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 10.4, Problem 10.13P
(a)
To determine
The value of
(b)
To determine
The value of
(c)
To determine
Show that the results are consistent with the answers to 10.4 in appropriate regime.
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Check out a sample textbook solutionStudents have asked these similar questions
Consider the "step" potential:
V(x) =
(a) Calculate the reflection coefficient, for the case E 0.
(b) Calculate the reflection coefficient for the case E > Vo.
(c) For a potential such as this, which does not go back to zero to the right of
the barrier, the transmission coefficient is not simply |F12/A2 (with A the
-Vo
AV(x)
Scattering from a "cliff"
incident amplitude and F the transmitted amplitude), because the transmitted
wave travels at a different speed. Show that
T =
E-Vo F1²
E |A|²
X
for E> Vo. Hint: You can figure it out using Equation
gantly, but less informatively-from the probability current (
What is T, for E Vo, calculate the transmission coefficient for the step potential, and
check that T + R = 1.
Consider the three-dimensional harmonic oscillator, for which the potential is
V ( r ) = 1/2 m ω2 r2
(a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies.
Answer: En = ( n + 3/2 ) ħ ω
(b) Determine the degeneracy d ( n ) of En
PROBLEM 3. Using the variational method, calculate the ground state en-
ergy Eo of a particle in the triangular potential:
U(r) = 0 r 0.
Use the trial function
v(x) = Cx exp(-ar),
where a is a variational parameter and C is a normalization constant to
be found. Compare your result for Eo with the exact solution, Eo
1.856(h? F/m)/3.
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