Problem 3 The harmonic oscillator eigenfunctions have the following properties 2 a (Vn + lyn+1 + Vnin-1} a. Using only the following eigenfunctions of the harmonic oscillator yo= 4 ax2 403 ax2 V1 = xe 2 TC 1/2 = ax2 (2ax2 - 1)e 2 in the appropriate manner, demonstrate the above expression. b. Using the same notation as the Xun, write the expression for X21 c. Using that property derived in b, show that for any eigenfunction of the harmonic oscillator, the average value of the displacement square is non-zero (x2 ) = un(x)x24n(x) dx # 0