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4300 results in General and Classical Physics

Page 40 of 215

  • 2 - Hydrogen

  • from Part I - Fundamental Principles
  • Mark Fox, University of Sheffield
  • Book:
    A Student's Guide to Atomic Physics
    Published online:
    21 June 2018
    Print publication:
    14 June 2018, pp 20-39

  • Summary

    The quantum theory of hydrogen is the starting point for the whole subject of atomic physics. Bohr's derivation of the quantized energies was one of the triumphs of early quantum theory, and makes a useful introduction to the notion of quantized energies and angular momenta. We, therefore, give a brief review of the Bohr model before moving to the main subject of the chapter, namely: the solution of the Schrödinger equation for the electron-nucleus system.

    The Bohr Model of Hydrogen

    The Bohr model is part of the “old” quantum theory of the atom (i.e., pre-quantum mechanics). It includes the quantization of energy and angular momentum, but uses classical mechanics to describe the motion of the electron. With the advent of quantum mechanics, we realize that this is an inconsistent approach, and therefore should not be pushed too far. Nevertheless, the Bohr model does give the correct quantized energy levels of hydrogen, and also gives a useful parameter (the Bohr radius) for quantifying the size of atoms. Hence, it remains a useful starting point to understand the basic structure of atoms.

    It is well known from classical physics that planetary orbits are characterized by their energy and angular momentum. We shall see that these are also key quantities in the quantum theory of the hydrogen atom. In 1911, Rutherford discovered the nucleus, which led to the idea of atoms consisting of electrons in classical orbits where the central forces are provided by the Coulomb attraction to the positive nucleus, as shown in Figure 2.1. The problem with this idea is that the electron in the orbit is constantly accelerating. Accelerating charges emit radiation called bremsstrahlung, and so the electrons should be radiating all the time, losing energy. This would cause the electron to spiral into the nucleus, like an old satellite crashing to Earth.

  • 12 - Atomic Physics in Astronomy

  • from Part II - Applications of Atomic Physics
  • Mark Fox, University of Sheffield
  • Book:
    A Student's Guide to Atomic Physics
    Published online:
    21 June 2018
    Print publication:
    14 June 2018, pp 226-247

  • Summary

    The subjects of atomic physics and astronomy have developed together over centuries. The understanding of stars, galaxies, nebulae, planets, and so on relies on detailed spectroscopic analysis of the atoms they contain. There is healthy feedback between the two disciplines, with astronomical observations prompting new research in atomic physics, and developments in atomic physics and spectroscopy leading to new understanding of astrophysical processes.

    It is not possible to do justice to such a broad subject in a single chapter such as this. The purpose here is to highlight some of the ways the principles developed in the book apply in the astrophysical context, and to point out where interesting differences are observed compared to lab-based experiments. The reader is referred to specialist books for a more comprehensive treatment of the subject.

    Astrophysical Environments

    The atoms in astrophysical sources are frequently found in extreme environments that are very different to those in Earth-based laboratories. The conditions inside an atomic discharge tube are usually benign compared to those found in stars. At the same time, the atom densities can be orders of magnitudes higher than those found in nebulae. All of this has an effect on the spectra that are observed. The underlying principles of atomic physics are the same, but the spectra can appear very different due to the change of the environment of the atoms. The two main differences that have to be considered relate to the temperature and the density.

    Astrophysical Temperatures

    The gas in an atomic discharge tube or an oven might reach a temperature of a few hundred degrees celsius. By contrast, the surface temperature of the sun is 5800 K, with the corona reaching 106 K. Other stars can be hotter. At such high temperatures, the thermal energy kBT is more than sufficient to dissociate molecules. A hydrogen cylinder on Earth will contain molecular hydrogen H2, but the temperatures in stars are sufficient to break the molecular bond and dissociate H2 into atomic hydrogen. Hence the spectra of stars are dominated by atomic hydrogen, whereas a standard hydrogen lamp will emit the spectrum of molecular hydrogen.

  • 1 - Preliminary Concepts

  • from Part I - Fundamental Principles
  • Mark Fox, University of Sheffield
  • Book:
    A Student's Guide to Atomic Physics
    Published online:
    21 June 2018
    Print publication:
    14 June 2018, pp 3-19

  • Summary

    Atomic physics is the subject that studies the inner workings of the atom. It remains one of the most important testing grounds for quantum theory and is therefore a very active area of research, both for its contribution to fundamental physics and to technology. Furthermore, many other branches of science rely heavily on atomic physics, especially astrophysics, laser physics, solid-state physics, quantum information science, and chemistry. So much so, that Richard Feynman once wrote (1964):

    If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or atomic fact, or whatever you wish to call it) that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence you will see an enormous amount of information about the world, if just a little imagination and thinking are applied.

    The task of atomic physics is to understand the structure of atoms, and hence to explain experimental observations such as the wavelengths of spectral lines. For all elements apart from hydrogen, we have to deal with a complicated many-body problem consisting of a nucleus and more than one electron. Atomic physics proceeds by a series of approximations that make this problem tractable. Before we set about this task, it is first necessary to cover a number of important basic concepts and definitions.

    Quantized Energy States in Atoms

    The first basic concept we need is that of bound states. Atoms are held together by the attractive force between the positively charged nucleus and the negatively charged electrons: the electrons are bound to the atom, rather than being free to move though space. In the limit where the electron is very far away from the nucleus, the attractive force is negligible; the electron is free to move with velocity (v) without any influence from the nucleus, as illustrated schematically in Figure 1.1(a). It is natural to define the energy (E) of this free (or unbound) state as being zero when v = 0.

  • 10 - Cold Atoms

  • from Part II - Applications of Atomic Physics
  • Mark Fox, University of Sheffield
  • Book:
    A Student's Guide to Atomic Physics
    Published online:
    21 June 2018
    Print publication:
    14 June 2018, pp 184-201

  • Summary

    The resonant force between atoms and light was first observed in 1933, when Otto Frisch measured the deflection of a sodium beam by a sodium lamp. The invention of lasers opened up new possibilities, leading to the development of the laser-cooling techniques that are the subject of this chapter.

    There are two aspects of laser cooling that make it particularly remarkable:

  • (i) It is highly surprising that the technique works at all. We would normally expect a powerful laser to cause heating rather than cooling. This makes us realize that the technique will only work when special conditions are satisfied.

  • (ii) The very low temperatures achieved by laser cooling are extremely impressive, but this in itself is not the main point, as techniques for achieving very low temperatures have been used for decades by condensed-matter physicists. For example, commercial dilution refrigerators routinely achieve temperatures in the milli-Kelvin range, and as early as the 1950s, Nicholas Kurti and coworkers at Oxford University used adiabatic demagnetisation to achieve nuclear spin temperatures in the micro-Kelvin range. The novelty of laser cooling is that it produces an ultracold gas of atoms, in contrast to the condensed-matter techniques that work on all liquids or solids. These ultracold atoms only interact weakly with each other, which makes it possible to study them with unsurpassed precision.

    • The ability to cool a gas of atoms to very low temperatures has given rise to a whole host of related benefits. Atomic clocks have been made with greater accuracy, and a whole range of new quantum phenomena have been discovered. The most spectacular of these is Bose–Einstein condensation, which was first observed in 1995 and is discussed in Section 10.7.

    The description of laser cooling and Bose–Einstein condensation in this chapter focuses on the basic principles. The reader is referred to specialized texts or articles for a more detailed discussion. See, for example, Foot (2004), Metcalf and van der Straten (1999), or Phillips (1998).

    Gas Temperatures

    In order to understand how laser cooling works, we first need to clarify how the temperature of a gas of atoms is measured. The key point is the link between the thermal motion of the atoms and the temperature. Starting from the Maxwell–Boltzmann distribution (see Eq. [3.39]), it is possible to define a number of different characteristic velocities for the gas.

Page 40 of 215