The shell model Hamiltonian is invariant under spatial rotations. This fact has profound implications for its eigenstates. Other physical operators which are not invariant may have well defined transformation properties under rotations. Much of the following is related to the behavior of various physical entities, operators as well as wave functions, under rotations in three-dimensional space. In the following pages a simple introduction to this subject is presented. First, single particle states and operators will be discussed. Subsequent sections will deal with transformation properties of more general systems of several nucleons.
The shell model Hamiltonian is invariant under spatial rotations. This fact has profound implications for its eigenstates. Other physical operators which are not invariant may have well defined transformation properties under rotations. Much of the following is related to the behavior of various physical entities, operators as well as wave functions, under rotations in three-dimensional space. In the following pages a simple introduction to this subject is presented. First, single particle states and operators will be discussed. Subsequent sections will deal with transformation properties of more general systems of several nucleons.
This chapter shows that the three operators are Hermitian and (in a rotationally invariant system) commute with the Hamiltonian; therefore they are observable constants of the motion. It finds out what their physical significance is. By knowing from classical mechanics, the constant of the motion arising from rotational invariance is the angular momentum. The angular momentum may split up into two parts: orbital and intrinsic. The orbital part can always be described by the dependence of a wave function, whereas the intrinsic part is by definition contained in the transformation property of the wave function. In the case of a single component (scalar), no intrinsic angular momentum exists. There are—apart from the vector addition and the scalar product—two other ways of combining two vectors: the direct sum and the direct product. Both occur in the theory of angular momentum, in fact whenever symmetry is involved.
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.
An Alternative Approach to Rotations In the preceding chapters, we have discussed the standard view of the quantum theory of angular momentum. The present chapter is concerned with developing a new view of the subject starting from the beginning with a fresh look at rotations based on a viewpoint adapted primarily from Hamilton [1], but also partly from Klein and Sommerfeld [2] and from Wigner [3] (see Note 1). This new survey will be rewarded by the concept of a novel geometric entity – the turn – which in its quantal version (Chapter 5) will be realized by an elementary operator structure. In discussing this alternative viewpoint, we are primarily interested in motivation and in suggesting concepts. Accordingly, we shall proceed, at first, intuitively. When reduced to fundamentals, the concept of an elementary (point) particle in physics (as discussed in Chapter 1) becomes synonymous with the fundamental symmetries assumed to characterize physical space. We have seen that, from the homogeneity and isotropy assumed for space, one induces the symmetries of spatial displacements (translations) and rotations; that is, one assumes that space is Euclidean and three-dimensional, having the isometry group, E (3), of rotations and translations. It is quite easy to characterize translations, since three-dimensional translations necessarily (Artin [4]) form an abelian (commutative) group. For physics, the translation generator is the (Hermitian) momentum operator p , and the associated operator generating finite displacements is realized by U ( a ) = exp(– i a · p / ħ ). This operator displaces the system by the (numerical) vector a .
This chapter discusses everything in terms of wave functions and describes the choice between active and passive rotations. It defines the representations of the rotation group by means of the abstract state vectors and by means of wave functions. That implies that the rotations are active rotations. The chapter finds that for a sequence of rotations the three-dimensional matrices have the property. It considers the change of the wave function in active and passive rotations respectively. In the active interpretation the physical system — characterized in space by the wave function — is bodily rotated. In the passive interpretation the frame of reference is rotated, whereas P and the local physical state at P remain unchanged. The chapter describes that there is a description which uses the new coordinates but describes the old situation.
Symmetries play a central role in physics. They dictate what one can change in a physical system without affecting any of its properties. You might have encountered symmetries like translational symmetry, where a system remains unchanged if it is spatially translated by an arbitrary distance. A system with rotational symmetry, however, is invariant under rotations. Some symmetries, like the ones mentioned above, give information about the structure of the system. Others have to do with the more fundamental physical framework that we adopt. An example for this is the invariance under Lorentz transformations in relativistic physics. Other types of symmetries can be even more subtle. For example, it is rather self-evident that physics should remain unchanged if we exchange two identical point-like particles. Nevertheless, this fundamental property that we call statistical symmetry gives rise to rich and beautiful physics. In three spatial dimensions it dictates the existence of bosons and fermions. These are particles with very different quantum mechanical properties. Their wave function acquires a +1 or a -1 phase, respectively, whenever two particles are exchanged. A direct consequence of this is that bosons can actually occupy the same state. In contrast, fermions can only be stacked together with each particle occupying a different state. When one considers two spatial dimensions, a wide variety of statistical behaviours is possible. Apart from bosonic and fermionic behaviours, arbitrary phase factors, or even non-trivial unitary evolutions, can be obtained when two particles are exchanged (Leinaas and Myrheim, 1977).
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.