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 2.2, Problem 2.9P
To determine
The expectation value of
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Write down the equations and the associated boundary conditions for
solving particle in a 1-D box of dimension L with a finite potential
well, i.e., the potential energy U is zero inside the box, but finite
outside the box. Specifically, U = U₁ for x L. Assuming that particle's energy E is less
than U, what form do the solutions take? Without solving the
problem (feel free to give it a try though), qualitatively compare with
the case with infinitely hard walls by sketching the differences in
wave functions and probability densities and describing the changes
in particle momenta and energy levels (e.g., increasing or decreasing
and why), for a given quantum number.
Consider the function
v(1,2) =(
[1s(1) 3s(2) + 3s(1) 1s(2)]
[x(1) B(2) + B(1) a(2)]
Which of the following statements is incorrect concerning p(1,2) ?
a.
W(1,2) is normalized.
Ob.
The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons.
OC.
y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with
eigenvalue = -5 hartree.
d.
The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit.
Oe.
The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.
A particle with mass m is moving along the x-axis in a potential given by the potential energy
function U(x) = 0.5mw²x². Compute the product (x, t)*U (x) V (x, t). Express your
answer in terms of the time-independent wave function, (x).
Chapter 2 Solutions
Introduction To Quantum Mechanics
Ch. 2.1 - Prob. 2.1PCh. 2.1 - Prob. 2.2PCh. 2.2 - Prob. 2.3PCh. 2.2 - Prob. 2.4PCh. 2.2 - Prob. 2.5PCh. 2.2 - Prob. 2.6PCh. 2.2 - Prob. 2.7PCh. 2.2 - Prob. 2.8PCh. 2.2 - Prob. 2.9PCh. 2.3 - Prob. 2.10P
Ch. 2.3 - Prob. 2.11PCh. 2.3 - Prob. 2.12PCh. 2.3 - Prob. 2.13PCh. 2.3 - Prob. 2.14PCh. 2.3 - Prob. 2.15PCh. 2.3 - Prob. 2.16PCh. 2.4 - Prob. 2.17PCh. 2.4 - Prob. 2.18PCh. 2.4 - Prob. 2.19PCh. 2.4 - Prob. 2.20PCh. 2.4 - Prob. 2.21PCh. 2.5 - Prob. 2.22PCh. 2.5 - Prob. 2.23PCh. 2.5 - Prob. 2.24PCh. 2.5 - Prob. 2.25PCh. 2.5 - Prob. 2.26PCh. 2.5 - Prob. 2.27PCh. 2.5 - Prob. 2.28PCh. 2.6 - Prob. 2.29PCh. 2.6 - Prob. 2.30PCh. 2.6 - Prob. 2.31PCh. 2.6 - Prob. 2.32PCh. 2.6 - Prob. 2.34PCh. 2.6 - Prob. 2.35PCh. 2 - Prob. 2.36PCh. 2 - Prob. 2.37PCh. 2 - Prob. 2.38PCh. 2 - Prob. 2.39PCh. 2 - Prob. 2.40PCh. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - Prob. 2.44PCh. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - Prob. 2.47PCh. 2 - Prob. 2.49PCh. 2 - Prob. 2.50PCh. 2 - Prob. 2.51PCh. 2 - Prob. 2.52PCh. 2 - Prob. 2.53PCh. 2 - Prob. 2.54PCh. 2 - Prob. 2.58PCh. 2 - Prob. 2.63PCh. 2 - Prob. 2.64P
Knowledge Booster
Similar questions
- Answer the following about an observable that is represented by the operator  = wo (3² + 3²). ħ (4) Is it possible to write a complete set of basis states that are simultaneously eigenstates of the operator Ĵ and Â? If so, explain how you know. If not, write an uncertainty principle for the observables A and J.arrow_forwardConsider the potential barrier illustrated in Figure 1, with V(x) = V₁ in the region 0 L. b) Identify the parts of your solutions that correspond to the incident, reflected and transmitted particles. Explain why the remaining term in the region > L can be set to zero. c) Determine the probability currents associated with the incident, reflected and transmitted particles.arrow_forwardConsider the half oscillator" in which a particle of mass m is restricted to the region x > 0 by the potential energy U(x) = 00 for a O where k is the spring constant. What are the energies of the ground state and fırst excited state? Explain your reasoning. Give the energies in terms of the oscillator frequency wo = Vk/m. Formulas.pdf (Click here-->)arrow_forward
- The eigenstates of the particle-in-a-box are written, n = √ sin (™T). If L = 10.0, what is the expectation value for the quantity 2ħ² + p² in the n = 3 eigenstate? Report your answer as a multiple of ħ². (Note: ô = −iħª) d dx'arrow_forwardProblem 1.17 A particle is represented (at time=0) by the wave function A(a²-x²). if-a ≤ x ≤+a. 0, otherwise. 4(x, 0) = { (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t = 0)? (c) What is the expectation value of p (at time t = 0)? (Note that you cannot get it from p = md(x)/dt. Why not?) (d) Find the expectation value of x². (e) Find the expectation value of p².arrow_forwardthat de/dx = 0 (x). **Problem 2.25 Check the uncertainty principle for the wave function in Equation 2.129. Hint: Calculating (p2) is tricky, because the derivative of has a step discontinuity at x = 0. Use the result in Problem 2.24(b). Partial answer: (p²) = (ma/h)².arrow_forward
- Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?arrow_forwardConsider a particle of mass, m, with energy, E, moving to the right from -o. This particle is subject to the potential energy V(x) = }V, for 0 V. x z a Sketch a picture that shows the potential energy. In this picture represent a particle moving to the right when x < 0. Solve the time independent Schrodinger equation to find g (x) on the domain -coarrow_forwardProblem 3.30 Derive the transformation from the position-space wave function to the “energy-space” wave function using the technique of Example 3.9. Assume that the energy spectrum is discrete, and the potential is time-independent.arrow_forwardThe wave function of a particle in two dimensions in plane polar coordinates is given by: T ¥(r,0) = A. r. sin0exp[- where A and ao are positive real constants. 1. Find the constant A A using the normalization condition in the form SS |¥(r,0)|²rdrd0 = 1 2. Calculate the expectation values of r, and ². 3. Assuming that the momentum operator in plane polar coordinate is giving in the form ħa p = calculate the expectation values of p and p². 1 Ər' 4. Find the standard deviations of r and p and show that their product is consistent with the Heisenberg uncertainty principle.arrow_forwardThe first four Hermite polynomials of the quantum oscillator areH0 = 1, H1 = 2x, H2 = 4x2 − 2, H3 = 8x3 − 12x. Let p(x) = 12x3 − 8x2 − 12x + 7. Using the basis H = {H0, H1, H2, H3}, find the coordinate vector ofp relative to H. That is, find [p]H. This is a textbook question, not a graded questionarrow_forwardConsider a particle in a one-dimensional rigid box of length a. Recall that a rigid box has U (x) = 00 for x a, and U () = 0 for 0 )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