As in classical mechanics, rotation in quantum mechanics is a transformation which deals with angular momentum. The difference with classical mechanics comes from the fact that angular momentum is a vector operator and not a usual vector and its components do not commute. As for any transformation in quantum mechanics, to each rotation we can associate an operator which acts in state space. The expression of this operator depends on whether the rotation is passive, that is we do a rotation of the coordinate axes and the physical system is left unchanged, or active, in which case the coordinate axes are unchanged and the rotation is performed on the physical system. In the first part (Chaps. 1 and 2) of this book, details concerning both aspects are given. Following the definition of the geometrical transformation associated with the most general rotation, we give the expression of the rotation operator for specific cases. Transformation laws for scalar fields, vector fields and spinor fields are given as well as transformation laws for scalar operators, vector operators and more generally, for operators of any rank. The second part (Chaps. 3 and 4) deals with angular momentum algebra. We define the coupling coefficients of 2, 3 and 4 angular momenta as well as the recoupling coefficients. The definition of the irreductible tensor operator, which is a generalisation of scalar and vector operators, is given as well as the Wigner-Eckart theorem. The application of this theorem to more complex cases is studied.
Chapter 11 is devoted to the quantum theory of angular momentum. As in classical mechanics, working with angular momentum requires leaving the one-dimensional space, with all the complications that this entails. The importance of angular momentum resides in at least three facts: i) central problems are of particular interest; (ii) many particles have an intrinsic angular momentum (the spin); and iii) the physics of angular momentum involves features such as the so-called space quantization. The first part of the chapter deals with the orbital angular momentum and its eigenfunctions, the spherical harmonics. Its algebraic properties, embodied in the matrix representation, allow for an extension of the theory to half-integer angular momentum in terms of Pauli matrices, as well as the addition of angular momenta.
This chapter presents key results on angular momentum in quantum mechanics, beginning with a theorem on the simultaneous diagonalization of commuting operators. The eigenvalues and eigenstates of both the total angular momentum operator and its z-component are derived. For a quantum rotator in the p-state with zero magnetic quantum number, it is shown that the most probable trajectory lies in a plane containing the z-axis, and the angular momentum vector is perpendicular to it. The procedure for angular momentum addition is explained in detail, including the use of Clebsch–Gordan coefficients to perform the explicit decomposition of tensor products of irreducible representations. The Bloch sphere representation of qubit spin states is introduced. The chapter then provides an experimental background to the discovery of spin, introduces the Pauli matrices, and demonstrates how they act on spin states. It concludes with the formulation of the Pauli equation, which incorporates spin into the framework of nonrelativistic quantum mechanics. Metadata/Keywords: angular momentum, simultaneous diagonalization, magnetic quantum number, Clebsch–Gordan coefficients, spin, Pauli matrices, Bloch sphere, spin states, tensor product, Pauli equation
In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum
j. In contrast, here we show that a spatial wavefunction,
jm(ϕ,θ,χ)= eimϕδ(θ−θm) ei(j+1/2)χ, which treats
j in the
∣jm> state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators;
ϕ,
θ,
χ are the Euler angles, and
cosθm=(m/∣j∣) is the Vector-Model polar angle. The
jm(ϕ,θ,χ) gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in
j and
m) which predicts the effective wavepacket angular uncertainty relations for
ΔmΔϕ,
ΔjΔχ, and
ΔϕΔθ, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive
j-rotation measurements. We also use the
jm(ϕ,θ,χ) to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles,
g=2, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to
j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that
jm(ϕ,θ,χ) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.
We perform an analysis of a graphical representation for the addition of two angular momenta, focusing our attention on the angle δ between the xy components of two angular momenta. Then we propose a new complete set of commuting observables, which differ from the usual sets that are connected by the Clebsch–Gordan coefficients. This set shows that the angle δ can be a well-defined variable in quantum mechanics. An empirical analysis of the graphical representations of the angular momenta relations, which may include the angle δ, followed by quantum mechanical considerations, leads to the vanishing of certain quantum angular momentum commutators for specific states. Therefore, although the commutators are not null in general, the quantum addition of angular momenta may be represented using classical-like diagrams.