*Problem 2.22 The gaussian wave packet. A free particle has the initial wave function (x, 0) = Ae-a.x² where A and a are constants (a is real and positive). (a) Normalize (.x. 0).
*Problem 2.22 The gaussian wave packet. A free particle has the initial wave function (x, 0) = Ae-a.x² where A and a are constants (a is real and positive). (a) Normalize (.x. 0).
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![*Problem 2.22 The gaussian wave packet. A free particle has the initial wave
function
(x, 0) = Ae¯
where A and a are constants (a is real and positive).
(a) Normalize (.r. 0).
(b) Find (x, 1). Hint: Integrals of the form
e-ax²
∞0+.
(x.t) =
can be handled by "completing the square": Let y = √a[x + (b/2a)], and
note that (ax² + bx) = y² – (b²/4a). Answer:
e-(ax²+bx) d.x
WE
2a 1/4-ax²/11+(2iħat/m)]
π
√1+(2iħat/m)
(c) Find |(x, t) |². Express your answer in terms of the quantity
.
a
1+ (2ħat/m)²
Sketch 2 (as a function of x) at t = 0, and again for some very large 1.
Qualitatively, what happens to |¥|², as time goes on?
(d) Find (.x), (p), (x²), (p²), σx, and σp. Partial answer: (p²) = aħ², but it
may take some algebra to reduce it to this simple form.
(e) Does the uncertainty principle hold? At what time does the system come
closest to the uncertainty limit?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00d97645-c8bf-4c97-b2bf-6271596bc983%2F0bdce5a4-952f-48a7-be4f-4cb677a9f6e2%2Fr3kzcjq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:*Problem 2.22 The gaussian wave packet. A free particle has the initial wave
function
(x, 0) = Ae¯
where A and a are constants (a is real and positive).
(a) Normalize (.r. 0).
(b) Find (x, 1). Hint: Integrals of the form
e-ax²
∞0+.
(x.t) =
can be handled by "completing the square": Let y = √a[x + (b/2a)], and
note that (ax² + bx) = y² – (b²/4a). Answer:
e-(ax²+bx) d.x
WE
2a 1/4-ax²/11+(2iħat/m)]
π
√1+(2iħat/m)
(c) Find |(x, t) |². Express your answer in terms of the quantity
.
a
1+ (2ħat/m)²
Sketch 2 (as a function of x) at t = 0, and again for some very large 1.
Qualitatively, what happens to |¥|², as time goes on?
(d) Find (.x), (p), (x²), (p²), σx, and σp. Partial answer: (p²) = aħ², but it
may take some algebra to reduce it to this simple form.
(e) Does the uncertainty principle hold? At what time does the system come
closest to the uncertainty limit?
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