Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 1

Equation 14.3 gives the form of the S matrix for a one-dimensional system in which a particle is scattered from an abrupt blip in the potential energy. Write down the analogous expression for scattering from a comparable dip in the potential energy. Proceed to compute the transmission probability for positive energies given that the particle is incident from the left.

Short Answer

Expert verified
The S matrix considering an abrupt dip in potential energy transforms to \[ S = \left( \begin{array}{cc}-t' & -r \-r' & -t \end{array} \right) \]\and the transmission probability for positive energies with particle approaching from the left becomes \( |t|^2 \).

Step by step solution

01

Define the Initial Reflection and Transmission Coefficients

Start by defining the transmission and reflection coefficients for the particle. In the given scenario, potential energy is abruptly dipped. The matrix for this can be defined following the pattern of the S matrix. Denote the reflection coefficient as \( r \) and the transmission coefficient as \( t \). Their initial values are dependent on the incident particle.
02

Apply the S Matrix Form

The S matrix can be defined as: \\[ S = \left( \begin{array}{cc}t' & r \r' & t \end{array} \right) \]\Given that the problem involves a dip in potential energy, this will introduce a phase change of \( \pi \) into the system. The modification therefore introduces negative signs into the S matrix. The system becomes:\\[ S = \left( \begin{array}{cc}-t' & -r \-r' & -t \end{array} \right) \]\These phase changes indicate that the wave is reflected upside down.
03

Determine the Transmission Probability

The transmission probability for positive energies, taking into account that the particle incident from the left, can be calculated with the formula: \\[ Transmission \: Probability = |t|^2 \]\The absolute value of the transmission coefficient is squared, giving the probability of a successful transmission.

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

Show that in the limit of low energies, the scattering phase shift for P-wave scattering by a hard sphere is proportional to \((k a)^{3}\) and therefore is negligible compared to the S-wave scattering phase shift. Hint. Use the asymptotic forms given in eqn \(14.32 \mathrm{c}\)

Show for the elastic scattering of a particle by a central potential \(V(r)\) that approaches zero more rapidly than \(1 / r\) as \(r \rightarrow \infty\) that the integral cross-section can be written as $$\sigma_{\mathrm{tot}}=\frac{4 \pi}{k} \mathrm{im} f_{k}(0)$$ where im \(f_{k}(0)\) is the imaginary part of the forward scattering amplitude \((\theta=0) .\) This is the so-called optical theorem. Hint. The Legendre polynomials are required to satisfy \(P_{l}(1)=1\) for all values of \(l\)

Consider the differential cross-section for elastic scattering given in eqn \(14.46 .\) At a given energy, sketch its dependence on the scattering angle \(\theta\) when the \(l=1\) partial wave dominates the scattering. Do the same for the \(l=0\) and \(l=2\) partial waves.

A particle of mass \(m\) is scattered off a central potential \(V(r)\) of the form $$V(r)=\left\\{\begin{array}{lll} \infty & \text { if } & r=0 \\\0 & \text { if } & 0V_{0},\) find an expression for the S-wave scattering phase shift \(\delta_{0} .\) Hint. Require the wavefunction and its first derivative to be continuous at \(r=a\) and at \(r=b\)

The incoming Green's function is given by $$G^{(-)}\left(r, r^{\prime}\right)=\frac{\mathrm{e}^{-i k\left|r-r^{\prime}\right|}}{\left|r-r^{\prime}\right|}$$ Show that \(G^{(-)}\) is a solution of the equation $$\left(\nabla^{2}+k^{2}\right) G\left(r, r^{\prime}\right)=-4 \pi \delta\left(r-r^{\prime}\right)$$ Hint. Use an analysis similar to that given in Further information 14.1. Although the incoming Green's function does not yield the desired asymptotic form of the stationary scattering state \((\mathrm{eqn} 14.14), G^{(-)}\) appears in some of the formal expressions of scattering theory.
See all solutions

Recommended explanations on Chemistry Textbooks

Ionic and Molecular Compounds

Read Explanation

Chemistry Branches

Read Explanation

Organic Chemistry

Read Explanation

Making Measurements

Read Explanation

Chemical Analysis

Read Explanation

Nuclear 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