Angular Momentum Coupling

10 Key excerpts on "Angular Momentum Coupling"

• 

  No longer available |Learn more

  Advances in Chemical Physics, Volume 136

  • Stuart A. Rice(Author)
  • 2008(Publication Date)
  • Wiley-Interscience

    (Publisher)

  As discussed next, cases involving coupled angular momenta and atoms in crossed fields give rise to less obvious isolated critical points, which are best understood by exploring the geometry of the angular momentum space. The following section derives from a series of papers by Sadovskii, Zhilinskii, Cushman, and co-workers [2–6]. 64 mark s. child A. Spin–Rotation Coupling The simplest example of Angular Momentum Coupling is provided by the scaled spin–rotation coupling Hamiltonian ^ H ¼ 1  g S ^ S z þ g NS ^ N  ^ S; 0  g  1 ð42Þ which was used by Sadovskii and Zhilinskii [2] as an early illustration of quantum monodromy. Here S and N are the magnitudes of the spin and rotational angular momenta and ^ J z ¼ S z þ N z is a constant of the motion [50]. It is interesting to follow changes in the qualitative eigenvalue pattern as the coupling strength g increases from Fig. 14a to 14d. Here the chains of small dots are the eigenvalues, the large dots are images of limiting points of the classical 20 10 0 –10 –20 J z J z –1 –0.5 0 0.5 1 E 20 10 0 –10 –20 –1 –0.5 0 0.5 1 E 20 10 0 –10 –20 –1 –0.5 0 0.5 1 20 10 0 –10 –20 –1 –0.5 0 0.5 1 γ = 0.5 γ = 0.1 γ = 0.3 γ = 0.8 D D D Figure 14. Eigenvalue structures in the EJ z map for s ¼ 4; n ¼ 16 and various coupling strengths, g ¼ 0:1, 0.3, 0.5, and 0.8. The eigenvalues appear as strings of small dots. Heavy dots mark the images of the special points A, B, C, and D in the left-hand panel of Fig. 16. The connecting lines are the relative equilibria. quantum monodromy and molecular spectroscopy 65 angular momentum space, at which S z ¼ S and N z ¼ N, and the joining lines are images of the relative equilibria of the classical Hamiltonian. Results are shown for total angular momentum quantum numbers, s ¼ 4 and n ¼ 16. To understand this diagram, note first that the spin–rotation eigenvalues reduce in the Hund’s case (a) limit g ¼ 0, to 2s þ 1 sets, each with degeneracy 2n þ 1.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Student's Guide to Atomic Physics

  • Mark Fox(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  5.5 Angular Momentum Coupling in Single-Electron Atoms If an atom has just a single electron, the addition of the orbital and spin angular momenta is relatively straightforward. The physical mechanism that couples the orbital and spin angular momenta together is the spin-orbit interaction, and the resultant total angular momentum vector j is defined by: j = l + s . (5.24) 94 Angular Momentum Vector j is described by the quantum numbers j and m j , which denote its magnitude and z -component according to the usual rules for quantum mechanical angular momenta, namely: | j | 2 = j ( j + 1 ) ¯ h 2 , (5.25) and j z = m j ¯ h , (5.26) where m j takes values of j , ( j − 1 ) , · · · , − j . The spin-orbit coupling of l and s to form the resultant j is illustrated by Figure 5.2 (b). The magnitudes of the vectors shown in Figure 5.2 (b) are, respectively: | j | = √ j ( j + 1 ) ¯ h , | l | = √ l ( l + 1 ) ¯ h , and | s | = √ s ( s + 1 ) ¯ h . The allowed values of j are worked out by applying Eq. (5.22) , with the knowledge that the spin quantum number s is always equal to 1 / 2. If the electron is in a state with orbital quantum number l , we then find j = l ⊕ s = ( l ± 1 / 2 ) , except when l = 0, in which case we just have j = 1 / 2. In the second case, the angular momentum of the atom arises purely from the electron spin. 5.6 Angular Momentum Coupling in Multi-Electron Atoms The Hamiltonian for an N -electron atom can be written in the form: ˆ H = ˆ H 0 + ˆ H 1 + ˆ H 2 , (5.27) where: ˆ H 0 = N i = 1 − ¯ h 2 2 m ∇ 2 i + V central ( r i ) , (5.28) ˆ H 1 = − N i = 1 Ze 2 4 π 0 r i + N i > j e 2 4 π 0 | r i − r j | − N i = 1 V central ( r i ) , (5.29) ˆ H 2 = N i = 1 ξ( r i ) l i · s i . (5.30) As discussed in Section 4.1 , ˆ H 0 is the central-field Hamiltonian and ˆ H 1 is the residual electrostatic potential. ˆ H 2 is the spin-orbit interaction summed over the electrons of the atom. In Chapter 4 we neglected both ˆ H 1 and ˆ H 2 , and just concentrated on ˆ H 0 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Symmetry, Broken Symmetry, and Topology in Modern Physics

  A First Course

  • Mike Guidry, Yang Sun(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  PART IV A VARIETY OF PHYSICAL APPLICATIONS 30 Angular Momentum Recoupling The basics of tensor methods for angular momentum operators were introduced in Section 6.4. In this chapter we expand that discussion to more ambitious cases of coupling three or four angular momenta. This topic is more complex than many in this book, often involving long equations with many indices. However, it is an essential issue for those fields like atomic or nuclear physics where one commonly deals with many-body states of good total angular momentum, with more than one way to couple particles to form those states (for example, L–S and J – J coupling schemes). This chapter is optional if your interests lie in fields like condensed matter or particle physics where many-body states of good total angular momentum are less important. 30.1 Recoupling of Three Angular Momenta If we have more than two angular momenta, the manner in which they can be coupled to a good total angular momentum is not unique. For example, three angular momenta j 1 , j 2 , and j 3 could be coupled in either of the following ways to a total angular momentum J , J j 1 j 2 j 3 j 12 j 23 J j 1 j 2 j 3 The explicit forms for these state vectors are   ( j 1 j 2 ) j 12 j 3 , JM  =  m 1 m 2  m 3 m 12  j 1 m 1 j 2 m 2 | j 12 m 12  ×  j 12 m 12 j 3 m 3 | J M | j 1 m 1  | j 2 m 2  | j 3 m 3  , (30.1)   j 1 ( j 2 j 3 ) j 23 , JM  =  m 1 m 2 m 3  m 23  j 2 m 2 j 3 m 3 | j 23 m 23  ×  j 1 m 1 j 23 m 23 | J M | j 1 m 1  | j 2 m 2  | j 3 m 3  . (30.2) Since these provide only different labelings of the same problem, the two coupling schemes are connected by a unitary transformation,   j 1 ( j 2 j 3 ) j 23 , JM  =  j 12  ( j 1 j 2 ) j 12 j 3 , JM   j 1 ( j 2 j 3 ) j 23 , JM    ( j 1 j 2 ) j 12 j 3 , JM  . (30.3) 535

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Atomic Structure and Lifetimes

  A Conceptual Approach

  • Lorenzo J. Curtis(Author)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  Clearly there are three choices for doing this: combining 1 and 2 to obtain (12) and then combining that to obtain (12)3; combining 1 and 3 to obtain (13) and then combining that to obtain (13)2; and combining 2 and 3 to obtain (23) and then combining that to obtain (23)1. For higher numbers of angular momenta the possibilities increase accordingly. Thus the mathematical coupling scheme that one selects is dependent upon the actual physical couplings that exist in the system, as described in the considerations above, and any scheme that is adopted is likely to be only an approximation. Two systems that have asymptotic applicability are the LS and the j j coupling models. 78 4 The vector model of angular momentum 4.5.1 LS coupling For electrons in an atom of low-Z , where spins tend to couple to other spins and orbits tend to couple to other orbits, the rotational dynamics can be described by the LS cou-pling approximation. In this limit, the atom as a whole possesses both a total spin angular momentum and a total orbital angular momentum S = electrons S i , L = electrons L i . (4.18) In this approximation (approached in light multielectron atoms, but truly attained only in the degenerate case of hydrogenlike atoms) the spin angular momenta of the individual electrons have eigenvalues for the lengths S 2 i = 1 2 1 2 + 1 ¯ h 2 = 3 4 ¯ h 2 , L 2 i = i ( i + 1) ¯ h 2 . (4.19) Their projections precess about their L and S resultants (vector sums) and not about an external magnetic field. Similarly, their resultants have eigenvalues for their lengths S 2 = S ( S + 1) ¯ h 2 ; L 2 = L ( L + 1) ¯ h 2 , (4.20) but in the presence of a moderate external magnetic field the spin and orbital moments will not separately precess about the field, but will first couple to each other to form a total angular momentum J J = L + S . (4.21) and J precesses as a whole about an external magnetic field B .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  A Concise Handbook of Mathematics, Physics, and Engineering Sciences

  • Andrei D. Polyanin, Alexei Chernoutsan(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  In turn, the orbital and spin angular momenta add up to form the atom total angular momentum hatwide J . Such a scheme is called the normal LS -coupling or the Russell–Saunders coupling . The number L is always integer, and the number S is integer if the shell contains an even number of electrons and is half-integer if the number of electrons is odd. We note that one must add up only the angular momenta of electrons in an unfilled shell. In heavy atoms, the angular momenta are added together according to the scheme called the jj -coupling . First, one adds up the orbital and spin angular momenta of each individual electron ( j i = l i + s i ) and then the electron angular momenta are added up to form the total angular momentum of the atom ( J = ∑ i j i ). ◮ Classification of terms of complex atoms. In the case of normal coupling, the total angular momentum is preserved (the value and the projection) and the values of the orbital and spin angular momenta. Each term is characterized by the quantum numbers L , S and J and, in the absence of a magnetic field, it is ( 2 J + 1 )-fold degenerate (the states that differ only in the value of the total angular momentum projection have the same energy). For given L and S , the quantum number J can take 2 S + 1 values from L – S to L + S , and hence the number 2 S + 1 is called the term mutliplicity . The difference between the energies of terms in the same multiplet is determined by a comparatively weak spin-orbit interaction. The spectral notation of terms is: the orbital quantum number of an atom is P6.3. S TRUCTURE AND S PECTRA OF C OMPLEX A TOMS 589 denoted by the capital letter S , P , D , F , G , and further in alphabetical order (to distinguish from one-electron states, where the lowercase letters are used). The subscript on the right shows the number J , which distinguished this term in the multiplet, and the superscript on the left indicates the term multiplicity 2 S + 1 .

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Unitary Symmetry And Combinatorics

  • James D Louck(Author)
  • 2008(Publication Date)
  • World Scientific

    (Publisher)

  There are, fortunately, general invariance properties of the Schr¨ odinger equation and its solution state functions for a complex composite phys-ical systems that can be used to classify the quantum states of physical systems into substates available to the system. Our focus here is on the properties of the total angular momentum of a physical system, which is a quantity L that has a vector expression L = L 1 e 1 + L 2 e 2 + L 3 e 3 in the right-handed frame ( e 1 , e 2 , e 3 ) and the expression L = L 1 e 1 + L 2 e 2 + L 3 e 3 in a second rotated right-handed frame ( e 1 , e 2 , e 3 ) . At a given instant of time, necessarily L = L , since these quantities are just redescriptions of the total angular momentum of the system at a given time. The total angular momentum is a conserved quantity; that is, d L /dt = 0 , for all time t, and this property makes the total angular momentum an important quantity for the study of the behavior of complex physical systems. For a system of n point particles, the total angular momentum relative to the origin of the reference frame ( e 1 , e 2 , e 3 ) is obtained by vector addition of that of the individual parti-cles by L = ∑ n i =1 L i , where L i is expressed by the vector cross product L i = x i × p i in terms of the vector position x i = x 1 i e 1 + x 2 i e 2 + x 3 i e 3 and the vector linear momentum p i = p 1 i e 1 + p 2 i e 2 + p 3 i e 3 of the particle labeled i. While angular momentum can be exchanged between interact-ing particles, the total angular momentum remains constant in time for an isolated physical system of n particles. The quantum-mechanical op-erator interpretation of such classical physical quantities is obtained by Schr¨ odinger’s rule p i → − i ∇ i , = h/ 2 π, where h is Planck’s constant. The reference frame vectors ( e 1 , e 2 , e 3 ) remain intact. 1.1. BACKGROUND AND VIEWPOINT 13 The viewpoints of Newtonian physics and quantum physics may be contrasted in many ways.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Physics, Volume 1

  • Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  This is the second of the major conservation laws we have discussed. Along with conservation of linear momentum, conservation of angular momentum is a general result that is valid for a wide range of systems. It holds true in both the relativistic limit and in the quantum limit. No excep- tions have ever been found. Like conservation of linear momentum in a system on which the net external force is zero, conservation of angular momentum applies to the total angular momentum of a sys- tem of particles on which the net external torque is zero. The angular momentum of individual particles in a system may change due to internal torques ( just as the linear mo- mentum of each particle in a collision may change due to internal forces), but the total remains constant. Angular momentum is (like linear momentum) a vector quantity so that Eqs. 10-15 is equivalent to three one- dimensional equations, one for each coordinate direction through the reference point. Conservation of angular mo- mentum therefore supplies us with three conditions on the motion of a system to which it applies. Any component of the angular momentum will be constant if the correspond- ing component of the torque is zero; it might be the case that only one of the three components of torque is zero, which means that only one component of the angular mo- mentum will be constant, the other components changing as determined by the corresponding torque components. For a system consisting of a rigid body rotating with an- gular speed  about an axis (the z axis, say) that is fixed in an inertial reference frame, we have (10-16) where L z is the component of the angular momentum along the rotation axis and I is the rotational inertia for this same axis. If no net external torque acts, then L z must remain constant.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Fundamentals of Quantum Mechanics

  • 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’.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  An Introduction to Quantum Physics

  • A.P. French(Author)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  l has a corresponding z component of magnetic moment given by

  μ z = − m l μ B ( orbital )

  (11-24)

  (The minus sign comes from the fact that the electron charge is negative.) But for the z. component of magnetic moment due to spin, we have11

  μ z = − g s m s μ B ( spin )

  (11-25)

  where the “g factor” g

  s

  is very close to 2.

  11-5  SPIN-ORBIT COUPLING ENERGY

  As we have discussed in the previous section, the splitting of a cesium beam into two components by a transverse magnetic field gradient (Section 10-3 ) tells us that the total angular momentum of a cesium atom in the ground state is that of its valence electron, with spin i and zero orbital angular momentum. The orbital and spin angular momenta of all the other electrons (an even number) pair off and so their magnetic moments cancel.

  When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from these two angular momenta interact to cause a splitting in the corresponding energy level. Roughly speaking, the energy of the system is slightly higher when the orbital and spin magnetic moments are “parallel” than when they are “antiparallel.” Since the interaction that leads to this energy difference is between the spin and the orbital electronic structure, this interaction is called spin-orbit coupling or L-S coupling.

  A simple semiclassical calculation of the order of magnitude of the energy of the spin-orbit coupling for a single electron can be made using the Bohr model. Suppose, to be specific, that an electron is orbiting a proton with one unit of angular momentum. (Recall that in the Bohr theory this is the smallest possible orbital angular momentum.) We then have

  M e

  υ r = ℏ

  Now, if we imagine ourselves sitting on the electron, we see the proton describing an orbit of radius r with speed v. This represents a circulating current I equal to ev/2πr

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Angular Momentum

  An Illustrated Guide to Rotational Symmetries for Physical Systems

  • William J. Thompson(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  Physics uses several kinds of language: natural language (such as English), graphical language (schematics such as Figure 5.1, diagrams, and graphs), and mathematics. When mathematics is used in physics it must be both appropriate and correct. Section 5.3.1 emphasizes that paradoxical results can be obtained if physi- cal concepts are not expressed appropriately and manipulated by the mathematics. As summarized in Figure 5.1, physics involves a loop of four themes: concepts, language, mathematics, and observations. We begin the chapter by discussing in Section 5.1 rotational symmetry and dy- namical angular momentum, especially the role of Planck‘s constant. In Section 5.2 we progress to the uncertainty relations for angular momentum as interpreted in quantum mechanics. We show that the semiclassical vector model (Section 5.3) is useful for visualizing angular momentum. The model also helps when considering how to combine two angular momenta (Section 7.1). Waves, particularly Schro- dinger wave mechanics, and their relation to angular momentum are discussed extensively in Section 5.4, where we also derive and present visualizations of par- tial-wave expansions involving Bessel functions. Section 5.5 summarizes the con- ceptual development of angular momentum from a historical perspective. The in- evitable problems round out the chapter. 5 . 1 ROTATIONAL SYMMETRY AND DYNAMICAL ANGULAR MOMENTUM Our aim in this section is to develop the connection between rotational symmetry (geometry) and dynamical angular momentum (physics). We begin by showing in Section 5.1.1 the correspondence between geometrical, quantal, and classical angular momentum. Then we consider in Section 5.1.2 the particular case of orbital angular momentum and the quantum-classical correspondence through an Ehrenfest theorem for orbital angular momentum, analogous to that often derived for the quantum analogue of Newton’s force equation.

  Sign up to read

  Learn more about book