C-1 8. We proved the triangle. inequality for complex numbers in Problem Al'i. For vectors, the triangle inequality takes the form In + til 5 ml + IVI Prove this inequality by starting with |n + VIE = |n|2 + l'rl2 + 2n - v and then using the Schwartz inequality (previous problem} Why do you think this is called the triangle inequality? 4-1. Which of the following candidates for wave functions are normalizable over the indicated intervals? (a) e \"'2\"- (m. 00) (b) e\" (30. oo) (o) e'"EJ (0. 211') (d) cosh x (U. GO} to} Jute"J {0, 00) Normalize those that can be normalized. Are the others suitable wave fLutctions'l 410. Suppose that a particle in a two-dimensional box (of. Problem 48) is in the state 30 $0.} y) = W101 51y\") J?) Show that ttntx, y) is normalized, and then calculate the value of (E) associated with the state described by #10:, y}. 48. Consider a free particle constrained to move over the rectangular region 0 5 x 5 n, D 5 y 5 b. The energy eigenfunctions of this system are _ 4 W _ \"r3; . nun); nI =1, 2, 3.... Prob. 4-8 IS shown here also, .r)= () Sln - H511: - H =1 2 3 . _ of; a for Informatlon only: The Hamiltonian operator for this system is