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 3.4, Problem 3.12P
To determine
The value of free particle
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
The figure shows two railway cars with a buffer spring. We want to investigate the transfer of momentum that occurs after car 1 with initial velocity v0 impacts car 2 at rest. The differential equation is given below. Show that the eigenvalues of the coefficient matrix A are λ1=0 and λ2=−c1−c2, with associated eigenvectors v1=
1
1
T
and
v2=
c1
−c2
T.
x′′=
−c1
c1
c2
−c2
x
with
ci=k /mi for i=1, 2
The coefficient matrix A is .
Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface.
i. sketch all forces acting on each mass
ii. explain how we get the following equation for the total energy
Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k >0. We know physically that our spring will eventually come to rest nomatter the initial conditions or the values of m, β, or k. If our modelis a good model, all solutions x(t) should approach 0 as t → ∞. Foreach of the three cases below, explain how we know that both rootsr1,2 =−β ± Sqrt(β^2 − 4km)/2mwill lead to solutions that exhibit exponentialdecay.(a) β^2 − 4km > 0.
(b) β^2 − 4km =0.
(c) β^2 − 4km >= 0.
Chapter 3 Solutions
Introduction To Quantum Mechanics
Ch. 3.1 - Prob. 3.1PCh. 3.1 - Prob. 3.2PCh. 3.2 - Prob. 3.3PCh. 3.2 - Prob. 3.4PCh. 3.2 - Prob. 3.5PCh. 3.2 - Prob. 3.6PCh. 3.3 - Prob. 3.7PCh. 3.3 - Prob. 3.8PCh. 3.3 - Prob. 3.9PCh. 3.3 - Prob. 3.10P
Ch. 3.4 - Prob. 3.11PCh. 3.4 - Prob. 3.12PCh. 3.4 - Prob. 3.13PCh. 3.5 - Prob. 3.14PCh. 3.5 - Prob. 3.15PCh. 3.5 - Prob. 3.16PCh. 3.5 - Prob. 3.17PCh. 3.5 - Prob. 3.18PCh. 3.5 - Prob. 3.19PCh. 3.5 - Prob. 3.20PCh. 3.5 - Prob. 3.21PCh. 3.5 - Prob. 3.22PCh. 3.6 - Prob. 3.23PCh. 3.6 - Prob. 3.24PCh. 3.6 - Prob. 3.25PCh. 3.6 - Prob. 3.26PCh. 3.6 - Prob. 3.27PCh. 3.6 - Prob. 3.28PCh. 3.6 - Prob. 3.29PCh. 3.6 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Prob. 3.32PCh. 3 - Prob. 3.33PCh. 3 - Prob. 3.34PCh. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - Prob. 3.40PCh. 3 - Prob. 3.41PCh. 3 - Prob. 3.42PCh. 3 - Prob. 3.43PCh. 3 - Prob. 3.44PCh. 3 - Prob. 3.45PCh. 3 - Prob. 3.47PCh. 3 - Prob. 3.48P
Knowledge Booster
Similar questions
- Consider a car moving at highway speed u. Is its actual kinetic energy larger or smaller than 1/2 mu2? Make an order-of-magnitude estimate of the amount by which its actual kinetic energy differs from 1/2 mu2. In your solution, state the quantities you take as data and the values you measure or estimate for them. You may find Appendix B.5 useful.arrow_forwardA pendulum of length l and mass m is mounted on a block of mass M. The block can movefreely without friction on a horizontal surface as shown in Fig 1 PHYS 24021. Consider a particle of mass m moving in a plane under the attractive force μm/r2 directedto the origin of polar coordinates r, θ. Determine the equations of motion.2. Write down the Lagrangian for a simple pendulum constrained to move in a single verticalplane. Find from it the equation of motion and show that for small displacements fromequilibrium the pendulum performs simple harmonic motion.3. A pendulum of length l and mass m is mounted on a block of mass M. The block can movefreely without friction on a horizontal surface as shown in Fig 1.Figure 1Show that the Lagrangian for the system isL =( M + m2)( ̇x)2 + ml ̇x ̇θ + m2 l2( ̇θ)2 + mgl(1 − θ22)arrow_forwardShow that the transformation for a system of one degree of freedom,Q = q cos α − p sin α,P = q sin α + p cos α,satisfies the symplectic condition for any value of the parameter α. Find agenerating function for the transformation. What is the physical significanceof the transformation for α = 0? For α = π/2? Does your generating functionwork for both of these casesarrow_forward
- 2.7 There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity v, that is, F = F(v). Write down Newton's second law and separate the variables by rewriting it as m dv/F(v) = dt. Now integrate both sides of this equation and show that t = m v dv' F(v¹) Vo Provided you can do the integral, this gives t as a function of v. You can then solve to give v as a function of t. Use this method to solve the special case that F(v) = Fo, a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and 2.9.arrow_forward. Consider the Lagrangian function 1 L = m (i² + j? + ?) + 16ý sin (æ – t), where m is a positive constant, (x, y, z) are generalised coordinates and t is time. (i) Write down the generalised momenta pa, Py and Pz. (ii) Does L have any cyclic coordinates? Justify your answer. (ii) Find the equations of motion. Can an analytic solution be found?arrow_forwardA particle is moving on top of a 2-dimensional. plane with its coordinates given incartesian system asx(t) = a sin ωt, y(t) = a cos ωt.Express the motion of the particle in terms of polar coordinates (ρ, φ). What is the minimum numberof generalised coordinates required to describe. its motion? Draw the. trajectory of the particle.Now if the particle trajectory is changed to the followings, repeat the exercise.x(t) = 2a sin ωt, y(t) = a cos 2ωtarrow_forward
- Consider an observer at rest in a frame where the ambient air is also at rest. Far away to the left, a speaker emits sound waves at angular frequency ω0; far away to the right, an identical speaker (emitting the same frequency waves in its rest frame) is moving at velocity u to the right. Let v denote the speed of sound in air. A: Assuming the amplitude of waves from each speaker is the same, we might expect that the (complex) amplitude of sound waves is given by (for all points x between the two speakers)u(x, t) = A(e^(ik(x−vt)) + e^(ik'(x+vt))). (1)Here A is an unimportant overall constant prefactor. Use the Doppler effect (if needed), and otherphysics of waves, to determine k and k'in terms of u, v and/or ω.B: If the observer is at x = 0, the amplitude of oscillations is given by u(0, t). Show that if u << v, thenRe[u(0, t)] = 2A cos(ω1t) cos(ω2t) (2)where |ω2| << |ω1|. Find explicit formulas for ω1,2 in terms of ω0, u, v. cos(ω2t) varies extremely slowly in time. This…arrow_forwardConsider the motion of a particle in two dimensions given by the Lagrangian m L = 2 (x+ where i>0 . The initial conditions are given as y(0)=0, x(0) = 42 meters, *(0) = y (0) = 0. What is the value of x(t)– y(t) at t= 25 seconds in meters?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
- Prove that F= m (2x/t^2) from equations (2.1) – (2.4).arrow_forwardYour character starts at position po with initial velocity v = à ||d || Each frame, it moves with constant acceleration a. Compute the position of the character with given the parameters. 4 4 What is Pi when pò = ( ₁ ), d = ( _^3 ), ª = (22) 3 -3 -2 and St = 0.23?arrow_forwardvative force, by explicitly showing that nds on only Q.n.4 Consider two particles of masses m1 and m2. Let ml be confined 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. Q.n.5 Oral Vivaarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
University 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 Learning
Lecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-Wesley
College 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