 1. (a) Use the trial variation function 1=(l2)1/2sin(lx) for 0xl to estimate the ground-state energy. Let your answer be in h,m and l. Compare with the true ground-state energy E=5.750345ml22 and comment. Hint: To save time in evaluation integrals, note that nAn=nT^n+nP^n and (nTn) equals the particle-in-a-box ground-state energy n^n=ml2n2h2. (b) Consider the above one-particle system in Figure 1. Treat the system as a perturbed particle in a box. Find the first-order energy correction for the groundstate energy E(1) and then E(0)+E(1), Let yout answer be in h,m and l. (c) Explain why E(0)+E(1) is the same as obtained by the variational treatment in 1(a). [Hint: Use standard integrals]