Topics in Modern Physics

Solutions to Problems

Paolo Amore, John Dirk Walecka

• Publisher: WSPC
• Year: 2014
• ISBN:
  9789814618977
• Topic: Physical Sciences
• Subtopic: Electromagnetism

Our understanding of the physical world was revolutionized in the twentieth century — the era of "modern physics". Two books by the second author entitled Introduction to Modern Physics: Theoretical Foundations and Advanced Modern Physics: Theoretical Foundations, aimed at the very best students, present the foundations and frontiers of today's physics. Many problems are included in these texts. A previous book by the current authors provides solutions to the over 175 problems in the first volume.

A third volume Topics in Modern Physics: Theoretical Foundations has recently appeared, which covers several subjects omitted in the essentially linear progression in the previous two. This book has three parts: part 1 is on quantum mechanics, part 2 is on applications of quantum mechanics, and part 3 covers some selected topics in relativistic quantum field theory. Parts 1 and 2 follow naturally from the initial volume. The present book provides solutions to the over 135 problems in this third volume.

The three volumes in this series, together with the solutions manuals, provide a clear, logical, self-contained, and comprehensive base from which students can learn modern physics. When finished, readers should have an elementary working knowledge in the principal areas of theoretical physics of the twentieth century.

Contents:

• Introduction
• Quantum Mechanics:
  • Solutions to the Schrödinger Equation
  • Formal Developments
• Applications of Quantum Mechanics 83:
  • Approximation Methods for Bound States
  • Scattering Theory
  • Time-Dependent Perturbation Theory
  • Electromagnetic Radiation and Quantum Electrodynamics
• Relativistic Quantum Field Theory:
  • Discrete Symmetries
  • Heisenberg Picture
  • Feynman Rules for QCD 231
• Problems and Appendices:
  • The Two-Body Problem
  • Charged Particle in External Electromagnetic Field
  • Advanced undergraduate and graduate students in physics, science and engineering, researchers and engineers. Key Features:
    • The three volumes in this modern physics series provide a clear, logical, self-contained, and comprehensive base from which readers can obtain an elementary working knowledge in the principal areas of theoretical physics of the twentieth century
    • The solutions manuals provide a teaching aid for instructors and a learning aid for students

PART 2

Applications of Quantum Mechanics

Chapter 4

Approximation Methods for Bound States

Problem 4.1 (a) Write the energy functional in the first of Eqs. (4.32) in spherical coordinates with a ψ(r) as

Now partially integrate the first term with the wave function in Eq. (4.30) to verify the second of Eqs. (4.32)

(b) Verify the numerical results in Fig. 4.3 and Eqs. (4.36).

Solution to Problem 4.1

(a) We are to evaluate the energy functional

with the normalized wave function

The radial part of the laplacian is

Substitute this into the energy functional

Now carry out a partial integration on the kinetic-energy term

The first term explicitly vanishes at both limits, and therefore

(b) Done in text.

Problem 4.2 Assume the potential is bounded from below. Show the energy functional satisfies Ɛ ≥V_min where V_min is the minimum value of V.

Solution to Problem 4.2

The energy functional is given by

As in Eq. (4.32), the hermiticity of p = (ħ/i)∇ allows this to be written as

The kinetic energy term is evidently positive definite, and hence

Now write this as

Since the last term in the second line is again positive definite, we have the desired inequality

Ɛ ≥V_min

Problem 4.3 Show through order H′ that the wave functions

in non-degenerate perturbation theory with an hermitian H′ form an orthonormal system

Solution to Problem 4.3

The wave function in first-order non-degenerate perturbation theory is given in Eq. (4.58) as

Now compute the inner product

The orthonormality of the unperturbed wave functions 〈φ_n|φ_m〉 = δ_mn, and the hermiticity of H′, then yield

The second and third terms cancel identically,¹ and this is the desired result

Problem 4.4 (a) Show the second-order wave function in non-degenerate perturbation theory is given by

(b) Verify that this expression is well-defined if the system is non-degenerate [see Eq. (4.59)].

Solution to Problem 4.4

(a) The second-order expression for the wave function

in non-degenerate perturbation theory can be obtained from Eqs. (4.56) in two steps:

(1)	Use the first-order expression for the wave function in Eqs. (4.58)^2 in the matrix...