24. A particle is defined in space by [x, (] = A e-(x/a) el(Kox-wi) where a, ko and w are constants. (a) Find the normalization constant, A. Assume A is real. (b) What is the probability of finding the particle in the interval x and x + dx at time ? Does this value depend on the value of t? (c) Sketch the probability density as a function of x. (d) Determine the uncertainty in position i.e. V((x -(x))2) 25. Using the wavefunction defined in Prob. 24 (a) Determine the k-space wavefunction o[k] and verify that it is normalized. (b) Determine the average value of k using the momentum space wavefunction o[k]. (c) Determine the uncertainty in momentum i.e. ((k -(k> >2). (d) Do these wavefunctions satisfy the uncertainty relationship A x Apx > h/2 26. By using the operational definition of the delta function ff[x] [x] dx = f(0) where f[x] is a "well-behaved" function and the rules of integration, verify the following operational statements: (a) 6[-x] = 6[x] (b) blax] = -, 6[x] where a # 0 (c) 8'[-x] = -8'[x] (d) f[x] 8'[x] = -f'[x] [x] where S'[x] = do[x] d x (e) S[f [x]] = 6[x -Xol [of [x]/ax] Ix where the only zero of f[x] is at xo i.e. f[xo] = 0 Hint: Consider the Taylor expansion of f