10. In 3-D, the Hamiltonian of the free wave (no potential energy) is: 2m2(x22+y22+z22) a. Show that writing the total wavefunction as: (x,y)=(x)+(y)+(z) is incorrect because it is not an eigenfunction of the Hamiltonian, i.e.: H(x,y,z)=E(x,y,z) where E=Ex+Ey+Ez. You can assume that: 2m2x22(x)=Ex(x),2m2y22(y)=Ey(y), and 2m2z22(z)=Ez(z) and Ex=Ey=Ez. Of course 2m2x22(y)=0, etc. (5pts) b. Now you can show that a proper wavefunction has the form: (x,y,z)=(x)(y)(z) because then H(x,y,z)=E(x,y,z) where E=Ex+Ey+Ez. (5 pts)