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 2.3P
To determine
Show that there is no acceptable solution to the Schrodinger equation for the infinite square well with
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Problem 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.)
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.
Consider an infinite well, width L from x=-L/2 to x=+L/2. Now
consider a trial wave-function for this potential, V(x) = 0 inside the well
and infinite outside, that is of the form (z) = Az. Normalize this
wave-function. Find , .
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
- A neutron of mass m of energy E a, V(x) = V, ) II. Calculate the total probability of neutron tunneling through the barrier?; from region I to II.arrow_forwardPROBLEM 2. Consider a spherical potential well of radius R and depth Uo, so that the potential is U(r) = -Uo at r R. Calculate the minimum value of Uc for which the well can trap a particle with l = 0. This means that SE at Uo > Uc has at least one bound ground state at l = 0 and E < 0. At Ug = Uc the bound state disappears.arrow_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_forward
- Consider 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_forwardA one-dimensional infinite potential well has a length of 2L. What are the energy eigenvalues? Calculate the ground state energy if ten protons are confined in the box. Assume that the protons don’t interact with each other. If the ten protons are replaced by ten neutral hydrogen atoms, what is the total ground state energy resulting from the confinement? Again, assume that the hydrogen atoms do not interact with each other. You can treat the mass of proton and hydrogen atom to be identical.arrow_forwardConsider the one-dimensional time-independent Schrödinger equation for some arbitrary potential V(x). Prove that if a solution p(x) has the property that (x) → 0 as r → ±00, then the solution must be nondegen- erate and therefore real, apart from a possible overall phase factor. Hint: Show that the contrary assumption leads to a contradiction.arrow_forward
- Let's consider the two-qubit state 3 |) = 100)+101) +110). a) Find the expectation values for the values of both qubits separately. b) The product of qubit values is represented by the operator b₁b2 = (ô× 1) (I Øô) = (ô ❀ô), where bn is the observable for the value of qubit n. Find the expectation value for the product. For statistically independent quantities the expectation value of their product is the product of their expectation values. Are the values of the qubits correlated in state |V)? c) Show that the state cannot be expressed as a product state, i.e., it is an entangled state.arrow_forwardA neutron of mass m with energy E a,V(x) =+Vo I. Draw the potential sketch! Write down the Schrödinger equation for: II. region I (0 a,V(x) =+Vo ), Calculate the total probability of nucleons tunneling through the barrier?; from region I to II (with E 2 0 and E < 0). II.arrow_forwardSolve the problem for a quantum mechanical particle trapped in a one dimensional box of length L. This means determining the complete, normalized wave functions and the possible energies. Please use the back of this sheet if you need more room.arrow_forward
- PROBLEM 2. The potential energy of a weakly anharmonic oscillator can be modeled by: m U(x) P²+Bx*, where the last quatric term describes a small anharmonic correction. The energy levels En of the anharmonic oscillator in the first order in the pa- rameter 3 are given by: En = hw 5) + B(n|z*\n). Calculate the energy of the ground state Eo of the anharmonic oscillator.arrow_forwardConsider a state function that is real, i.e., such that y (x) = y* (x). Show that (p) Under what conditions on p (x), would the function o (p) turn out to be real, and if so, what is (x) worth? = 0. What happens in that case with (p2) and with (x) ?;arrow_forwardWrite 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.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_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