Physics
Quantum Spin
Quantum spin is an intrinsic property of elementary particles, such as electrons, that is not related to actual spinning motion. It is a fundamental aspect of quantum mechanics and is characterized by the particle's angular momentum. Quantum spin can take on discrete values, typically expressed in units of ħ/2, where ħ is the reduced Planck constant.
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12 Key excerpts on "Quantum Spin"
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 - PDFQuantum Untangling
An Intuitive Approach to Quantum Mechanics from Einstein to Higgs
- Simon Sherwood(Author)
- 2023(Publication Date)
- Wiley(Publisher)
The idea of versus Φ makes no sense. Quantum Untangling: An Intuitive Approach to Quantum Mechanics from Einstein to Higgs, First Edition. Simon Sherwood. © 2023 John Wiley & Sons Ltd. Published 2023 by John Wiley & Sons Ltd. 162 17 Spin Instead of worrying about classical preconceptions, we must embrace the quantum definition of angular momentum as the angular variation in the wave function. has no more to do with the electron spinning than has with the electron orbiting the nucleus. What is called intrinsic spin is an underlying angular variation in the wave function that exists for almost all elementary particles. In Figure 15.3, we compared angular excitation with radial excitation for the electron in hydrogen. Using the Schrödinger equation we derived its ground state ( = 1, = 0, = 0) as having radial excitation but no angular excitation. This is not the full story. The electron’s intrinsic spin means that in the ground state it actually has ℏ 2 of angular momentum. The angular momentum can be clockwise or anticlockwise around the axis so there are two distinct ground states that we label ( = 1, = 0, = 0, = + 1 2 ) and ( = 1, = 0, = 0, = − 1 2 ). We distinguish between the electron spin states using the names spin-up and spin-down. This means that for each energy-momentum eigenstate of hydrogen (listed in the table in Section 16.2) there are two distinct electron states, one spin-up and one spin-down. Hence each one can accommodate two electrons. Why does the electron have this intrinsic angular momentum ? You will learn in the next chapter that Schrödinger’s equation is a good approximation, but not quite right for the electron. The mass-energy-momentum balance that sits behind the correct equation (the Dirac equation) is subtly different and requires that this intrinsic angular momentum exist. It is perhaps unfortunate that physicists called it spin which is misleading, but the name has stuck.
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.
- David A. B. Miller(Author)
- 2008(Publication Date)
- Cambridge University Press(Publisher)
Spin Spin is a property intrinsic to each elementary particle that behaves like an angular momentum but that cannot be written as a function of space. Whereas orbital angular momentum has integer values l = 0, 1, 2, …, for the electron, the magnitude s of the spin is ½. 7 See P. A. M. Dirac, The Principles of Quantum Mechanics (4 th Edition, revised) (Oxford, 1967), Chapter XI. 8 The same equation he derived also proposed the positron, the positive antiparticle to the electron. 9 W. Greiner, Quantum Mechanics (Third Edition) (Springer-Verlag, Berlin, 1994), Chapter 13. 12.8 Summary of concepts 309 State vectors for electron spin A general state of electron spin can be represented by a linear combination of two basis states, one corresponding to the spin-up state, written as ↑ , 1/ 2,1/ 2 , or 1 0 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ , and the other corresponding to a spin-down state, written as ↓ , 1/ 2, 1/ 2 − , or 0 1 ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ . The “up” and “down” refer to the z direction, conventionally, though any axis in space can be chosen. A general electron spin state can, therefore, be written as 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2 1/ 2,1/ 2 1/ 2, 1/ 2 a s a a a a a − − − ⎡ ⎤ = + − ≡ ↑ + ↓ ≡ ⎢ ⎥ ⎣ ⎦ (12.13) Spin operators By analogy with orbital angular momentum operators, spin operators ˆ x S , ˆ y S , and ˆ z S can be defined with analogous commutation relations.
eBook - PDFAtomic Structure and Lifetimes
A Conceptual Approach
- Lorenzo J. Curtis(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
For example, the language by which atomic levels are described is based on hydrogenic quantum numbers, and the characterization of spin and space properties is made as if they were the independent quantities that are obtained for single-electron systems of low nuclear charge. These concepts and notational assumptions must be continually re-examined in complex atomic systems. 4.2 The intrinsic angular momentum and magnetic moment of the electron The ratio of the magnetic moment to the angular momentum of an electron is approximately twice the so-called “classical” value e / 2 m that occurs when mass and charge are assumed 74 4 The vector model of angular momentum to have identical distributions. There is really nothing “nonclassical” about such a value, since the same gyromagnetic ratio of two occurs for any uniform solid cylinder of mass that spins about its axis, and has a uniform charge confined to its outer cylindrical surface. However, attempts to apply this macroscopic model to the electron invariably lead to self-contradictory results. The minimum radius for a mechanically spinning electron model that will yield a value for the electromagnetic inertia that does not exceed its observed mass (the so-called “classical electron radius”) leads to a tangential velocity much greater than the speed of light. Thus, any attempt to gain conceptual insights by considering the electron as anything other than a point particle are ill-conceived and counter-pedagogic. However, a very attractive model does exist that provides a clear conceptual picture of a mechanism by which a point particle can possess both an angular momentum and a magnetic moment. This lies in the formulation of the Foldy–Wouthuysen transformation. In 1950, L. L. Foldy and S. A. Wouthuysen reported [98] on a phenomenon similar to that encountered in Schr¨ odinger’s first and second equations, in that it involved differing choices in evaluating a nonrelativistic limit.
- Andrei D. Polyanin, Alexei Chernoutsan(Authors)
- 2010(Publication Date)
- CRC Press(Publisher)
It turned out that the intrinsic magnetic moment of an electron has anomalous gyromagnetic ratio g s = 2 : the ratio of the spin magnetic moment to the spin angular momentum is equal not to e/ 2 m but to e/m . The magnetic moment projection on the axis z takes the values ( e/m ) planckover2pi1 m s = ± μ B . The anomalous values of the gyromagnetic ratio are confirmed by the results of the Einstein–de Haas experiments (1914), in which magnetomechanical effects were studied (formation of a mechanical angular momentum in magnetization of an iron sample). All subatomic particles have an intrinsic angular momentum associated—the spin. The spin quantum number of proton, neutron, and μ -mesons is equal to 1/2, that of photon is equal to 1, and that of π -mesons is 0. ◮ Fine structure of spectral lines. The fine structure of lines of hydrogen-like atoms and alkali metals can be explained by interaction between the spin magnetic moment and the orbital angular momentum. The easiest way to understand the appearance of such an interaction is to use the reference frame associated with the electron. In this reference frame, the moving nucleus creates a magnetic field with which the spin magnetic moment interacts. With the spin-orbit interaction (also known as spin-orbit coupling ) taken into account, the following operators are preserved (commute with the Hamiltonian): the total 586 Q UANTUM M ECHANICS . A TOMIC P HYSICS angular momentum hatwide J 2 , its projection hatwide J z , and the value of the orbital angular momentum hatwide L 2 . The position of the levels E n , l , j is determined by three quantum numbers, n , l , and j .
eBook - PDFGroup Theory
And its Application to the Quantum Mechanics of Atomic Spectra
- Eugene Wigner(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
(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. The question of the probability for the different spatial directions of spin conse-quently has no meaning, i.e., will be unanswerable by experiment, and only 220 ELECTRON SPIN 221 the component of the spin in a single direction can be measured. These measurements, which are exemplified by the Stern-Gerlach experiment, can give only two answers : either that the spin is in the direction in question, or that it is in the opposite direction. The possible experimental results for the component of the angular momentum in the direction in question are H h -— or . If the former result has been obtained when measuring the 2 2 component of the spin in the Z-direction, then a second measurement of the Z-component carried out immediately afterwards will yield --Z with certainty, and with certainty does not yield — Z. A measurement of the T-component, on the other hand, yields the two possible results + T and — Y with equal probabilities. It is important, therefore, to ascribe indepen-dent probabilities to all directions of the spin; even if the spin is certainly in the Z-direction (i.e., if the Z-component of the angular momentum is +T&/2 with certainty) the probability for the --Y direction is 1/2, and for all directions except — Ζ the probability is different from zero. Spin obtains an even more symbolic character in Dirac's relativistic theory of the electron, as was emphasized particularly by N. Bohr. According to this theory (which we shall not discuss here) the existence of the magnetic moment is entirely a relativistic effect which appears automatically when space and time are treated equivalently.
eBook - PDF- Steven Weinberg(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
Any component of angular momentum s would take 2s + 1 values, so the quantity s correspond- ing to for the internal angular momentum would have to have the unusual value 1/2. This internal angular momentum came to be called the electron’s spin. At first this idea was widely disbelieved. As we saw in Section 2.1, orbital angular momentum cannot have the non-integer value = 1/2. Another worry was that if a sphere with the mass of the electron and with angular momentum 1 S. Goudsmit and G. Uhlenbeck, Naturwissenschaften 13, 953 (1925); Nature 117, 264 (1926). 104 4 Spin et cetera 105 /2 has a rotation velocity at its surface less than the speed of light, then its radius must be larger than /2m e c 2 × 10 −11 cm, and it was presumed that an electron radius that large would not have escaped observation. Electron spin became more respectable a little later, when several authors 2 showed that the coupling between the electron’s spin and its orbital motion accounted for the fine structure of hydrogen – the splitting of states with = 0 into doublets. (This is discussed in Section 4.2.) The worries about models of spinning electrons were due to the lingering wish to understand quantum phenomena in classical terms. Instead, we should think of the existence of both spin and orbital angular momenta as consequences of a symmetry principle. We saw in Sections 3.4–3.6 how symmetry principles imply the existence of conserved observables such as energy and momentum. There is another classic symmetry of both non-relativistic and relativistic physics, invariance under spatial rotations. In Section 4.1 we will show how rotational invariance leads in quantum mechanics to the existence of a conserved angular- momentum three-vector J. The commutation relations of these operators will be used in Section 4.2 to derive the spectrum of eigenvalues of J 2 and J 3 , and to find how all three components of J act on the corresponding eigenstates.
eBook - PDF- J.C. Anderson, Keith D. Leaver, Rees D. Rawlings, Patrick S. Leevers(Authors)
- 2004(Publication Date)
- CRC Press(Publisher)
Pictorially we may represent the spinning electron as a fuzzy distribution of mass rotating about its own centre of gravity, rather as a planet spins on its axis while orbiting the Sun. It is difficult to couple this image with that of the wave motion around the nucleus and the reader is not advised to try to do so! However, a spinning wave packet is not too difficult to imagine, although more difficult to draw. Fig. 3.7 The components of angular momentum for the precessing wave, (b) shows that the component meh/2ir can sometimes point in the opposite direction to a magnetic flux density B, when me has a negative value. 48 THE SIMPLEST ATOM: HYDROGEN Fig. 3.8 A pictorial spinning electron (a) and equivalent motion of a 'particle electron' (b). As might be expected, the angular momentum of this spinning motion is quantized and in units of h/27r. To distinguish it from the orbital angular momentum, £h/27r, we call it the spin angular momentum and its magnitude is m s h/27r, where m s is the spin quantum number. Unlike £ and ma, however, m s does not have integral values, but can only take one of two values, + and -. Thus, when a magnetic field is present, the motion can be either clockwise or anticlockwise about the magnetic field direction, the spin angular momentum being equal to ±m s h/27r. Because the vector representing this points along the axis of rotation, the two states are often referred to as 'spin up' and 'spin down' (Fig. 3.8). An explanation of the reason for the non-integral values of m s is beyond the scope of this book. 3.4 Electron clouds In the hydrogen atom Because the shape of the wave function ip is defined by the quantum numbers n, £ and m¿, so too is the probability density distribution for the electron. Thus we may picture an 'electron cloud' that represents the charge distribution in a hydrogen atom using the value of eip 2 as a measure, just as described in Chapter 2.
eBook - PDF- Henrik Smith(Author)
- 1991(Publication Date)
- WSPC(Publisher)
152 Introduction to Quantum Mechanics 7 ANGULAR MOMENTUM According to classical mechanics the angular momentum of a particle with respect to a given point in space is conserved, when the potential energy of the particle only depends on the distance from the particle to the given point. Such a potential is called a central field. Since the force on the particle moving in a central field is directed towards or away from the given point, it follows that the moment of the force with respect to this point must be zero, and the angular momentum with respect to the same point is therefore independent of time. The conservation of angular momentum is a consequence of the symmetry of the system, the potential energy being invariant under rotations about the given point. As we shall see, the existence of this symmetry means that the operators for each component of the angular momentum commute with the Hamiltonian. However, the individual components of the angular momentum operator do not commute with each other. This may be seen from the classical expression for the angular momentum L, L = r x p , (7.1) where r and p in quantum mechanics are represented by operators which do not commute. The commutation relations for the components of the angular momentum will be derived in Section 7.1.2 below, starting from the classical expression (7.1). These commutation relations imply that the three components cannot take on definite values at the same time. It is however possible to find states which are simultaneous eigenstates for one of the components of the angular momentum and the square of its length. The eigenvalues for one of the com-ponents of the angular momentum turn out to be an integer or a half-integer times the Planck constant ft. The half-integer values cannot be derived from the orbital angular momentum alone. They are a consequence of the Lorentz invariance that must apply to the wave equation describing the motion of the particle.
eBook - PDF- Supriyo Bandyopadhyay, Marc Cahay(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
The question now is how to include “spin” in Equation (2.4)? This was investigated by Wolfgang Pauli. He derived an equation to replace Equation (2.4) which bears his name and is known as the Pauli Equation . But before we discuss this equation, we need to understand an important concept, namely, Pauli spin matrices , since they appear in the Pauli equation. 2.1 Pauli spin matrices In quantum mechanics, any physical observable is associated with an operator (which would be a linear operator in the Schr¨ odinger formalism, or a matrix in the Heisenberg formalism). The eigenvalues of the linear operator, or the eigenvalues of the matrix, are the expectation values of the physical quan-tity, i.e.,, the values we expect to find if we measure the physical quantity in an experiment † . Spin is a physical observable since the associated angular momentum can be measured, as was done unwittingly by Stern and Gerlach. Consequently, there must be a quantum mechanical operator associated with spin. Pauli derived the quantum mechanical operators for the spin compo-nents along three orthogonal axes – S x , S y and S z . They are 2 × 2 complex † Repeated measurements of a physical observable will produce a distribution of values whose average will be the expectation value. 20 Introduction to Spintronics matrices that came to be known as the Pauli spin matrices. Pauli’s approach was based on the premise that: (1) the measurement of the spin angular mo-mentum component along any coordinate axis for an electron should give the results + planckover2pi1 2 or − planckover2pi1 2 , and (2) the operators for spin components along three mu-tually orthogonal axes should obey commutation rules similar to those obeyed by the operators associated with components of the orbital angular momen-tum. This would put spin angular momentum and orbital angular momentum on the same footing.
eBook - PDF- Steven Weinberg(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
12 N. M. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949). 13 J. B. French and V. F. Weisskopf, Phys. Rev. 75, 1240 (1949). 162 5 Quantum Mechanics the particle is in a state with orbital angular momentum , the parity of its state is (−1) π n . In our discussion above we have implicitly taken the electron to have positive intrinsic parity. This is a matter of definition; if the electron had negative intrinsic parity we could redefine the parity operator as = exp(iπQ/e), where Q is the operator for total electric charge. The one-electron state is an eigenstate of Q with eigenvalue −e, so it is an eigenstate of exp(iπQ/e) with eigenvalue −1; if it were an eigenstate of with eigenvalue −1 it would be an eigenstate of with eigenvalue +1. Since Q as well as commutes with the Hamiltonian, so does and it can be called the operator of space inversion just as well as . In the same way, because of the conservation of another quantity known as baryon number (described in Section 6.2) we can define the parity of the proton as +1. But the intrinsic parities of most particles have to be determined experimentally. Hyperfine Structure We must not forget the atomic nucleus, for if it has spin this produces a magnetic field felt by orbiting electrons. This effect is most important for the s -wave electrons that are not prevented from getting close to the nucleus by the centrifugal barrier that is present for = 0. In hydrogen the spin 1/2 of the nucleus combines with the spin 1/2 of the electron in its = 0 ground state to split the energy of the ground state into components with total spin s = 0 and s = 1, separated in energy by 5.9 × 10 −6 eV. The transition between these states produces the famous 21-cm absorption and emission spectral lines, discussed in Section 3.5.
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