Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 3.5, Problem 3.19P
(a)
To determine
To show that
(b)
To determine
To prove any normalized wave packet denoting a particle in harmonic oscillator potential oscillates at classical frequency.
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Check out a sample textbook solutionStudents have asked these similar questions
Problem 2.15 In the ground state of the harmonic oscillator, what is the probability
(correct to three significant digits) of finding the particle outside the classically
allowed region? Hint: Classically, the energy of an oscillator is E = (1/2)ka² =
(1/2)mw²a², where a is the amplitude. So the "classically allowed region" for an
oscillator of energy E extends from -√2E/mw² to +√2E/mw². Look in a math
table under "Normal Distribution" or "Error Function" for the numerical value of
the integral.
Problem 1.17 A particle is represented (at time=0) by the wave function
A(a²-x²). if-a ≤ x ≤+a.
0,
otherwise.
4(x, 0) = {
(a) Determine the normalization constant A.
(b) What is the expectation value of x (at time t = 0)?
(c) What is the expectation value of p (at time t = 0)? (Note that you cannot
get it from p = md(x)/dt. Why not?)
(d) Find the expectation value of x².
(e) Find the expectation value of p².
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.…
Chapter 3 Solutions
Introduction To Quantum Mechanics
Ch. 3.1 - Prob. 3.1PCh. 3.1 - Prob. 3.2PCh. 3.2 - Prob. 3.3PCh. 3.2 - Prob. 3.4PCh. 3.2 - Prob. 3.5PCh. 3.2 - Prob. 3.6PCh. 3.3 - Prob. 3.7PCh. 3.3 - Prob. 3.8PCh. 3.3 - Prob. 3.9PCh. 3.3 - Prob. 3.10P
Ch. 3.4 - Prob. 3.11PCh. 3.4 - Prob. 3.12PCh. 3.4 - Prob. 3.13PCh. 3.5 - Prob. 3.14PCh. 3.5 - Prob. 3.15PCh. 3.5 - Prob. 3.16PCh. 3.5 - Prob. 3.17PCh. 3.5 - Prob. 3.18PCh. 3.5 - Prob. 3.19PCh. 3.5 - Prob. 3.20PCh. 3.5 - Prob. 3.21PCh. 3.5 - Prob. 3.22PCh. 3.6 - Prob. 3.23PCh. 3.6 - Prob. 3.24PCh. 3.6 - Prob. 3.25PCh. 3.6 - Prob. 3.26PCh. 3.6 - Prob. 3.27PCh. 3.6 - Prob. 3.28PCh. 3.6 - Prob. 3.29PCh. 3.6 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Prob. 3.32PCh. 3 - Prob. 3.33PCh. 3 - Prob. 3.34PCh. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - Prob. 3.40PCh. 3 - Prob. 3.41PCh. 3 - Prob. 3.42PCh. 3 - Prob. 3.43PCh. 3 - Prob. 3.44PCh. 3 - Prob. 3.45PCh. 3 - Prob. 3.47PCh. 3 - Prob. 3.48P
Knowledge Booster
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Partial answer: For Earrow_forwardProblem 2.11 (a) Compute (x). (p). (x²), and (p²), for the states yo (Equation 2.60) and 1 (Equation 2.63), by explicit integration. Comment: In this and other problems involving the harmonic oscillator it simplifies matters if you introduce the variable = √mo/hx and the constanta (m/h)¹/4 (b) Check the uncertainty principle for these states. (c) Compute (T) and (V) for these states. (No new integration allowed!) Is their sum what you would expect?arrow_forwardProblem 3.7 (a) Suppose that f(x) and g(x) are two eigenfunctions of an operator Q, with the same eigenvalue q. Show that any linear combination of f and g is itself an eigenfunction of Q. with eigenvalue q. (b) Check that f(x) = exp(x) and g(x) = exp(-x) are eigenfunctions of the operator d?/dx², with the same eigenvalue. 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