Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 9

A particle of mass \(m\) is incident from the left on a wall of infinite thickness and which may be represented by a potential energy \(V\). Calculate the reflection probability for (a) \(E \leq V,\) (b) \(E>V\). For electrons incident on a metal surface \(V=10 \mathrm{eV} .\) Evaluate and plot the reflection probability. Hint. Proceed as in Problems 2.6 and 2.7 but consider only two domains, inside the barrier and outside it. The reflection probability is the ratio \(|B|^{2} /|A|^{2}\) in the notation of eqn 2.21 a.

Short Answer

Expert verified
The reflection probabilities for the given problem are as follows: For E ≤ V, Reflection probability = 1, i.e., the entire incident wave is reflected. For E > V, Reflection probability = \(V^2 / E^2\), that decreases as the energy of the particle increases above the potential barrier. These are then plotted for a range of values greater than the potential barrier of 10 eV.

Step by step solution

01

- Setting Up The Problem

The reflected amplitude, B, divided by the incident amplitude, A gives the reflection coefficient. The reflection probability is the square of this, \(|B|^2/|A|^2\). The expression for the reflection coefficient can be derived from Schrodinger's equation. In this case, we will proceed as indicated and consider only two domains, that is inside the barrier (V) and outside it (E).
02

- Calculating Reflection Probability For E ≤ V

If E ≤ V, then we are considering a case where the energy of the particle is less than the potential barrier it encounters. The potential barrier is treated as an impenetrable wall for the particle in this case. Hence, we can say, no particle can penetrate through the barrier and so all of the particles are reflected back. This gives a reflection probability of 1.
03

- Calculating Reflection Probability For E > V

If E > V, the energy of the particles is greater than the potential barrier. In this case, the particles penetrate through the barrier. For this case, the reflection probability is given by \(|B|^2/|A|^2 = V^2 / E^2\). Substituting the given values, \(V = 10 eV\), and taking a range of values greater than 10 for E, we get the reflection probabilities.
04

- Plotting the Reflection Probability

With the reflection probabilities calculated for a range of energy levels E greater than the potential barrier, we can now plot these against their corresponding energy levels to get the reflection probabilities as a function of the incident energy.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The oscillation of the atoms around their equilibrium positions in the molecule HI can be modelled as a harmonic oscillator of mass \(m \approx m_{\mathrm{H}}\) (the iodine atom is almost stationary \()\) and force constant \(k_{\mathrm{f}}=313.8 \mathrm{N} \mathrm{m}^{-1} .\) Evaluate the separation of the energy levels and predict the wavelength of the light needed to induce a transition between neighbouring levels.

Evaluate the matrix elements (a) \(\langle v+1|x| v\rangle\) and (b) \(\left\langle v+2\left|x^{2}\right| v\right\rangle\) of a harmonic oscillator by using the relations given at the bottom of Table 2.1 for the Hermite polynomials.

A particle was prepared travelling to the right with all momenta between \(\left(k-\frac{1}{2} \Delta k\right) \hbar\) and \(\left(k+\frac{1}{2} \Delta k\right) \hbar\) contributing equally to the wavepacket. Find the explicit form of the wavepacket at \(t=0,\) normalize it, and estimate the range of positions, \(\Delta x,\) within which the particle is likely to be found. Compare the last conclusion with a prediction based on the uncertainty principle. Hint. Use eqn 2.13 with \(g=B\) a constant, inside the range \(k-\frac{1}{2} \Delta k\) to \(k+\frac{1}{2} \Delta k\) and zero elsewhere, and eqn 2.12 with \(t=0\) for \(\Psi_{k} .\) To evaluate \(\int\left|\Psi_{k}\right|^{2} \mathrm{d} \tau\) (for the normalization step) use the integral \(\int_{-\infty}^{\infty}(\sin x / x)^{2} \mathrm{d} x=\pi .\) Take \(\Delta x\) to be determined (numerically) by the locations where \(|\Psi|^{2}\) falls to half its value at \(x=0\) For the last part use \(\Delta p_{x} \approx \hbar \Delta k\)

Energy is required to compress the box when a particle is inside: this suggests that the particle exerts a force on the walls. (a) On the basis that when the length of the box changes by dL the energy changes by \(\mathrm{d} E=-F \mathrm{d} L,\) find an expression for the force. (b) At what length does \(F=1 \mathrm{N}\) when an electron is in the state \(n=1 ?\)

2.32 Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators. 2.33 Consider a particle of mass \(1.00 \times 10^{-25} \mathrm{g}\) moving freely in a three-dimensional cubic box of side \(10.00 \mathrm{nm}\) The potential energy is zero inside the box and is infinite at the walls and outside of the box. (a) Evaluate the zero-point energy of the particle. (b) Consider the energy level that has an energy 9 times greater than the zero-point energy. What is the degeneracy of this level? Identify all the sets of quantum numbers that correspond to this energy. (The energy levels of the cubic box were deduced in Problem \(2.18 .\) (c) Compute the wavelength, frequency, and wavenumber of the photon responsible for the transition from the ground state of the particle to the energy level of part (b).
See all solutions

Recommended explanations on Chemistry Textbooks

Making Measurements

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Chemistry Branches

Read Explanation

Nuclear Chemistry

Read Explanation

Chemical Analysis

Read Explanation

Organic Chemistry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free