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 7, Problem 7.39P
(a)
To determine
The exact eigenvalues for the perturbed Hamiltonian
(b)
To determine
The expansion of the results from part (a) as a power series in
(c)
To determine
The results obtained by applying non-degenerate perturbation theory to find the energies of all three states.
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The Hamiltonian of a certain system is given by
0 0
H = ħw|0 _0 0
0 1
1
Two other observables A and B are represented by
A = a|-i 0 0
0 0 1
[1 0 0
B = b|0 _2 0
Lo o 2
w, a, b are positive constant.
a. Find the eigenvalues and normalized eigenvectors of H
b. Suppose the system is initially in the state
2c
ly(0) >=
2c
where c is a real constant. Determine the normalized state |4(t) >.
c. What are the eigenvectors of B?
d. Find the expectation values of A and B in the state [Þ(t) >, and hence determine if A and B are
conservative observables
6. Consider a mass m which is constrained to move on the frictionless surface of a vertical cone
(Fig. 1) (in cylindrical polar coordinates p. p, z with z>0) in a uniform gravitational field g
vertically down. Find Hamiltonian function of the cone and deduce Hamilton's equations.
z
SohCahtoa
sing
cos
r = pcosd + psind
Fig. 1
The Hamiltonian of a certain system is given by
[1
H = ħw|0 0 0
Lo o 1
Two other observables A and B are represented by
i 0
A = a|-i 00
0 1
0 0
[1
B = b|0
2 0
0 2
w, a, b are positive constant.
Suppose the system is initially in the state
2c
lf(0) >=
-c
2c
where c is a real constant. Determine the normalized state |4(t) >.
What are the eigenvectors of B?
Find the expectation values of A and B in the state |(t) >, and hence determine if A and B are
conservative observables
Chapter 7 Solutions
Introduction To Quantum Mechanics
Ch. 7.1 - Prob. 7.1PCh. 7.1 - Prob. 7.2PCh. 7.1 - Prob. 7.3PCh. 7.1 - Prob. 7.4PCh. 7.1 - Prob. 7.5PCh. 7.1 - Prob. 7.6PCh. 7.2 - Prob. 7.8PCh. 7.2 - Prob. 7.9PCh. 7.2 - Prob. 7.10PCh. 7.2 - Prob. 7.11P
Ch. 7.2 - Prob. 7.12PCh. 7.2 - Prob. 7.13PCh. 7.3 - Prob. 7.15PCh. 7.3 - Prob. 7.16PCh. 7.3 - Prob. 7.17PCh. 7.3 - Prob. 7.18PCh. 7.3 - Prob. 7.19PCh. 7.3 - Prob. 7.20PCh. 7.3 - Prob. 7.21PCh. 7.3 - Prob. 7.22PCh. 7.4 - Prob. 7.23PCh. 7.4 - Prob. 7.24PCh. 7.4 - Prob. 7.25PCh. 7.4 - Prob. 7.26PCh. 7.4 - Prob. 7.27PCh. 7.4 - Prob. 7.28PCh. 7.4 - Prob. 7.29PCh. 7.5 - Prob. 7.31PCh. 7.5 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - Prob. 7.38PCh. 7 - Prob. 7.39PCh. 7 - Prob. 7.40PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - Prob. 7.46PCh. 7 - Prob. 7.47PCh. 7 - Prob. 7.49PCh. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - Prob. 7.52PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.56PCh. 7 - Prob. 7.57P
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