2 The Square Barrier Let's work out the transmission coefficient T for a rectangular barrier defined...
Consider the finite rectangular barrier described by the potential: where Vo is a real and positive constant. 1. How many bound states does this potential admit? 2. Find the transmission coefficient for a scattering state with energy E Vo 0
Consider a finite square barrier potential shown below. Figure A. For a<x<b, the space part of the electron wave function has the form: k? = 2mE/h? and gu2m(V,-E)/h2 (a) Aeikx (b) Aegn (c) Ae*** + Be** (d) Ae* (e) Aelkx + Be-ika For the finite square barrier potential shown below, Figure A. For x<a, the space part of the electron wave function has the form: k = 2mE/h? and g=2m(Vo-E) /h (a) Aeikx (b) Aetex (c) Ae*EN + Bet* (d)...
Problem 16.1 P16.1 In this problem, you will calculate the transmission probability through the barrier illustrated in Figure 16.10. We first go through the mathematics leading to the solution. You will then carry out further calculations. The domain in which the calculation is carried out is divided into three regions for which the potentials are Aetikx + Be-ikx Region I ψ(x)-cexpFPWh-x] - 1 V(x) =0 for x 0 V(x) = Vo for 0 < x < a V(x) =0 for...
Derive the equation about and and plot the transmission coefficient as a function of E. T= - 1 Vo 1+1_12 +4EE - V.) sinềkza
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that incident particles of energy E> v are coming from-X. Find the stationary states (the equations for region . 2 and 3 and the main equation for the all regions). Apply the matching limit conditions in the figure. Explain and find all the constants used in the equations in terms of the parameters provided and Planck's constant -(6) Find the transmission and reflection coefficients. -(4)
2. An electron with energy E= 1 eV is incident upon a rectangular barrier of potential energy Vo = 2 eV. About how wide must the barrier be so that the transmission probability is 10-37 Electron mass is m=9.1 x 10-31 kg. (Hint: note the word "about". A quick sensible approximation is accepted for full credit. The exact calculation is feasible in an exam, but long and perilous - avoid at all costs.]
A square wave of amplitude A and period T can be defined as -A, 5<t<0, with f(t) = f(t + T), since the function is periodic. Compute the Fourier series for the function in the form f(t) = aneinwot, n=- where wo = 21/T and the coefficients an are the complex Fourier coefficients. Show all your work. Make a simple sketch of the signal and its series. The FIR filter is defined by the filter coefficients bk = [3,-1,2,1] Write...
Consider a traveling (electrons) wave moving in the +x direction approaching a step barrier of height 1 eV; that is V = 0 for x < 0, V = 1.0 eV for x ≥ 0. For x < 0, there will be both the traveling wave in the +x direction. For x ≥ 0, only a solution corresponding to motion in the +x direction exists. By solving the Schrödinger wave equation in both x < 0 and x ≥ 0...
2. Goal of this problem is to study how tunnelling in a two-well system emerges. In particular, we are interested in determining how the tunnelling rate T' of a particle with mass m scales as a function of the (effective) height Vo - E and width b of an energy barrier separating the two wells. The following graphics illustrates the set-up. Initially the particle may be trapped on the left side corresponding to the state |L〉, we are now interested...
1. Consider a 1D finite square well potential defined as follows. Vo-a<x<a V(x) = 0otherwise a) What are the energy eigenfunctions n of the Hamiltonian for a single particle bound in this potential? You may write your answer in piece-wise form, with an arbitrary normalization. b) Derive the characteristic equation that the energy eigenvalues E, must satisfy in order to satisfy the eigenvalue equation Hy,-EnUn for eigen function Un c) Write a computer program1 to find the eigenvalues E, for...