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 6.8, Problem 6.26P
To determine
Work out the value of
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
Q4.1 Determine explicitly (i.e. give all the details of the derivation), the energy eigenvalues En
and the normalised energy eigenfunctions {on (x)} for a particle moving in a one dimensional 'box'
where the potential energy is U (x) = 0 for 0 a.
Consider two particles m1 and m2. Let mlbe confined to move on a circle of radius R1
in the z-0 plane and centered at x-0, y-0. Let m2 be confined to move on a circle of
radius R2 in the z-a plane and centered at x-0, y-0. A massless spring of spring
constant C is joining the two particles.
www
Set up the Lagrangian for the system.
Set up the equations of constraints.
Set up the Lagrange equations using Lagrange multipliers.
Q4.1 Determine explicitly (i.e. give all the details of the derivation), the energy eigenvalues En
and the normalised energy eigenfunctions {n (x)} for a particle moving in a one dimensional 'box'
where the potential energy is U (x) = 0 for 0 a.
Show that the average or mean value for the position x of a particle in a state represented by the
nth eigenfunction is
a
If the particle's wavefunction is represented by the eigenfunction øn (x), find an integral expression
for the probability of finding the particle somewhere in the interval b
Chapter 6 Solutions
Introduction To Quantum Mechanics
Ch. 6.1 - Prob. 6.1PCh. 6.2 - Prob. 6.2PCh. 6.2 - Prob. 6.3PCh. 6.2 - Prob. 6.4PCh. 6.2 - Prob. 6.5PCh. 6.2 - Prob. 6.7PCh. 6.4 - Prob. 6.8PCh. 6.4 - Prob. 6.9PCh. 6.4 - Prob. 6.10PCh. 6.4 - Prob. 6.11P
Ch. 6.4 - Prob. 6.12PCh. 6.4 - Prob. 6.13PCh. 6.5 - Prob. 6.14PCh. 6.5 - Prob. 6.15PCh. 6.5 - Prob. 6.16PCh. 6.5 - Prob. 6.17PCh. 6.6 - Prob. 6.18PCh. 6.6 - Prob. 6.19PCh. 6.7 - Prob. 6.20PCh. 6.7 - Prob. 6.21PCh. 6.7 - Prob. 6.22PCh. 6.7 - Prob. 6.23PCh. 6.7 - Prob. 6.25PCh. 6.8 - Prob. 6.26PCh. 6.8 - Prob. 6.27PCh. 6.8 - Prob. 6.28PCh. 6.8 - Prob. 6.30PCh. 6 - Prob. 6.31PCh. 6 - Prob. 6.32PCh. 6 - Prob. 6.34PCh. 6 - Prob. 6.35PCh. 6 - Prob. 6.36PCh. 6 - Prob. 6.37P
Knowledge Booster
Similar questions
- Q. A particle of mass m. dimension Such that it has the Lagrangian moves. in one ,4 2. 12 function of where V is a x' Tind the egiation of Motion for 7) and intetpret the physical nature of the System differentiablearrow_forwardA particle of mass m slides down in a smooth rigid bowl (hemispherical surface) as shown in Figure 3. R Figure 3. Derive for the particle the (a) Newtonian equations of motion, (b) Lagrange's equations of motion and the reaction of the bowl on m, and (c) Hamilton's equations of motion.arrow_forwardConsider a Lagrangian given by: L(r, i) = mi2-k. %3D Here x is the generalized coordinate. Evaluaté the conjugate momentum p.arrow_forward
- For Problem 8.35, how do I prove, or perhaps verify, what it is they're asking for?arrow_forwardWhat does your result for the potential energy U(x=+L) become in the limit a→0?arrow_forwardQ.n.4 Consider two particles of masses m1 and m2. Let m1 be contined to move on a circle of radius a in the z = 0 plane, centered at x =y = 0. Let m2 be confined to move on a circle of radius b in the z=c plane, centered at x = y =0. A light (massless) spring of spring constant k is attached between the two particles. a) Find the Lagrangian for the system.arrow_forward
- Problem 3 A ball of mass m, slides over a ev sliding inclined plane of mass M and angle a. Denote by X, the coordinate of O' with re- spect to 0, and by (x.y) the coordinate of m with respect to O'(see figure below) m M a 1. Calculate the degree of freedom of the system 2. Find the velocity of m with respect to 0. 3. Write the expression of the Lagrangian function 4. Derive the Euler Lagrange equations 5. Find r" and X" in terms of the masses (m,M), angle a and garrow_forwardConsider a particle of mass m moving in 1-dimension under a piecewise-constant po- tential. In region I, that corresponds to x 0. In region II, that corresponds to x > 0 the potential energy is V1(x) = 0. The particle is shot from = -∞ in the positive direction with energy E > Vo > 0. See the figure in the next page for a representation of V(x) as a function of x. Also shown in the graph (green dashed line) the energy E of the particle. (a) Which of the following functions corresponds to the wavefunction 1(x) in region I? (a1) Aeikiæ + Be-iki¤ ; (а2) Ае\1 + Bе-кӕ (a3) Aeikræ (а4) Ве- кта (b) Which of the following functions corresponds to the wavefunction 1(x) in region II? (b1) Сеkп* + De-ikr (62) C'e*I1* + De-*1¤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
- CASE 2 Let three equations of the model take these forms: p = 1 1 -3U + dn 3 (р — п) - 3- dt 4 dU (т — р) dt a. Find p(t), T(t), and U(t) b. Are the time path convergent? Fluctuating? explainarrow_forwardProblem 9. For a system described by the Hamiltonian H = p°/2m + V(x), obtain an expression for d (p/2m)Idt. Discuss the relation of your result to the classical work-energy theorem.arrow_forwardA particle of mass m moves in the one-dimensional potential x2 -x/a U(x) = U, Sketch U (x). Identify the location(s) of any local minima and/or maxima, and be sure that your sketch shows the proper behaviour as x → t00. Here a is a constant.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning