For the Gaussian wave packet (6.62). (pl(0)) is given by (6.68) and thus (x, t) is given by ¥ (x, t) = (x|\/(t)) = [dpe=ip²1/2mh (x\p)(p(0)) (6.75) If you are proficient at carrying out Gaussian integrals (see Appendix D), it is straightforward to show that 1 √(x,t) = e-x²/12a211+(iht/ma²)]} √√la+(iht/ma)] (6.76) Comparing *(x, t) (x, t) with its form (6.63) at r = 0, we see that the position uncertainty is given by 12,2 1/2 √(1+1)" Ax= 2 m2a4 (6.77) 6.4. (a) Show for a free particle of mass m initially in the state 1 v(x)=(x|y) √(x) = (xy) = -x²/2a² πα that 1 (x,t) = (x(t)) = e-x²/{2a² 11+ (iht/ma²] √√√[a+(iht/ma)] and therefore ht Ax 1+ √2 ma2 Page 254 (metric system) Problems 239 Suggestion: Start with (6.75) and take advantage of the Gaussian integral (D.7), but in momentum space instead of position space. (b) What is Ap, at time t? Suggestion: Use the momentum-space wave function to evaluate Ap

For the Gaussian wave packet (6.62). (pl(0)) is given by (6.68) and thus (x, t) is given by ¥ (x, t) = (x|\/(t)) = [dpe=ip²1/2mh (x\p)(p(0)) (6.75) If you are proficient at carrying out Gaussian integrals (see Appendix D), it is straightforward to show that 1 √(x,t) = e-x²/12a211+(iht/ma²)]} √√la+(iht/ma)] (6.76) Comparing *(x, t) (x, t) with its form (6.63) at r = 0, we see that the position uncertainty is given by 12,2 1/2 √(1+1)" Ax= 2 m2a4 (6.77) 6.4. (a) Show for a free particle of mass m initially in the state 1 v(x)=(x|y) √(x) = (xy) = -x²/2a² πα that 1 (x,t) = (x(t)) = e-x²/{2a² 11+ (iht/ma²] √√√[a+(iht/ma)] and therefore ht Ax 1+ √2 ma2 Page 254 (metric system) Problems 239 Suggestion: Start with (6.75) and take advantage of the Gaussian integral (D.7), but in momentum space instead of position space. (b) What is Ap, at time t? Suggestion: Use the momentum-space wave function to evaluate Ap