Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 7, Problem 7.46P
(a)
To determine
To construct the
(b)
To determine
The Eigen values and their degeneracies.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
For Problem 9.18, how do I determine part A & B? This is from a chapter titled, "Electron Spin." This chapter is part of quantum mechanics.
Problem 9.4
For the 2D LHO with K1 = K2 show that
and
[ê, ²] = 2ihxy,
(ê, p}] = -2ihxy
Problem 9.5
It follows from the above that
[ê., Ĥ] = 0
if K1 = K2 only
Work out the equivalent commutator for ê and é, with the
Hamiltonian. What do these mean?
O Consider the kinetic energy matrix elements between Hydrogen states
(n' = 4, l', m'| |P|²|
m
-|n = 3, l, m),
=
for all the allowed l', m', l, m values. What kind of operator is the the kinetic energy (scalar or
vector)? Use this to determine the following. For what choices of the four quantum numbers
(l', m', l, m) can the matrix elements be nonzero (e.g. (l', m', l, m) (0, 0, 0, 0),...)? Which
of these nonzero values can be related to each other (i.e. if you knew one of them, you could
predict the other)? In this sense, how many independent nonzero matrix elements are there?
(Note: there is no need to calculate any of these matrix elements.)
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
Knowledge Booster
Similar questions
- Draw an energy level diagram for a nonrelativistic particle confined inside a three-dimensional cube-shaped box, showing all states with energies below 15· (h2/8mL2). Be sure to show each linearly independent state separately, to indicate the degeneracy of each energy level. Does the average number of states per unit energy increase or decrease as E increases?arrow_forwardHow do I prove that the transition is successful in problem 7.22?arrow_forwardH2) Particle in a finite well: Let us consider the following potential. V(x) = -Vo for |x| L %3Darrow_forward
- Consider N distinguishable 3-dimensional harmonic oscillators of Hamiltonian H(q, p) = P mw²q? 2m 2 3N i=1 This is a simple model of the vibrations of atoms in a solid. 1. Calculate the umber of accessible states 2 for a given energy E. Since N is very large, you may assume that the number of state of energy E is the same as the number of states of energyarrow_forwardFor Problem 8.16, how do I prove the relations and give the correct expressions?arrow_forwardShow that the operator = sin e cos o Sx + sin 0 sin o Sy +cos 6 Sz, (1.84) which represents the spin operator for the component of spin along a direction determined by the spherical polar angles 0 and ø, has eigenvalues ± and eigenstates of the form cos(0/2) I↑) = ( sin(@/2) e* eiø ) (1.85) sin(8/2) =(- cos(9/2) e* (1.86) Convince yourself that these results agree with the Rie- mann sphere representation in Fig. 1.8. Show further that 1 0 0 1 (1.87) 14arrow_forwardQuestion 2 2.1 Consider an infinite well for which the bottom is not flat, as sketched here. If the slope is small, the potential V = 6 |x|/ a may be considered as a perturbation on the square- well potential over -a/2 ≤x≤a/2. -8 W V(x) a/2 -a/2 X Calculate the ground-state energy, correct to first order in perturbation theory. Given (0) = √²/co COS Ground state of box of size a: = Ground state energy: E(0) = 4²k² 2ma². 0 Yarrow_forwardpoblem 11.13 cies wx # 0, express the angular momentum operatorl, in terms of creation and annihilation operators. Consider the limiting transition to the isotropic case. For a two-dimensional harmonic oscillator in the xy-plane with different frequen- and show that this operator becomes a constant of motion, in agreement with Section 11.6. OProve that I mn = Vn 2^n! Smn Find fow I Yes> =L[21>+i12>e -iwt Find for1 Yes> Find the time-deperden t uncert arty la Hint APe) = -arrow_forward8.8 Calculate by direct integration the expectation values (r) and (1/r) of the radial position for the ground state of hydrogen. Compare your results to the quoted expressions in Eq. (8.89) and discuss your results. Did you expect that (1/r) # 1/(r)? Use your result for (1/r) to find the expectation value of the kinetic energy of the ground state of hydrogen and discuss your result. 8.9 Calculate by direct integration the expectation value of the radial position for each of thearrow_forwardProve the following commutator identity: [AB, C] = A[B. C]+[A. C]B.arrow_forwardHow might I be able to prove that the Hamiltonian is linear in Problem 11.16? This problem is in a chapter called "Atomic transitions and Radiation." The chapter is under Quantum Mechanics.arrow_forwardThe Hamiltonian of a system has the form 1 A = -1¹ ²³² + ²x² + Y₁x² = H₁ + V₁x¹ 2 dx2 2 Let un(x) = |n) be the eigenstates of Fo, with Fo|n) = (n + ½) |n), n = 0, 1, 2, ... . In this problem, we will first utilize the linear variational method to set up the secular determinant for finding the lowest energy state for the trial function, lp) = co10) + C₂12). (a) In setting up the secular determinant for this problem, we will need to evaluate the Hamiltonian and overlap matrix elements, Hmn = (m|Â|n) and Smn = (m[n), respectively, for m, n = 0, 2. Find the overlap (S) matrix, assuming [0) and 12) are both normalized.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
- College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
- Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage LearningUniversity Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University PressPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningLecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-WesleyCollege Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON