3-16. Show that (p) = 0 for all states of a one-dimensional box of length a. 3-21. Using the trigonometric identity sin a sin = 1 cos(- B) - cos(a + B) show that the particle-in-a-box wave functions (Equations 3.27) satisfy the relation S" & " (x) Vm (x) dx = 0 m #n (The asterisk in this case is superfluous because the functions are real.) If a set of functions satisfies the above integral condition, we say that the set is orthogonal and, in particular, that (x) is orthogonal to y,(x). If, in addition, the functions are normalized, then we say that the set is orthonormal. 3-28. Consider a particle of mass m in a one-dimensional box of length a. Its average energy is given by 1 (E) =1 -(p) 2m Because (p) = 0, (p) =2, where p can be called the uncertainty in p. Using the uncertainty principle, show that the energy must be at least as large as h/8ma because x, the uncertainty in x, cannot be larger than a.