*Problem 2.22 The gaussian wave packet. A free particle has the initial w function (x, 0) = Ae-ar² where A and a are constants (a is real and positive). (a) Normalize (.x. 0). (b) Find (x, 1). Hint: Integrals of the form +x [to ₂-(ax²+bx) dx can be handled by "completing the square": Let y = √a [x + (b/2a)], note that (ax² + bx) = y² - (b²/4a). Answer: (2) 1/4-ax²/11+(2iħat/m)]
*Problem 2.22 The gaussian wave packet. A free particle has the initial w function (x, 0) = Ae-ar² where A and a are constants (a is real and positive). (a) Normalize (.x. 0). (b) Find (x, 1). Hint: Integrals of the form +x [to ₂-(ax²+bx) dx can be handled by "completing the square": Let y = √a [x + (b/2a)], note that (ax² + bx) = y² - (b²/4a). Answer: (2) 1/4-ax²/11+(2iħat/m)]
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![*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).
(b) Find (x, 1). Hint: Integrals of the form
+∞
[+0° e-(ax²+bx)
can be handled by "completing the square": Let y = √a [x + (b/2a)], and
note that (ax² + bx) = y² - (b²/4a). Answer:
e
(.x. 1) =
2a 1/4-ax²/11+(2ihat/m)]
√1 + (2iħat/m)
(c) Find (x. 1)². Express your answer in terms of the quantity
ພ =
d.x
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 1², as time goes on?
(d) Find (v), (p), (x²), (p²), ox, and op. 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%2F668ff5f2-d557-46a3-9b4c-7a2704e0b63a%2F0bcf35d5-c5b9-40e1-816f-e542683c23c4%2Ftil27mq_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-a.x²
where A and a are constants (a is real and positive).
(a) Normalize (.x. 0).
(b) Find (x, 1). Hint: Integrals of the form
+∞
[+0° e-(ax²+bx)
can be handled by "completing the square": Let y = √a [x + (b/2a)], and
note that (ax² + bx) = y² - (b²/4a). Answer:
e
(.x. 1) =
2a 1/4-ax²/11+(2ihat/m)]
√1 + (2iħat/m)
(c) Find (x. 1)². Express your answer in terms of the quantity
ພ =
d.x
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 1², as time goes on?
(d) Find (v), (p), (x²), (p²), ox, and op. 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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