Application-Driven Quantum and Statistical Physics

A Short Course for Future Scientists and Engineers - Volume 2: Equilibrium

Jean-Michel Gillet

• Publisher: WSPC (EUROPE)
• Year: 2018
• ISBN:
  9781786345592
• Topic: Naturwissenschaften
• Subtopic: Quantentheorie

Bridging the gap between traditional books on quantum and statistical physics, this series is an ideal introductory course for students who are looking for an alternative approach to the traditional academic treatment.

This pedagogical approach relies heavily on scientific or technological applications from a wide range of fields. For every new concept introduced, an application is given to connect the theoretical results to a real-life situation. Each volume features in-text exercises and detailed solutions, with easy-to-understand applications.

Building on the principles introduced in Volume 1, this second volume explains the structure of atoms, the vibration and rotation of molecules. It describes how this is related to thermodynamics through statistical physics. It is shown that these fundamental achievements help to understand how explosives and CO₂ can be detected, what makes a gecko stick to the ceiling, why old stars do not necessarily collapse, where nuclear energy comes from, and more.

Contents:

• Model Hamiltonians and Approximations:
  • Vibrating Systems
  • Rotating Systems
  • Spin, a New Degree of Freedom
  • Central Coulombic Potential
  • The N -electron Atom
• Statistical Treatment of Large Assemblies at the Classical Limit:
  • Thermodynamics in the Macroworld
  • Isolated Systems of Particles
  • Regulated Systems of Classical Particles

Readership: Undergraduate students who need a concise introduction to quantum and statistical physics. Graduate students who want to return and learn about the subject from a different perspective.
Key Features:

• Conciseness
• Emphasis on easy-to-understand applications
• Exercises in the text with detailed solutions
• Focus on applicable concepts (very little physics for its own sake)
• Detailed index
• Additional explanations on essential points which (over the years) have proved to be a recurrent difficulty for the student

Part I

Model Hamiltonians and Approximations

1

Vibrating Systems

The content of this chapter will help you understand Mössbauer spectroscopy, how pollutants and explosives can be detected, what the common denominator between a gecko and an atomic force microscope is, and related topics.

The quantum behaviour of a single particle in a piecewise potential, as described in Volume 1, is a simple but instructive example of how a real situation can be modelled and solved exactly. However the number of real-life problems which can be simplified to such an extremity are sparse. Other types of situations therefore need to be explored and, fortunately, the quadratic potential is often encountered and exactly solvable. The quantum study of harmonic oscillator physics will be the chance to see other types of potentials which, under certain conditions, can legitimately be limited to their quadratic component. Furthermore, in a broader perspective, it will also give the opportunity to discover a powerful, and quite general, approximation method for solving a much larger class of problems.

1.1.On the Role of Harmonic Oscillators in Physics

1.1.1.The pendulum example

Beyond the need to give a rigorous treatment to harmonic motions, the resolution of the harmonic oscillator problem should be seen as a way to approximately describe the motion of a system around its equilibrium point. The first example is often encountered in mechanics for the movement of a pendulum within the limit of small displacements. It is well known that, for a pendulum length ℓ, the angular undamped equation of motion is (Fig. 1.1):

The problem is easily solved providing that θ can substitute to sin θ, i.e. when displacements are much smaller than the length ℓ or, equivalently, when the angle variation is close to zero.

Going up a notch in the deduction of the equation, this leads to the assumption that the tension in the string is constant and the tangential component of weight can simply be expressed by −mgθ. The force becomes proportional to the angular displacement. It can also be written as −(mg/ℓ)ℓθ and takes the familiar Hooke’s law form −kx with x = ℓθ and the effective stiffness k = mg/ℓ as in the harmonic oscillator case (i.e. a weight attached to a spring¹). According to this approximation, the pendulum reduces to the movement of a solid mass m immersed in a potential of the form: (mgℓ)θ²/2. It is this potential with a quadratic dependence on the displacement which is called a harmonic potential.

Fig. 1.1. At the small displacement limit, there is equivalence between the simple pendulum (left) and the harmonic oscillator (right).

We therefore appreciate how important it is to solve the generic problem of the “harmonic oscillator”: whenever the description of a phenomenon can be limited to the study of small deviations from an equilibrium position, the motion is liable to reduce to the resolution of an harmonic oscillator.

1.1.2.A more general perspective

Extending our considerations from the previous section, we consider a particle (or a set of particles) subjected to any one-dimensional (1D) potential V(x) for which there is a local minimum V(x₀) at position x₀. It is thus ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• About the Author
• Contents
• Part I Model Hamiltonians and Approximations
• Part II Statistical Treatment of Large Assemblies at the Classical Limit
• Bibliography
• Index