*Problem 2.22 The gaussian wave packet. A free particle has the initial wav function (x, 0) = Ae-ar² where A and a are constants (a is real and positive). (a) Normalize (x. 0). K
*Problem 2.22 The gaussian wave packet. A free particle has the initial wav function (x, 0) = Ae-ar² where A and a are constants (a is real and positive). (a) Normalize (x. 0). K
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![*Problem 2.22 The gaussian wave packet. A free particle has the initial wav
function
(x, 0) = Ae-ax²
where A and a are constants (a is real and positive).
(a) Normalize (.x. 0).
(b) Find (x, 1). Hint: Integrals of the form
Ste-(ax²+bx) dx
can be handled by "completing the square": Let y = √a [x + (b/2a)], and
note that (ax² + bx) = y₁² — (b²/4a). Answer:
1/4-ax²/11+(2ihat/m)}
(²)*
√1+ (2iħat/m)
(c) Find |(x, 7)|². Express your answer in terms of the quantity
(.x. 7) =
w=
e
a
V1+ (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?
K](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00d97645-c8bf-4c97-b2bf-6271596bc983%2Ff0bf2b1a-40ed-4309-bc0f-acff836cc595%2F8u8zbvb_processed.png&w=3840&q=75)
Transcribed Image Text:*Problem 2.22 The gaussian wave packet. A free particle has the initial wav
function
(x, 0) = Ae-ax²
where A and a are constants (a is real and positive).
(a) Normalize (.x. 0).
(b) Find (x, 1). Hint: Integrals of the form
Ste-(ax²+bx) dx
can be handled by "completing the square": Let y = √a [x + (b/2a)], and
note that (ax² + bx) = y₁² — (b²/4a). Answer:
1/4-ax²/11+(2ihat/m)}
(²)*
√1+ (2iħat/m)
(c) Find |(x, 7)|². Express your answer in terms of the quantity
(.x. 7) =
w=
e
a
V1+ (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?
K
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