John L Wood
John L Wood is a Professor Emeritus in the School of Physics at Georgia Institute of Technology. He continues to collaborate on research projects in both experimental and theoretical nuclear physics. Special research interests include nuclear shapes and systematics of nuclear structure.
IOP Publishing
Quantum Mechanics for Nuclear Structure, Volume 2
An intermediate level view
Kris Heyde and John L Wood
Chapter 1
Representation of rotations, angular momentum and spin
The various representations of rotations in physical space, (3,R) and Hilbert space (n,C) are developed in detail. This leads to an in-depth treatment of the representation of states of well-defined spin and angular momentum in quantum systems. The peculiarities of the physics of spin-12 systems (spinors) are outlined. The tensorial character of representations is implicit in the treatment. The Schwinger and Bargmann representations are introduced in some detail; and this leads to SU(2) coherent states (which are important for more advanced group representation theory).
Concepts: Euler angles; matrix representations; Pauli spin matrices; ket rotations; SU(2) and SO(3) tensor representations; Schwinger representation; spherical harmonics as Cartesian tensors; spin-12 neutron interferometry; Bargmann space; measure of a space; SU(2) coherent states; non-unitary representations.
Angular momentum and spin are dynamical variables that are fundamental to finite systems in quantum mechanics, i.e. for molecules, atoms, nuclei and hadrons. To fully handle the quantum mechanics of these systems, the mathematical representation of rotations is fundamental. Some elements of these issues in quantum mechanics are introduced in Volume 1. Namely, the concept of a group, the use of matrices, the distinction between rotations in physical space, (3, R), and Hilbert space is presented in chapter 10; and the basic quantization of spin and angular momentum, using algebraic methods, is presented in chapter 11. Further, the facility with which these methods reduce the solution of central force problems in quantum mechanics to simple algebraic problems in terms of a single (radial) degree of freedom is presented in chapter 12.
The mathematical representation of rotations is a rich paradigm for the whole of quantum mechanics. In this chapter, a wide range of mathematical tools is introduced. Matrix algebra and the algebra of polynomials in real and complex variables feature prominently. The peculiar physics of spin-12 particles and spinors is presented. But, the primary aim is to initiate a language that is suitable for the theoretical formulation of finite many-body quantum systems. Group theory and Lie algebras are implicit in the material presented in this chapter: the groups SO(2), SO(3) and SU(2) feature prominently in their behind-the-scene role. The road into many-body systems necessitates more complicated groups such as SU(3): some of the material in this chapter is intended to ‘pave’ this road.
1.1 Rotations in (3, R)
One way to describe a rotation in (3, R) is in terms of rotation in a plane through a specified angle1. This is defined in terms of an axis of rotation nˆ and an angle ϕ. The axis nˆ is perpendicular to the plane defined by the initial and final orientations of the vectors V⃗, V⃗′: R(ϕ)V⃗=V⃗′. The difficulty lies in ascertaining the direction of nˆ. Although this ‘axis-angle’ parameterisation or Darboux parameterisation is simple in principle, it is difficult to use in practice.
The most widely used practical parameterisation of rotations in (3, R) is in terms of Euler rotations. Consider a space-fixed coordinate frame Oxyz and a body-fixed coordinate frame Ox¯y¯z¯. The orientation of an object can be specified by the rotation R that rotates the Ox¯y¯z¯ frame into the Oxyz frame. This can be done in three steps as illustrated in figure 1.1.
Figure 1.1. The Euler angles (α,β,γ) defined in terms of a three-step sequence of rotations that take an intrinsic or body-fixed frame Ox¯y¯z¯ into a space-fixed frame Oxyz. Note that the axes of rotation are: Oz¯; the line of intersection of the Ox¯y¯ and Oxy planes, OY; and Oz. In Step I, y¯ rotates to Y and z¯ remains fixed; in Step II, z¯ rotates to z and Y remains fixed; in Step III, Y rotates to y and z remains fixed. Further, note that the ranges of the angles are: 0⩽α<2π, 0⩽β<π, 0⩽γ<2π. This results in an ambiguity for the rotation β=0, (α,0,γ)≡(α′,0,γ′) if α+γ=α′+γ′: this is referred to as ‘gimbal lock’ (where ‘gimbal’ refers to the rotation device or mechanical operator). This figure is adapted from that found on the Easyspin website.
Figure 1.1 depicts the following:
R(α,β,γ)=Rz(γ)RY(β)Rz¯(α).(1.1)
Note the order of the three rotations. The problem is that these three rotations are about axes belonging to three different frames of reference. The three rotations on the right-hand side of equation (1.1) can be restated in terms of a single frame of reference using similarity transformations, specifically
Rz(γ)=RY(β)Rz¯(γ)RY−1(β)(1.2)
and
RY(β)=Rz¯(α)Ry¯(β)Rz¯−1(α).(1.3)
Thus,
∴R(α,β,γ)=RY(β)Rz¯(γ)Rz¯(α);(1.5)
and, since Rz¯(γ) and Rz¯(α) commute,
∴R(α,β,γ)=RY(β)Rz¯(α)Rz¯(γ),(1.6)
∴R(α,β,γ)=Rz¯(α)Ry¯(β)Rz¯(γ).(1.8)
Note the new order of the three rotations (cf. equation (1.1)).
1.2 Matrix representations of spin and angular momentum operators
The matrix elements of Jˆz,Jˆ± in the {∣jm〉;j=0,12,1,32,…;m=+j,+j−1,…,−j} basis are (cf. Volume 1, chapter 11):
〈j′m′∣Jˆz∣jm〉=mℏδj′jδm′m,(1.9)
〈j′m′∣Jˆ±∣jm〉=(j∓m)(j±m+1)ℏδj′jδm′m±1.(1.10)
Matrix elements of Jˆx and Jˆy follow from:
Jˆx=12(Jˆ++Jˆ−),(1.11)
Jˆy=12i(Jˆ+−Jˆ−),(1.12)
where, recall Jˆ±≔Jˆx±iJˆy. Thus, the matrix representations of Jˆx,Jˆy, and Jˆz in a ∣jm〉 basis are:
Jˆx↔ℏ200000011000002020202000000300302002030030⋱,(1.13)
Jˆy↔ℏ2i0000001−10000020−2020−2000000300−30200−20300−30⋱,(1.14)
Jˆz↔ℏ200000100−100020000000−200003000010000−10000−3⋱.(1.15)
Note the ‘block-diagonal’ form of Jˆx and Jˆy. These blocks correspond to j=0,12,1,32,…. The matrix representation of Jˆz is diagonal with eigenvalues 0;12ℏ,−12ℏ;ℏ,0,−ℏ;32ℏ,12ℏ, −12ℏ,−32ℏ;…. It is normal practice