Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 10.3, Problem 10.5P
(a)
To determine
The reflected wave for the given potential.
(b)
To determine
The verification that the amplitude of the reflected wave is same as that of the incident wave.
(c)
To determine
The phase shift for a very deep well.
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Since the initial potential energy and final kinetic energy are zero, our equation now becomes
EP,f = EK,i.
We can then substitute the formula for kinetic energy,
EK = 1/2mv2,
and the formula for gravitational potential energy,
EP = mgh,
mghf = 1/2mvi2.
Now it's just a matter of doing the algebra, solving for the final height
hf,
and substituting values to find
hf.
Notice that the mass m divides out of both sides of the equation, so the value of the mass is not needed to find the final height.
Calculate the maximum height of the ball in meters.
hf = __________ m
Consider the equation for kinetic energy: KE = 1/2mv^2 = 1/2 * m * v^2. If I ask you to take the derivative of kinetic energy, you should ask "the derivative with respect to what?"
a) Suppose mass m is constant. Compute the derivative of KE with respect to v, (d(KE)/dv).
b) Who takes derivatives with respect to velocity? No one. Except you, just now. Sorry.
The rate of change of energy with respect to time is more important: it is the Power. Now, consider velocity v to be a function of time, v(t). We will rewrite KE showing this time dependance: KE= 1/2 * m * v(t)^2. Show that (d(KE)/dt) = F(t)v(t). Hint: use Newton's second law, F = ma, to simplify.
c) In the computation above, we assumed m was constant, and v was changing in time. Think of a physical situation in which both m and v are varying in time.
d) Compute the Power when both mass and velocity are changing in time. (First rewrite KE(t) showing time dependence, then compute (d(KE)/dt).
Particle A had 16 times the mass of particle B, but they both have the same value for the kinetic energy. What is the ratio of their speeds Vb/Va
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