Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Question
Chapter 11.5, Problem 11.18P
(a)
To determine
The density matrix for an electron that is either in the state spin up along x or in the spin down along y.
(b)
To determine
The value of
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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.
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 satisfy the boundary conditions.)
A particle of mass m confined to an infinite potential well of length L from x= 0 to x=L is in the ground state.
(a) Is this state an eigenfunction of the momentum operator, px? Justify your answer.
(b) If an observation of momentum is made, what value, or values, could be obtained? Justify your answer.
Chapter 11 Solutions
Introduction To Quantum Mechanics
Ch. 11.1 - Prob. 11.1PCh. 11.1 - Prob. 11.2PCh. 11.1 - Prob. 11.3PCh. 11.1 - Prob. 11.4PCh. 11.1 - Prob. 11.5PCh. 11.1 - Prob. 11.6PCh. 11.1 - Prob. 11.7PCh. 11.1 - Prob. 11.8PCh. 11.1 - Prob. 11.9PCh. 11.3 - Prob. 11.10P
Ch. 11.3 - Prob. 11.11PCh. 11.3 - Prob. 11.12PCh. 11.3 - Prob. 11.13PCh. 11.3 - Prob. 11.14PCh. 11.3 - Prob. 11.15PCh. 11.3 - Prob. 11.16PCh. 11.4 - Prob. 11.17PCh. 11.5 - Prob. 11.18PCh. 11.5 - Prob. 11.19PCh. 11.5 - Prob. 11.20PCh. 11.5 - Prob. 11.21PCh. 11.5 - Prob. 11.22PCh. 11 - Prob. 11.23PCh. 11 - Prob. 11.24PCh. 11 - Prob. 11.25PCh. 11 - Prob. 11.26PCh. 11 - Prob. 11.27PCh. 11 - Prob. 11.28PCh. 11 - Prob. 11.29PCh. 11 - Prob. 11.30PCh. 11 - Prob. 11.31PCh. 11 - Prob. 11.33PCh. 11 - Prob. 11.35PCh. 11 - Prob. 11.36PCh. 11 - Prob. 11.37P
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- Determine the transmission coefficient for a rectangular barrier (same as Equation 2.127, only with +Vo in the region -a Vo (note that the wave function inside the barrier is different in the three cases). Partial answer: For Earrow_forwardConsider a particle in a box of length L with one end coinciding with the origin. Consider the initial state to be given by the wavefunction: ψ(x,0)=Asin2nπx/L for 0≤x≤L/2 and ψ(x,0)=0 for x>L/2. 1. Normalize this wavefunction. 2. Compute the time evolution of this system. 3. Find the probability of measuring Ei at some time t.arrow_forwardProblem 1: Simple Harmonic oscillator (a) Find the expectation value of kinetic energy T for the nth state of a simple harmonic oscillator. (b) Write p² in terms of a+ and a_ (c) Construct 2 from o state. Vo is given in equation (2.60). Use a+ operators (d) Find ,,,, σx and σy for state 2 found in part (c). Check if the uncer- tainty principle works for this state.arrow_forwardSolve 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.arrow_forwardLegrende polynomials The amplitude of a stray wave is defined by: SO) =x (21+ 1) exp li8,] sen 8, P(cos 8). INO Here e is the scattering angle, / is the angular momentum and 6, is the phase shift produced by the central potential that performs the scattering. The total cross section is: Show that: 'É4+ 1)sen² 8, .arrow_forward1 W:0E Problem 1.17 A particle is represented (at time t = 0) by the wave function | A(a? – x²). if -a < x < +a. 0, Y (x, 0) = otherwise. (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 p?. (f) Find the uncertainty in x (ox).arrow_forwardProblem 3.10 Is the ground state of the infinite square well an eigenfunction of momentum? If so, what is its momentum? If not, why not?arrow_forwardInfinite/finite Potential Well 1. Sketch the solution (Wave function - Y) for the infinite potential well and show the following: (a) Specify the boundary conditions for region I, region II, and region III. (i.e. U = ?, and x = ?) (b) Specify the length of the potential well (L=10 cm) (c) Which region will have the highest probability of finding the particle?arrow_forwardDetermine the probability of finding a quantum particle (restricted to a one-dimensional box of length L), in the interval 0 < x < L/8, if L = 6.59 nm and the wave function describing the particle is the following. y(x) : A sin for < X < 2 elsewhere Did you apply the normalization condition to the wave function? What are the limits of integration for the normalization condition? What are the limits of integration for finding the probability? Review integral calculus and check your work.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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