Quantum Model of Hydrogen Atom

12 Key excerpts on "Quantum Model of Hydrogen Atom"

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  Modern Physics

  • Kenneth S. Krane(Author)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Chapter 7 THE HYDROGEN ATOM IN WAVE MECHANICS These computer-generated distributions represent the probability to locate the electron in the n = 8 state of hydrogen for angular momentum quantum number l = 2 (top) and l = 6 (bottom). The nucleus is at the center, and the height at any point gives the probability to find the electron in a small volume element at that location in the xz plane. This way of describing the motion of an electron in hydrogen is very different from the circular orbits of the Bohr model. © John Wiley & Sons, Inc. 208 Chapter 7 The Hydrogen Atom in Wave Mechanics In this chapter, we study the solutions of the Schrödinger equation for the hydrogen atom. We will see that these solutions, which lead to the same energy levels calculated in the Bohr model, differ from the Bohr model by allowing for the uncertainty in localizing the electron. Other deficiencies of the Bohr model are not so easily eliminated by solving the Schrödinger equation. First, the so-called “fine structure” of the spectral lines (the splitting of the lines into close-lying doublets) cannot be explained by our solutions; the proper explanation of this effect requires the introduction of a new property of the electron, the intrinsic spin. Second, the mathematical difficulties of solving the Schrödinger equation for atoms containing two or more electrons are formidable, so we restrict our discussion in this chapter to one-electron atoms, in order to see how wave mechanics enables us to under- stand some basic atomic properties. In Chapter 8, we discuss the structure of many-electron atoms. 7.1 A ONE-DIMENSIONAL ATOM Quantum mechanics gives us a view of the structure of the hydrogen atom that is very different from the Bohr model. In the Bohr model, the electron moves about the proton in a circular orbit.

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  Electrons, Neutrons and Protons in Engineering

  A Study of Engineering Materials and Processes Whose Characteristics May Be Explained by Considering the Behavior of Small Particles When Grouped Into Systems Such as Nuclei, Atoms, Gases, and Crystals

  • J. R. Eaton(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  While an accurate geometric description of the atom is impossible, it may be loosely stated that the shape and size of the atom is dependent on the values of the several quantum numbers. In the Bohr model of the hydrogen atom, the electron moves in a circular path known as an orbit , and a specific orbit is associated with each of the several values of n. In the wave mechanical model of the hydrogen atom, the electron moves through three-dimensional space in a pattern known as an orbital, and a specific orbital is associated with each of the several sets of quantum numbers. 5.8. ORBITALS The preceding section indicates that it is impossible to describe the hydrogen atom by defining the path of the electron as it moves about the nucleus. About the best that can be done is to present a visualization of the orbital in terms of the probability density or electron cloud in the three dimensional space sur-rounding the nucleus. A discussion of the characteristics of the hydrogen atom in its several states aids in giving a general picture of the electron orbitals of atoms of many other elements. For example, an understanding of the general characteristics of the hydrogen atom in the 2-p state, gives a qualitative insight into the behavior of the 2-p electrons of another element such as carbon. STRUCTURE OF THE ATOM 71 (a) Description of Orbitals The three-dimensional orbitals to be discussed here will be described in terms of spherical co-ordinates, Fig. 5.4. The wave mechanical solution of the prob-ability function can be worked out such that the function itself is the product of three independent coordinate functions. U = U,(r) ϋ 2 (φ) ϋ 3 (θ). (5.25) Here C/is the probability function such that UAV is the probability of finding the electron in the small incremental volume A V. U x {r) is a factor which is a function of the radial distance r only, ϋ 2 (φ) is a function of the angle φ only, and ϋ 3 (θ) is a function of the angle 0 only.

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  Absolutely Small

  • Michael D. Fayer(Author)
  • 2010(Publication Date)
  • AMACOM

    (Publisher)

  The Hydrogen Atom: Quantum Theory

  IN 1925 SCHRÖDINGER AND HEISENBERG

  separately developed quantum theory. Their two formulations are mathematically different, but their theories are rigorous and form the underpinning of modern quantum theory. At about the same time, Dirac made major contributions as well. First, he presented a unified view of quantum theory that showed that the Schrödinger and Heisenberg theories, while mathematically different, were equivalent representations of quantum mechanics. In addition, he developed a quantum theory for the hydrogen atom that is also consistent with Einstein’s Theory of Relativity. The formulation by Schrödinger is the most often used to describe atoms and molecules. Therefore, most of our discussions, starting with the hydrogen atom and then going on to larger atoms and molecules, will be based on the concepts and language that is inherent in the Schrödinger approach.

  THE SCHRÖDINGER EQUATION

  We used a very simple but correct mathematical method for obtaining the energy levels of the particle in the box and the wavefunctions, but the method we used is not general. For example, it cannot be used to find the energy levels and the wavefunctions for the hydrogen atom. In fact, the language we have been using, that is, wavefunctions and probability amplitude waves, comes from Schrödinger’s formulation of quantum theory. In 1925 Schrödinger presented what has come to be known as the Schrödinger Equation. The Schrödinger Equation is a complicated differential equation in three dimensions. We will not do the mathematics necessary to solve the Schrödinger equation for the hydrogen atom or other atoms or molecules. However, we will use many of the results to learn about atoms and molecules, beginning with the hydrogen atom.

  The solution of the hydrogen atom problem using the Schrödinger Equation is particularly important because it can be solved exactly. The hydrogen atom is a “two-body” problem. There are only two interacting particles, the proton and the electron. The next simplest atom is the helium atom, which has a nucleus with a charge of +2 and two negatively charged electrons. This is a three-body problem that cannot be solved exactly. In classical mechanics, it is also not possible to solve a three-body problem. The problem of determining the orbits of the Earth orbiting the Sun with the Moon orbiting the Earth cannot be solved exactly with classical mechanics. However, in both quantum mechanics and classical mechanics, there are very sophisticated approximate methods that permit very accurate solutions to problems that cannot be solved exactly. The fact that a method is approximate does not mean it is inaccurate. Nonetheless, because the hydrogen atom can be solved exactly with quantum theory, it provides an important starting point for understanding more complicated atoms and molecules.

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  Physics in the Modern World

  • Jerry Marion(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  18

  THE MODERN VIEW OF ATOMS

  Publisher Summary

  Atoms emit radiation only with certain definite wavelengths, and each atomic species has its own characteristic spectrum of emitted radiation. This chapter explains the Bohr model of the hydrogen atom. It describes the relative energies for several of the hydrogen atom states. The state of an electron in an atom is completely specified only when all four quantum numbers are given. The chapter discusses the quantum numbers and their significance. It focuses on the quantum theory of the hydrogen atom. The chapter explains the need of new principle in the periodic table and Pauli’s exclusion principle. Light from an ordinary source consists of spontaneous photons that are emitted in random directions. On the other hand, laser photons are emitted in a narrow beam that retains its small size even though it travels a substantial distance through space.

  In the preceding chapter we traced the development of quantum theory from the viewpoint of particles and photons. Actually, there was a parallel historical development concerned with the quantum properties of atoms. We now examine this approach to the problem, and we see, finally, how the two lines of attack merge into a single successful quantum theory.

  By 1912 new and important discoveries concerning atoms and radiation had been accumulating for about 15 years. The electron had been identified by Thomson in 1897. Planck had made his quantum hypothesis in 1900, and Einstein had adopted this idea in 1905 to explain the photoelectric effect. Rutherford’s nuclear model of the atom was proposed in 1911. And in that year, a young Dane, Niels Bohr (1885–1962), came to work in Thomson’s laboratory at Cambridge.

  Bohr wondered what connection there could be between the quantized nature of radiation and the structure of atoms. Experiments had shown that atoms emit radiation only with certain definite wavelengths and that each atomic species has its own characteristic spectrum of emitted radiation. Bohr concluded that if an atom could emit radiation only with definite wavelengths (that is, with discrete energies), the internal energy of the atom must also be quantized. Thomson, Bohr’s host at the Cavendish Laboratory, would not accept the idea that atoms possess a quantized structure–he much preferred a classical atomic model. Several sharp arguments over the matter took place and this unpleasantness caused Bohr to decide to leave Cambridge and spend the remainder of his fellowship in a more forward-looking atmosphere. Bohr chose Manchester, where Rutherford and his colleagues were investigating atomic structures with radioactivity methods.

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  Introductory Chemistry

  An Active Learning Approach

  • Mark Cracolice, Edward Peters, Mark Cracolice(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  The last detail of the quantum mechanical model of the atom comes from the Pauli exclusion principle, named after the Austrian scientist Wolfgang Pauli (Figure 11.21). Its effect is to limit the population of any orbital to two electrons. At any instant, an orbital may be (1) unoccupied, (2) occupied by one electron, or (3) occupied by two electrons. No other occupancy is possible. Summary of the Quantum Mechanical Model of the Atom The quantum mechanical model of the atom is summarized in the a summary of… box that follows. Refer back to it as you study the next section, in which we will tie together the quantum model and the periodic table. You will find that this connec- tion will help you better understand the quantum model. Figure 11.22 presents an analogy that will also help you learn the quantum model. x y z 3s s atomic orbitals x y z 2s x y z 1s Figure 11.20 Relative sizes of the 1s, 2s, and 3s orbitals according to the quantum mechanical model of the atom. Figure 11.21 Wolfgang Pauli (1900–1958) discovered the exclusion principle and was awarded the 1945 Nobel Prize in Physics for his work. Bettmann/Getty Images a summary of... The Quantum Mechanical Model of the Atom Principal Energy Levels Principal energy levels are identified by the principal quantum number, n, in a series of counting num- bers: n 5 1, 2, 3, p , 7. Generally, energy increases with increasing n: n 5 1 , n 5 2 , n 5 3, p . Sublevels Each principal energy level—each value of n—has n sublevels. These sublevels are identified by the principal quantum number followed by the letter s, p, d, or f. Sublevels that are not needed for the ground state electron configurations of elements known today appear in blue.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  30.5 The Quantum Mechanical Picture of the Hydrogen Atom 865 According to the Bohr model, the nth orbit is a circle of radius r n , and every time the position of the electron in this orbit is measured, the electron is found exactly at a distance r n away from the nucleus. This simplistic picture is now known to be incorrect, and the quantum mechanical picture of the atom has replaced it. Suppose that the electron is in a quantum mechanical state for which n = 1, and we imagine making a number of measurements of the electron’s position with respect to the nucleus. We would find that its position is uncertain, in the sense that there is a probability of finding the electron sometimes very near the nucleus, sometimes very far from the nucleus, and sometimes at intermediate locations. The probability is determined by the wave function ψ, as Section 29.5 discusses. We can make a three-dimensional picture of our findings by marking a dot at each location where the electron is found. More dots occur at places where the probability of finding the electron is higher, and after a sufficient number of measurements, a picture of the quantum mechanical state emerges. Figure 30.12 shows the spa- tial distribution for the position of an electron in a state for which n = 1, ℓ = 0, and m ℓ = 0. This picture is constructed from so many measurements that the individual dots are no longer visible but have merged to form a kind of probability “cloud” whose density changes gradually from place to place. The dense regions indicate places where the probability of finding the electron is higher, and the less dense regions indicate places where the probability is lower. Also indicated in Figure 30.12 is the radius where quantum mechanics predicts the greatest probability per unit radial distance of finding the electron in the n = 1 state. This radius matches exactly the radius of 5.29 × 10 −11 m found for the first Bohr orbit.

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  The Bohr model does not correctly represent this aspect of reality at the atomic level. Check Your Understanding (The answers are given at the end of the book.) 4. In the Bohr model for the hydrogen atom, the closer the electron is to the nucleus, the smaller is the total energy of the electron. Is this also true in the quantum mechanical picture of the hydrogen atom? 5. In the quantum mechanical picture of the hydrogen atom, the orbital angular momentum of the electron may be zero in any of the possible energy states. For which energy state must the orbital angular momentum be zero? 6. Consider two different hydrogen atoms. The electron in each atom is in a different excited state, so that each electron has a different total energy. Is it possible for the electrons to have the same orbital angular momentum L, according to (a) the Bohr model and (b) quantum mechanics? 7. The magnitude of the orbital angular momentum of the electron in a hydrogen atom is observed to increase. According to (a) the Bohr model and (b) quantum mechanics, does this necessarily mean that the total energy of the electron also increases? 30.6 | The Pauli Exclusion Principle and the Periodic Table of the Elements Except for hydrogen, all electrically neutral atoms contain more than one electron, with the number given by the atomic number Z of the element. In addition to being attracted by the nucleus, the electrons repel each other. This repulsion contributes to the total energy of a multiple-electron atom. As a result, the one-electron energy expression for hydrogen, E n 5 2(13.6 eV) Z 2 /n 2 , does not apply to other neutral atoms. However, the simplest approach for dealing with a multiple-electron atom still uses the four quantum numbers n, ,, m , , and m s . n = 2  = 1 m  = 0 (a) (b) n = 2  = 0 m  = 0 Figure 30.13 The electron probability clouds for the hydrogen atom when (a) n 5 2, , 5 0, m , 5 0 and (b) n 5 2, , 5 1, m , 5 0.

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  Chemistry

  The Molecular Nature of Matter

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

    (Publisher)

  b. The Bohr Model of Hydrogen The first theoretical model of the hydrogen atom that successfully accounted for the Rydberg equation was proposed in 1913 by Niels Bohr (1885–1962), a Danish physicist. In his model, Bohr likened the electron moving around the nucleus to a planet circling the sun. He sug- gested that the electron moves around the nucleus along fixed paths, or orbits. This model broke with the classical laws of physics by placing restrictions on the sizes of the orbits and the NOTE Niels Bohr won the 1922 Nobel Prize in physics for his work on his model of the hydrogen atom. 7.3 The Bohr Theory 323 7.3 The Bohr Theory 323 energies that electrons could have in given orbits. This ultimately led Bohr to propose an equa- tion that described the energy of the electron in the atom. The equation includes a number of physical constants such as the mass of the electron, its charge, and Planck’s constant. It also contains an integer, n, that Bohr called a quantum number. Each of the orbits is identified by its value of n. When all the constants are combined, Bohr’s equation becomes E = − b _ n 2 (7.3) where E is the energy of the electron and b is the combined constant (its value is 2.18 × 10 –18 J). The allowed values of n are whole numbers that range from 1 to ∞. From this equation, the energy of the electron in any particular orbit could be calculated. Because of the negative sign in Equation 7.3, the lowest (most negative) energy value occurs when n = 1, which corresponds to the first Bohr orbit. The energy of the electron is negative because the electron is attracted to the nucleus, and energy is released due to the attraction. The lowest energy state of an atom that is, the most negative, is the most stable one and is called the ground state. For hydrogen, the ground state occurs when its electron has n = 1. According to Bohr’s theory, this orbit brings the electron closest to the nucleus.

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  Theoretical Spectroscopy of Transition Metal and Rare Earth Ions

  From Free State to Crystal Field

  • Mikhail G. Brik, Ma Chong-Geng, Mikhail G. Brik, Ma Chong-Geng(Authors)
  • 2019(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  Being based on these postulates and employing classical physics, N. Bohr was able to derive explicit expressions for the energy levels of the hydrogen atom and the wavelengths of possible transitions (this derivation will not be repeated here, since it can be found even in the school textbooks on physics). The obtained results agreed perfectly with the experimental data, and in 1922 N. Bohr was awarded the Nobel Prize in physics for his pioneering works.

  Bohr’s theory—though it was a real breakthrough in developing our knowledge about the fundamental properties of matter—was still not very consistent. It was, in fact, “a happy alliance” between the classical physics and new quantum-mechanical postulates. Obvious intrinsic inconsistency of Bohr’s theory, on one side, and its remarkable success in explanation of the observed spectra, on the other side, turned out to be a strong driving force for creating a new theory, with new physical concepts and a new mathematical apparatus. Such a theory was developed in the late 1920s and was called quantum (or wave) mechanics. Its application to the description of the hydrogen atom finally provided physicists with strictly consistent explanation of the spectral series and N. Bohr’s postulates.

  3.3    Solution of the Schrödinger Equation for the Hydrogen-Like Atom

  The hydrogen-like atom (or an ion) has only one electron, and all others have been removed by various ways of ionization. An essential feature of these systems is the presence of one electron only. The Coulomb interaction between this electron and the nucleus is spherically symmetrical. Later, we shall see that the situation becomes considerably more complicated in the case of multielectron atoms.

  In the present paragraph, we shall be concerned with finding a solution of the following physical problem: determination of a spectrum (possible energetic states and corresponding wave functions) of an electron with the mass m and the charge −e revolving around a nucleus with the mass M and the charge +Ze (Fig. 3.1

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  Introduction to Modern Physics

  • John Mcgervey(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R The Hydrogen Atom One of the greatest and earliest triumphs of the Schrodinger equation was the solution of the problem of the hydrogen atom, which was achieved without the assumption of arbitrary rules like those of the old quantum theory. Although it was still impossible to obtain exact solutions for atoms containing more than one electron, the new quantum theory permitted approximations which could be applied, in principle, to any problem and to any desired degree of accuracy. The ultimate result was almost unbelievable accuracy in theoretical calculations of hydro-gen energy levels, taking into account such factors as the spin of the electron, a quantity which was unheard of in the older theory. Experimenters have responded with equally accurate experiments to test these calculations. This accuracy has not been sought simply to demonstrate the prowess of physics and physicists; proof of the existence of each small contribution to the energy in the amount predicted by the theory is an indication of the correctness of our fundamental ideas concerning the nature of matter. In 1930, Dirac showed that electron spin emerges as a natural consequence of a relativistic wave equation for the electron. We close this chapter with a brief account of the Dirac equation. We cannot treat the Dirac equation in the detail with which we have treated the Schrodinger equation, but we can see that two fundamental properties of matter— internal spin of particles and the existence of antimatter— .can be 4 'derived'' directly from this equation. 239 240 THE HYDROGEN ATOM 7.1 WAVEFUNCTIONS FOR MORE THAN ONE PARTICLE If the proton were of infinite mass, it would be a fixed center of force for the electron in the hydrogen atom, and we could solve the problem by the methods of Chapter 6. The wavefunction would be a function of the co-ordinates of a single particle, the electron. However, because the proton also moves, we must incorporate this fact into our wavefunction.

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  Physical Chemistry

  • David Ball(Author)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 376 Chapter 11 | Quantum Mechanics: Model Systems and the Hydrogen Atom Unless otherwise noted, all art on this page is © Cengage Learning 2014. where a is the same constant previously defined for the R functions. Because it is defined as a group of constants, a itself is a constant and has units of length (shown in equation 11.69 in units of Å). The constant a is called the Bohr radius . This most probable distance is exactly the same distance that an electron of Bohr’s theory would have in its first orbit. Quantum mechanics does not constrain the distance of the electron from the nucleus as did Bohr’s theory. But it does predict that the distance Bohr calculated for the electron in its lowest energy state is in fact the most probable distance of the electron from the nucleus. (A similar constant is a 0 , which is defined similarly but uses the mass of the electron instead of the reduced mass of the hydrogen atom. The difference is very slight.) EXAMPLE 11.24 a. What is the probability that an electron in the C 1 s orbital of hydrogen will be within a radius of 2.00 Å from the nucleus? b. Calculate a similar probability, but now for an electron within 0.250 Å of a Be 3 1 nucleus. SOLUTION a. For a normalized wavefunction, the probability P is equal to P 5 3 b a C * C d t where a and b are the limits of the space being considered.

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  Modern Physics for Scientists and Engineers

  • Stephen Thornton, Andrew Rex, Carol Hood, , Stephen Thornton, Stephen Thornton, Andrew Rex, Carol Hood(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Although we generally confine ourselves to hydrogen in this chapter, we occasion- ally digress—for example, for the Stern–Gerlach experiment in Section 7.4—in order to incorporate an important experimental result that we need for our un- derstanding of the hydrogen atom. Two sections in this chapter (Sections 7.2 and 7.6) are advanced topics and may be skipped without losing continuity. 7.1 Application of the Schrödinger Equation to the Hydrogen Atom The hydrogen atom is the first system we shall consider that requires the full complexity of the three-dimensional Schrödinger equation. To a good approxi- mation the potential energy of the electron-proton system is electrostatic: V sr d 5 2 e 2 4pe 0 r (7.1) 7 The Hydrogen Atom Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 246 Chapter 7 The Hydrogen Atom We rewrite the three-dimensional time-independent Schrödinger Equation (6.43) as 2 " 2 2m 1 c sx, y, zd 3 − 2 c sx, y, zd −x 2 1 − 2 c sx, y, zd −y 2 1 − 2 c sx, y, zd −z 2 4 5 E 2 V sr d (7.2) As discussed in Chapter 4, the correct mass value m to be used is the reduced mass m of the proton-electron system. We can also study other hydrogen-like (called hydrogenic) atoms such as He 1 or Li 11 by inserting the appropriate re- duced mass m and by replacing e 2 in Equation (7.1) with Z e 2 , where Z is the atomic number. We note that the potential V(r) in Equation (7.2) depends only on the distance r between the proton and electron. To take advantage of radial sym- metry, we transform to spherical polar coordinates.

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