Problem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: A, if -a < x < a, V (x, 0) = 0, otherwise, where A and a are positive real constants. p. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine ø(k) (Equation 2.86). (c) Comment on the behavior of o(k) for very small and very large values of a. How does this relate to the uncertainty principle?
Problem 2.21 Suppose a free particle, which is initially localized in the range -a < x < a, is released at time t = 0: A, if -a < x < a, V (x, 0) = 0, otherwise, where A and a are positive real constants. p. 2 The Time-Independent Schrödinger Equation (a) Determine A, by normalizing V. (b) Determine ø(k) (Equation 2.86). (c) Comment on the behavior of o(k) for very small and very large values of a. How does this relate to the uncertainty principle?
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![Problem 2.21 Suppose a free particle, which is initially localized in the range
-a < x < a, is released at time t = 0:
А, if -a < х <а,
otherwise,
(x, 0) =
where A and a are positive real constants.
50
Chap. 2 The Time-Independent Schrödinger Equation
(a) Determine A, by normalizing V.
(b) Determine (k) (Equation 2.86).
(c) Comment on the behavior of (k) for very small and very large values of a.
How does this relate to the uncertainty principle?
*Problem 2.22 A free particle has the initial wave function
(x, 0) =
Ae ax
where A and a are constants (a is real and positive).
(a) Normalize (x, 0).
(b) Find V(x, t). Hint: Integrals of the form
e-(ax?+bx)
dx
can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note
that (ax? + bx) = y? – (b²/4a). Answer:
1/4 e-ax?/[1+(2ihat/m)]
2a
Y (x, t) =
VI+ (2iħat/m)
(c) Find |4(x, t)2. Express your answer in terms of the quantity w
Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again
for some very large t. Qualitatively, what happens to |1² as time goes on?
(d) Find (x), (p), (x2), (p²), ox, and op. Partial answer: (p) = aħ?, but it may
take some algebra to reduce it to this simple form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80d5f871-429b-41ff-a7f8-0e2603cbce95%2Fbe274f06-0f35-4fa6-a383-7d544bf9a20f%2F5t4gf9p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 2.21 Suppose a free particle, which is initially localized in the range
-a < x < a, is released at time t = 0:
А, if -a < х <а,
otherwise,
(x, 0) =
where A and a are positive real constants.
50
Chap. 2 The Time-Independent Schrödinger Equation
(a) Determine A, by normalizing V.
(b) Determine (k) (Equation 2.86).
(c) Comment on the behavior of (k) for very small and very large values of a.
How does this relate to the uncertainty principle?
*Problem 2.22 A free particle has the initial wave function
(x, 0) =
Ae ax
where A and a are constants (a is real and positive).
(a) Normalize (x, 0).
(b) Find V(x, t). Hint: Integrals of the form
e-(ax?+bx)
dx
can be handled by "completing the square." Let y = Ja[x+(b/2a)], and note
that (ax? + bx) = y? – (b²/4a). Answer:
1/4 e-ax?/[1+(2ihat/m)]
2a
Y (x, t) =
VI+ (2iħat/m)
(c) Find |4(x, t)2. Express your answer in terms of the quantity w
Va/[1+ (2hat/m)²]. Sketch |V|? (as a function of x) at t = 0, and again
for some very large t. Qualitatively, what happens to |1² as time goes on?
(d) Find (x), (p), (x2), (p²), ox, and op. Partial answer: (p) = aħ?, but it may
take some algebra to reduce it to this simple form.
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