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 9.1, Problem 9.2P
(a)
To determine
Show that
(b)
To determine
Show that
(c)
To determine
The expression for
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Students have asked these similar questions
A quantum mechanical particle is confined to a one-dimensional infinite potential well
described by the function V(x) = 0 in the region 0 < x < L, V(x) = ∞ elsewhere.
The normalised eigenfunctions for a particle moving in this potential are:
Yn(x)
=
√
2
Nπ
sin -X
L L
where n = 1, 2, 3, ..
a) Write down the expression for the corresponding probability density function. Sketch
the shape of this function for a particle in the ground state (n = 1).
b) Annotate your sketch to show the probability density function for a classical particle
moving at constant speed in the well. Give a short justification for the shape of your
sketch.
c) Briefly describe, with the aid of a sketch or otherwise, the way in which the quantum
and the classical probability density functions are consistent with the correspondence
principle for large values of n.
Start by defining
1(1) = N1 sin(7r/a)
(1)
b2(x) = N2 sin(2ñr/a)
(2)
for the infinite square well. Fix N1 and N2 so that
%3D
2)
You should find that p(r) is periodic in time. That is p(x, t + T) = p(x,t). Find
that T, and draw p(x) for at t = 0, t = T/4, t = T/2, and T = 3T/4.
A ID harmonic oscillator of angular frequency w and charge q is in its ground state at
time t=0. A perturbation H'(t) = qE eA3 (where E is ekctric field and ß is a constant) is
%3D
applied for a time t = t. Cakulate the probability of transition to the first and second
excited state. (hint: you may expand exponential in perturbation and keep it only up to
linear term)
Chapter 9 Solutions
Introduction To Quantum Mechanics
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