Wave Mechanical Model

6 Key excerpts on "Wave Mechanical Model"

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  eBook - PDF

  A Textbook of Physical Chemistry

  • Arther Adamson(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  CHAPTER SIXTEEN WAVE MECHANICS 16-1 Introduction The chapter on wave mechanics is probably subject to more variation in style and content than any other in a textbook of physical chemistry. Wave mechanics is of central importance to the physical chemist: Many of its results are of great utility, as in statistical thermodynamics, chemical bonding, and molecular spectro-scopy. It is also the most nearly correct theory of mechanics that we have and its very language has permeated chemistry. On the other hand, it is difficult to present. Wave mechanics is virtually unique in the history of scientific theories in that it introduces a mathematical rather than a physical model of nature—and its mathematics becomes so fiercely complicated for systems of chemical interest that its practice has developed into the very specialized field of discovering and evaluating various approximation methods. It is true, however, that the wave equation can be solved exactly for certain simple or model cases: A particle confined to a box or potential well, the hydrogen atom, and the harmonic oscillator are important illustrations. Most of the approxi-mation methods use combinations or modifications of the exact solutions for such cases, and the procedure in this chapter will be to describe some of these exact solutions in sufficient detail that the rationale of the more advanced approaches can be appreciated. Moreover, most of the language and qualitative thinking in wave mechanics draws on the results for the simple situations. This will be true in Chapter 17 on chemical bonding and again in Chapter 19, on molecular spectroscopy. The detailed treatment of the important model situations is thus both highly utilitarian and within the spirit of a course in physical chemistry. It might be thought, from all this, that the older quantum chemistry, as exemplified by the Bohr model, would be useless.

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  Chemistry

  The Molecular Nature of Matter

  • Neil D. Jespersen, Alison Hyslop(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  However, the theory was not able to explain quantitatively the spectra of atoms with more than one electron, and all attempts to modify the theory to make it work failed. It became clear that Bohr’s picture of the atom was flawed and that another theory would have to be found. Nevertheless, the concepts of quantum num- bers and fixed energy levels were important steps forward. 7.4 The Wave Mechanical Model Bohr’s efforts to develop a theory of electronic structure were doomed from the very begin- ning because the classical laws of physics—those known in his day—simply do not apply to objects as small as the electron. Classical physics fails for atomic particles because matter at the atomic level is below the limit that our physical senses can perceive. When physicists were proposing that photons (particles) and the waves of electromagnetic radiation were the same NOTE All the objects that had been studied by scientists up until the time of Bohr’s model of the hydrogen atom were large and massive in comparison with the electron, so no one had detected the limits of classical physics. 7.4 The Wave Mechanical Model 325 7.4 The Wave Mechanical Model 325 thing, it became apparent that particles such as electrons may behave as waves. This idea was proposed in 1924 by a young French graduate student, Louis de Broglie. In Section 7.1 we discussed that light waves are characterized by their wavelengths and their frequencies. The same is true of matter waves. De Broglie suggested that the wavelength of a matter wave, λ, is given by the equation λ = h _ mv (7.4) where h is Planck’s constant, m is the particle’s mass, and v is its velocity. Notice that this equa- tion allows us to connect a wave property, wavelength, with mass, which is characteristic of a particle, allowing us to describe the electron as either a particle or a wave. When first encountered, the concept of a particle of matter behaving as a wave rather than as a solid object is difficult to comprehend.

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  Chemistry

  The Molecular Nature of Matter

  • James E. Brady, Neil D. Jespersen, Alison Hyslop(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Figure 7.34 | A closer look at the variation in ionization energy for the Period 2 elements Li through Ne. Summary 345 Using the Bohr model of hydrogen, explain how a line spectrum of hydrogen is generated Niels Bohr recognized this and, although his theory was later shown to be incorrect, he was the first to propose a model that was able to account for the Rydberg equation. Bohr was the first to introduce the idea of quantum numbers. Describe the main features of the Wave Mechanical Model of an atom The wave behavior of electrons and other tiny particles, which can be demonstrated by diffraction experiments, was suggested by de Broglie. Schrödinger applied wave theory to the atom and launched the theory we call wave mechanics or quantum mechanics. This theory tells us that electron waves in atoms are standing waves whose crests and nodes are stationary. Define and use the three major quantum numbers and their possible values Each standing wave, or orbital, is characterized by three quan- tum numbers, n, ℓ, and m ℓ (principal, secondary, and mag- netic quantum numbers, respectively). Shells are designated by n (which can range from 1 to ¥), subshells by ℓ (which can range from 0 to n - 1), and orbitals within subshells by m ℓ (which can range from -ℓ to +ℓ). Describe electron spin and apply it to explain paramagnetism and diamagnetism along with the Pauli exclusion principle The electron has magnetic properties that are explained in terms of spin. The spin quantum number, m s , can have values of +½ or -½. The Pauli exclusion principle limits orbitals to a maximum population of two electrons with paired spins. Substances with unpaired electrons are paramagnetic and are weakly attracted to a magnetic field. Substances with only paired electrons are diamag- netic and are slightly repelled by a magnetic field.

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  Atomic and Nuclear Chemistry

  Atomic Theory and Structure of the Atom

  • T. A. H. Peacocke, J. E. Spice(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  C H A P T E R 5 THE DEVELOPMENT OF THE Wave Mechanical Model OF THE ATOM Introduction T H E Rutherford-Bohr atom, as discussed in the last chapter, gave a rational explanation of the scattering of a-particles by metal foils and also agreed with Planck's quantum theory as applied to the spectre of hydrogen-like atoms. In its simple form, however, it proved to be inadequate to account, even in a qualitative way, for the spectra of other elements and modifications became necessary. Extension of the Bohr theory To explain the fine structure of the spectral lines (p. 69), Sommerfeld proposed that the electron could rotate in eUiptical as well as circular orbits and introduced an azhnuthal or subsidiary quantum number determining the ratio of the major to the minor axis of the eUipse. Two further quantum numbers had to be added called the magnetic quantum number and the spin quantum number. The magnetic quantum number was to account for the Zeeman effect — the spUtting of the spectral Unes in a magnetic field. The spin quantxun number referred to the spin of the electron which was supposed to rotate like a top one way or the other, and so to cause the multipUcity of the spectral Unes. Detailed electronic structure of the Rutherford-B5hr-SommerfeId atom This theory supposes the nucleus to be surrounded by electrons, particles spinning on their axes and moving in orbits arranged in 80 The Wave Mechanical Model of the Atom 8 1 energy levels at varying distances from the nucleus. Reckoned outwards the levels are given principal quantum numbers of 1, 2, 3, 4, etc., and each level can accommodate a maximum of two, eight, eighteen, and thirty-two electrons respectively. The outer-most level in a free atom never contains more than eight electrons and the next level never more than eighteen. The electrons in each main level are divided into sub-levels called s, p, d and / ; the energy of each sub-level is determined by the value of the sub-sidiary or azimuthal quantum number.

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  Quantum Physics

  An Introduction

  • J Manners(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  In particular, we shall make use of concepts from Newtonian mechanics, electricity and magnetism, and the study of waves. You should note that the major proportion of the material in this chapter is contained in Section 4 and you should reserve at least half of your study time for it. It is very important to note at the outset that the laws of quantum mechanics are completely different from the laws of classical mechanics. And, whereas classical mechanics is based on intuitively reasonable ideas, quantum-mechanical ideas are very unintuitive and may therefore appear somewhat harder to grasp. Nevertheless, it is essential to come to grips with these new ideas in order to understand the behaviour of matter at the atomic level. 2 Towards quantum mechanics In this section, we start by studying some idealized experiments, the results of which motivate the need for a quantum-mechanical description of matter. In Section 2.2, we shall very briefly discuss the scope of quantum mechanics, and we shall contrast some aspects of this theory with the more familiar ideas of classical mechanics. Section 2 ends with a brief description of the kind of waves that are required to give a quantum-mechanical description of matter. 2.1 Electron diffraction experiments In Section 6 of the last chapter we saw how Louis de Broglie suggested a model of nature which assigned wave-like attributes to what our experience of reality had formerly designated a particle (say, an electron). Certain observations can be understood by associating with the particle a de Broglie wavelength given by (Eqn 1.15) where p is the magnitude of the particle’s momentum and h (= 6.626 x 10~34Js) is Planck’s constant. Although we can’t go into details, it is worth noting that a large number of experiments have amply demonstrated that particles such as neutrons, helium atoms and much heavier atoms, and molecules, all show wave-like properties.

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  The Sciences

  An Integrated Approach

  • James Trefil, Robert M. Hazen(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  In a sense, then, the wave-particle duality exists in our minds, and not in nature—nature has arranged things so that what we think doesn’t matter. (a) (b) (c) FIGURE 9-5 A vibrating string adopts a regular pattern, known as a standing wave. These diagrams illustrate fixed patterns with (a) 1 ⁄ 2, (b) 1, and (c) 3 ⁄ 2 wavelengths. 201 9.4 WAVE-PARTICLE DUALITY AND THE BOHR ATOM Quantum Weirdness he fact that quantum objects behave so diferently from objects in our everyday experience causes many people to worry that nature has somehow become “weird” at the subatomic level. he description of particles in terms of a wave deies our commonsense. Situations in which a photon or an electron seems to “know” how an apparatus will be arranged before the arranging is done seem wrong and unnatural. Many people, scientists and nonscientists alike, ind the conclusions of quantum mechanics to be quite unsettling. he American physicist Richard Feynman stressed this point when he said, “I can safely say that nobody understands quantum mechanics. . . . Do not keep saying to yourself, ‘But how can it be like that?’ . . . Nobody knows how it can be like that.” In spite of this rather disturbing situation, the success of quantum mechanics provides ample evidence that there is a correct way to describe an atomic-scale system. If you ignore this fact, you can get into a lot of trouble. Newtonian notions such as position and velocity just aren’t appropriate for the quantum world, which must be described from the beginning in terms of waves and probabilities. Quantum mechanics thus becomes a way of predicting how subatomic objects change in time. If you know the state of an electron now, you can use quan- tum mechanics to predict the state of that electron in the future. his process is identical to the application of Newton’s laws of motion in the macroscopic world. he only diference is that in the quantum world, the “state” of the system is a probability.

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