Physics
Spin Properties
Spin properties refer to intrinsic angular momentum possessed by elementary particles, such as electrons and quarks. This property is a fundamental aspect of quantum mechanics and is characterized by the spin quantum number, which can take on values of 1/2, 1, 3/2, and so on. Spin plays a crucial role in determining the behavior of particles in magnetic fields and is a key factor in understanding particle interactions.
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6 Key excerpts on "Spin Properties"
eBook - PDF- Supriyo Bandyopadhyay, Marc Cahay(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
1 Early History of Spin 1.1 Spin Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechan-ical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can-not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e 2 / (4 πǫ 0 m 0 c 2 ) (where e is the electron’s charge, m 0 is the mass, c is the speed of light in vacuum, and ǫ 0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quan-tum mechanics is required to understand how the spin property comes about.
eBook - PDF- Ajit Kumar(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
Chapter 8 Quantum Mechanical Theory of the Spin Angular Momentum 8.1 Spin Spin angular momentum or simply spin is a fundamental property of all particles, irrespective of whether they are elementary or composite. It belongs to an internal degree of freedom (completely independent of the spatial degrees of freedom) and manifests itself as some intrinsic angular momentum of the particle. It was introduced in quantum mechanics as an attempt to explain the experimentally observed fine structures of the spectral lines in the emission spectra of alkali metals and the peculiarities involved in the anomalous (complex) Zeeman effect that showed the unusual splitting pattern of atomic energy levels in the presence of a weak external magnetic field. Note that all efforts, prior to the conjecture about spin, to explain the aforementioned experimental results on the basis of the Schr¨ odinger equation without spin had miserably failed. An atom of any of the alkali metals has an almost inert core, consisting of the nucleus and (Z - 1) inner electrons, together with a single outer electron. The transitions of the outer electron between energy levels are responsible for the aforementioned spectral lines. Therefore, any additional property required to be postulated for the explanation of the fine structures of the spectral lines or anomalous Zeeman effect, had to be attributed to the valence electron. It is because of this reason that Uhlenbeck and Goudsmit put forward their conjecture about electron’s spin. They assumed that, similar to Earth’s spinning motion about its axis, an electron, in addition to its orbital motion about the nucleus, also possessed a spinning motion about its axis of symmetry. The angular momentum related to this spinning motion was given the name ‘spin’.
eBook - PDF- Benjamin Schumacher, Michael Westmoreland(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
12 Spin and rotation 12.1 Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often dis-tinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle . 1 The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-1/2 particle. Our approach began with the empirical observation that a measurement of any spin com-ponent of a spin-1/2 particle could yield only the results + / 2 or − / 2. We introduced the basis states | z ± for the two-dimensional Hilbert space H . We also gave other basis states such as { | x ± } and { | y ± } in terms of the | z ± states. From basis states and meas-urement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56) . Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic. We now begin with spin component operators that are assumed to satisfy the same commutation relations we obtained for the spin-1/2 operators. Amazingly, this will be a sufficient foundation to derive everything – the eigenvalues, eigenvectors, and matrix representations for all of the spin operators. Later, in Section 12.3 , we will see how the commutation relations themselves follow naturally from the geometry of 3-D rotations.
eBook - PDFNuclear Magnetic Resonance Spectroscopy
An Introduction to Principles, Applications, and Experimental Methods
- Joseph B. Lambert, Eugene P. Mazzola, Clark D. Ridge(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The magnetic moment 𝝁 is a vector, because it has both magnitude and direction, as defined by its axis of spin in the figure. In this context, boldface symbols connote a vectorial parameter; when only the magnitude is under consideration, the symbol is depicted without boldface, as 𝜇 . The NMR experi-ment exploits the magnetic properties of nuclei to provide information on the molecular structure. The Spin Properties of protons and neutrons in the nuclei of heavier elements combine to define the overall spin of the nucleus. When both the atomic number (the number of protons) and the atomic mass (the sum of the protons and neutrons) are even, the nucleus has no magnetic properties, as signified by a zero value of its spin quantum number , I (Figure 1.2). Such nuclei are considered not to be spinning. Common non-magnetic (nonspinning) nuclei are carbon ( 12 C) and oxygen ( 16 O), which therefore are invisible to the NMR experiment. When either the atomic number or the atomic mass is odd, or when both are odd, the nucleus has magnetic properties that correspond to spin. Nuclear Magnetic Resonance Spectroscopy: An Introduction to Principles, Applications, and Experimental Methods, Second Edition. Joseph B. Lambert, Eugene P. Mazzola, and Clark D. Ridge. © 2019 John Wiley & Sons Ltd. Published 2019 by John Wiley & Sons Ltd. 2 Nuclear Magnetic Resonance Spectroscopy Spinning spherical nucleus Charge moving in a circle μ μ Figure 1.1 Analogy between a charge moving in a circle and a spinning nucleus. No spin l = 0 Spinning sphere l = 2 1 Spinning ellipsoid l = 1, 2 3 , 2, ... Figure 1.2 Three classes of nuclei. For spinning nuclei, the spin quantum number can take on only certain values, which is to say that it is quantized. Those nuclei with a spherical shape have a spin I of 1 / 2 , and those with a nonspherical, or quadrupolar, shape have a spin of 1 or more (in increments of 1 / 2 ). Common nuclei with a spin of 1 / 2 include 1 H, 13 C, 15 N, 19 F, 29 Si, and 31 P.
eBook - PDFGroup Theory
And its Application to the Quantum Mechanics of Atomic Spectra
- Eugene Wigner(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
20. ELECTRON SPIN THE PHYSICAL BASIS FOR THE PAULI THEORY 1. In the preceding chapter the most important properties of atomic spectra which could be treated without the introduction of electron spin have been discussed. However, many of the less obvious characteristics—among which the fine structure is perhaps the most prominent—could not be described since they are closely related to another property of the electron, its magnetic moment. The hypothesis that the electron has a magnetic moment and an angular momentum, in short a spin, was suggested by Goudsmit and Uhlenbeck. They noted, even before the discovery of quantum mechanics, that complete description of spectra was not possible unless a magnetic moment and a mechanical moment were ascribed to the electron—the concept of an electron as a point charge was insufficient. As is well known, in classical electro-dynamics a magnet is equivalent to a point charge rotating about the axis of the magnetic moment. The vector of the magnetic moment SR is then calculated from the angular momentum vector £ by efi m = = η2 (20.E.1) 2mc where e is the charge of the rotating particle and m its mass. However, according to Goudsmit and Uhlenbeck, Eq. (20.E.1) does not apply to the magnetic moment resulting from spin if one uses the normal electronic charge and mass. Rather, one must assume that the angular momentum is of amount | β | = έ · » , (20.1) whereas the magnetic moment is a whole Bohr magneton |9R| = eh/2 mc = (e/rac)|S| = 2η&. (20.1a) The quantum mechanics of electron spin shows that these statements cannot be taken literally. Even the Pauli theory requires that no experiment can be performed which allows the determination of the direction (and thus, say, of the direction cosines) of the mechanical or magnetic moment. It is possible only to differentiate between one direction and its opposite.
eBook - PDF- J.C. Anderson, Keith D. Leaver, Rees D. Rawlings, Patrick S. Leevers(Authors)
- 2004(Publication Date)
- CRC Press(Publisher)
momentum (b) magnetic properties of the electron (c) the spin of the electron (d) precession of the electron orbit (wave function) 12 The spin quantum number of the electron determines (a) the angular momentum about the nucleus (b) the total angular momentum of the electron (c) the angular momentum of the electron about its own centre of mass 13 The principal quantum number n may have only the values (a) 0,1,2,... (b) 0,±1,±2,±3,... (c) 1,2,3,... 14 The angular momentum quantum number Í may take only the values (a) 0,1,2,3,... (n-1) (b) 0,1,2,3,... n (c) 1,2,3,...ii (d) 1,2,3,...(n-1) SELF-ASSESSMENT QUESTIONS 53 15 The magnetic quantum number m¡ may have only the values (a) 0,±1,±2,...±£ (b) 0 , ± l , ± 2 , . . . ± n (c) 0,±1,±2,...±(£-1) (d) 0 , ± l , ± 2 , . . . ± ( n -l ) 16 The spin quantum number m s may have only the values (a) 0,±± (b) 0 , ± ± ± l , ± § .
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