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 9.2, Problem 9.3P
To determine
The approximate transmission probability for a particle of energy
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Problem 2.14 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) mo²a², where a
is the amplitude. So the “classically allowed region" for an oscillator of energy E extends
from –/2E/mw² to +/2E/mo². Look in a math table under “Normal Distribution" or
"Error Function" for the numerical value of the integral, or evaluate it by computer.
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 2.15 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.
Chapter 9 Solutions
Introduction To Quantum Mechanics
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