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 5.2, Problem 5.12P
(a)
To determine
A solution to a completely symmetric function and a completely antisymmetric function which also satisfy the Schrodinger equation with same energy and determine what happens to the completely antisymmetric function if
(b)
To determine
Show that a completely antisymmetric spin state for
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Solve the time-independent Schrödinger equation with appropriate
boundary conditions for an infinite square well centered at the origin [V (x) = 0, for
-a/2 < x < +a/2; V (x) = 00 otherwise]. Check that your allowed energies are
consistent with mine (Equation 2.23), and confirm that your y's can be obtained from
mine (Equation 2.24) by the substitution x x - a/2.
Problem #1
(Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and
derive the differential form that reveals A as a potential:
dA < -SdT – pdV [Eqn 5.20]
A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space.
Find the components of the force on the particle in spherical polar coordinates, on the basis of the equation for the components of the generalized force Qj:
Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)
Chapter 5 Solutions
Introduction To Quantum Mechanics
Ch. 5.1 - Prob. 5.1PCh. 5.1 - Prob. 5.2PCh. 5.1 - Prob. 5.3PCh. 5.1 - Prob. 5.4PCh. 5.1 - Prob. 5.5PCh. 5.1 - Prob. 5.6PCh. 5.1 - Prob. 5.8PCh. 5.1 - Prob. 5.9PCh. 5.1 - Prob. 5.10PCh. 5.1 - Prob. 5.11P
Ch. 5.2 - Prob. 5.12PCh. 5.2 - Prob. 5.13PCh. 5.2 - Prob. 5.14PCh. 5.2 - Prob. 5.15PCh. 5.2 - Prob. 5.16PCh. 5.2 - Prob. 5.17PCh. 5.2 - Prob. 5.18PCh. 5.2 - Prob. 5.19PCh. 5.3 - Prob. 5.20PCh. 5.3 - Prob. 5.21PCh. 5.3 - Prob. 5.22PCh. 5.3 - Prob. 5.23PCh. 5.3 - Prob. 5.24PCh. 5.3 - Prob. 5.25PCh. 5.3 - Prob. 5.26PCh. 5.3 - Prob. 5.27PCh. 5 - Prob. 5.29PCh. 5 - Prob. 5.30PCh. 5 - Prob. 5.31PCh. 5 - Prob. 5.32PCh. 5 - Prob. 5.33PCh. 5 - Prob. 5.34PCh. 5 - Prob. 5.35PCh. 5 - Prob. 5.36PCh. 5 - Prob. 5.38PCh. 5 - Prob. 5.39P
Knowledge Booster
Similar questions
- Using the eigenvectors of the quantum harmonic oscillator Hamiltonian, i.e., n), find the matrix element (6|X² P|7).arrow_forwardProblem 6.25 Express the expectation value of the dipole moment pe for an electron in the hydrogen state 1 4 = (211 +210) √2 in terms of a single reduced matrix element, and evaluate the expectation value. Note: this is the expectation value of a vector so you need to compute all three components. Don't forget Laporte's rule!arrow_forwardWrite down the energy eigenfunctions for a particle in an infinitely deep one- dimensional square well extending from Z = -L/2 to z = +L/2 and check that they are eigenfunctions of parity operator (that maps z →-z) corresponding to the eigenvalue (-1)^n-1, where n labels the energy level.arrow_forward
- A point particle moves in space under the influence of a force derivablefrom a generalized potential of the formU(r, v) = V (r) + σ · L,where r is the radius vector from a fixed point, L is the angular momentumabout that point, and σ is the fixed vector in space. Find the components of the force on the particle in Cartesian coordinates, on the basis of the equation for the components of the generalized force Qj: Qj = −∂U/∂qj + d/dt (∂U/∂q˙j)arrow_forward(a_)*(a_V)ax- J-o Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < oo, if y itself is a normalized solution to the Schrödinger equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that y*(a,a_) dx. = -00 Then invoke the Schrödinger equation (Equation 2.46) to obtain la-yl² dx E - hw, -0- where E is the energy of the state y. **Problem 2.12 (a) The raising and lowering operators generate new solutions to the Schrödinger equation, but these new solutions are not correctly normalized. Thus a Vn is proportional to yn+1, and a n is proportional to yn-1, but we'd like to know the precise proportionality constants. Use integration by parts and the Schrödinger equation (Equations 2.43 and 2.46) to show that roo | la+ Vl² dx = (n+ 1)hw, la- Vnl? dx = nhw, -00 -00 and hence (with i's to keep the wavefunctions real) a+ Vn = iv(n + 1)hw yn+1, [2.52] a_n = -ivnhw n-1. [2.53] Sec. 2.3: The Harmonic…arrow_forwardProblem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation (for the infinite square well) with E = 0 or E < 0. (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation and showing that you cannot meet the boundary conditions.)arrow_forward
- A particle of mass M is constrained to move on a sphere of radius R (a) What is the spectrum of the Hamiltonian? (b) What is the probability at time T, to find the particle in the southern hemisphere if the initial state of the particle is |w(t = 0)) ll = 1, m = 1) + |l =0, m = 0). V2arrow_forwardThe Hamiltonian for a harmonic oscillator can be written in dimension- less units (m = h = w = 1) as Ĥ = âtâ + 1/2, where â = (îr + ip)/v2, ât = (â – ip)/V2. %3D %3D One unnormalized energy eigenfunction is Va = (2x – 3x) exp(-x²/2). Find two other (unnormalized) eigenfunctions which are closest in en- ergy to fa.arrow_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_forwardProblem 2.3 Show that there is no acceptable solution to the (time-independent) Schrödinger equation for the infinite square well with E = 0 or E < 0. (This is a special case of the general theorem in Problem 2.2, but this time do it by explicitly solving the Schrödinger equation, and showing that you cannot satisfy the boundary conditions.)arrow_forwardconditions.) Problem 2.4 Solve the time-independent Schrödinger equation with appropriate boundary conditions for an infinite square well centered at the origin [V (x) = 0, for -a/2 < x < +a/2; V (x) = ∞ otherwise]. Check that your allowed energies are consistent with mine (Equation 2.23), and confirm that your y's can be obtained from mine (Equation 2.24) by the substitution x x - a/2. Droblo m 25 Celaulnte lu) .2arrow_forwardO 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.)arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended 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 Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON