Problem 2.21 The gaussian wave packet. A free particle has the initial wave function Y (x, 0) = Ae¯ax². where A and a are (real and positive) constants. (a) Normalize 4 (x, 0). (b) Find Y (x, t). Hint: Integrals of the form e-(ar²+bx)dx can be handled by “completing the square": Let y = Ja [x + (b/2a)], and note that (a.x² +bx) = y² – (b²/4a). Answer: %3D 2a\/4 1 Y (x, t) = -ax?ly*, where y = /1+(2i hat/m). (2.111) (c) Find|4 (x, t)|². Express your answer in terms of the quantity w = Ja/[1+ (2ħat/m)*]. Sketch | y|2 (as a function of x ) at 1 = 0, and again for some very large t. Qualitatively, what happens to |², as time goes on? (d) Find (x), (p), (x2). (p²), a» and O p. Partial answer: (p²) = ah?, but take some algebra to reduce it to this simple form. it may (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit?

Problem 2.21 The gaussian wave packet. A free particle has the initial wave function Y (x, 0) = Ae¯ax². where A and a are (real and positive) constants. (a) Normalize 4 (x, 0). (b) Find Y (x, t). Hint: Integrals of the form e-(ar²+bx)dx can be handled by “completing the square": Let y = Ja [x + (b/2a)], and note that (a.x² +bx) = y² – (b²/4a). Answer: %3D 2a\/4 1 Y (x, t) = -ax?ly, where y = /1+(2i hat/m). (2.111) (c) Find|4 (x, t)|². Express your answer in terms of the quantity w = Ja/[1+ (2ħat/m)]. Sketch | y|2 (as a function of x ) at 1 = 0, and again for some very large t. Qualitatively, what happens to |², as time goes on? (d) Find (x), (p), (x2). (p²), a» and O p. Partial answer: (p²) = ah?, but take some algebra to reduce it to this simple form. it may (e) Does the uncertainty principle hold? At what time t does the system come closest to the uncertainty limit?