Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 9, Problem 9.15P
To determine

The allowed energies of a particle in the logarithmic potential V(r)=V0ln(ra) and show that the spacing between the levels is independent of mass.

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A triangle in the xy plane is defined with corners at (x, y) = (0,0), (0, 2) and (4, 2). We want to integrate some function f(x, y) over the interior of this triangle. Choosing dx as the inner integral, the required expression to integrate is given by: Select one: o Sro S-o f(x, y) dx dy x=0 2y y=0 O S-o So F(x, y) dæ dy O o S f(x, y) dy dæ O So So F(x, y) dx dy x/2 =0
Consider the "step" potential: V(x) = (a) Calculate the reflection coefficient, for the case E 0. (b) Calculate the reflection coefficient for the case E > Vo. (c) For a potential such as this, which does not go back to zero to the right of the barrier, the transmission coefficient is not simply |F12/A2 (with A the -Vo AV(x) Scattering from a "cliff" incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed. Show that T = E-Vo F1² E |A|² X for E> Vo. Hint: You can figure it out using Equation gantly, but less informatively-from the probability current ( What is T, for E Vo, calculate the transmission coefficient for the step potential, and check that T + R = 1.
Problem 7. 1. Calculate the energy of a particle subject to the potential V(x) = Vo + câ?/2 if the particle is in the third excited state. 2. Calculate the energy eigenvalues for a particle moving in the potential V(x) = câ2/2+ bx. %3!
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