Answer Happy

3. Properties of energy eigenstates. a) (5 points) Prove the following: If u and I are two normalized eigenfunctions of a Hamiltonian, ie.. Hum = Emlm and Hun= En Un and Em En, then fox,Aqdır = 8mm (1) This shows in general, solutions to Schrodinger's equation for different energy eigenvalues are orthonormal. b) (5 points) Prove the following: We can always choose the energy eigenstates e(x) we work with to be purely real functions (unlike the physical wave function (3,1), which is necessarily complex). (Hint: If og(x) is an energy eigenstate with energy eigenvalue E, what can be said about 7(x)?) c) (5 points) Prove the following: If V(x) is an even function i.e. V(-x) = V(x)), then the energy eigenfunctions of(x) can always be taken to be either even or odd. (Hint: If oe(r) is an energy eigenstate with energy E, what can be said about cel-)?)