There are unfortunately a large variety of possibilities for writing down a rotation; also there are two different interpretations of the word ‘rotation’. Obviously, if one rotates first the coordinate frame and then, by the same rotation, the space, all points will obtain again the same coordinate as before these rotations; the same is true in the reverse order. Rotational invariance means then that the result of this rotation, seen by an observer who did not participate in the rotation, is again a possible state of the unrotated system. One says the rotations in the physical three-dimensional space induce unitary transformations in the Hilbert space, whose matrices give rise to (infinitely many independent) representations of the rotation group. The active interpretation means carrying out the sequence of rotations using a fixed system of rotation axes; the passive interpretation means carrying out the rotations using a system of rotation axes which participates in the rotation.
There are unfortunately a large variety of possibilities for writing down a rotation; also there are two different interpretations of the word ‘rotation’. Obviously, if one rotates first the coordinate frame and then, by the same rotation, the space, all points will obtain again the same coordinate as before these rotations; the same is true in the reverse order. Rotational invariance means then that the result of this rotation, seen by an observer who did not participate in the rotation, is again a possible state of the unrotated system. One says the rotations in the physical three-dimensional space induce unitary transformations in the Hilbert space, whose matrices give rise to (infinitely many independent) representations of the rotation group. The active interpretation means carrying out the sequence of rotations using a fixed system of rotation axes; the passive interpretation means carrying out the rotations using a system of rotation axes which participates in the rotation.
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.
A symmetry is an equivalence of different physical situations, and is therefore strictly associated with the concept of invariance . One of the main aims of physics is the search in nature for symmetries and, therefore, for observables that are invariant under certain classes of transformation. In this context, it becomes particularly interesting to look for the conditions under which a certain symmetry is eventually broken. We have already discussed some symmetries of quantum mechanics: time translations (Ch. 3), space translations (Ch. 2) and rotations (Ch. 6). Another example of invariance is represented by the indistinguishability of identical particles, which we discussed in the previous chapter. Symmetries are induced by physical transformations that generate some invariant properties (see Sec. 8.1). In Sec. 8.2, the continuous transformations induced by rotations and space translations are analyzed, whereas in Sec. 8.3, space reflection and time reversal ( discrete symmetries) are discussed. As we shall see in the next chapter, time reversibility is lost in the measurement process. The rigorous mathematical framework of a theory of transformations and of their invariance is called group theory (the subject of Sec. 8.4). Quantum transformations and symmetries Active and passive transformations We have seen that time evolution may be represented either in the Schrödinger picture (state vectors evolve and observables are kept constant) or in the Heisenberg picture (states vectors remain constant and observables evolve) (see Subsec. 3.6.1 and Sec. 5.3).
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.
For n particles moving in three-dimensional space, each coordinate should carry an index i = 1, 2,3 denoting the vector component and an index l = 1, 2,..., n denoting the particle number. Among the linear point transformations acting on these coordinates there are two commuting subgroups of transformations. The elements of the first one are given by product matrices M 3 × I n where the 3 × 3 matrix M 3 acts on the vector components. The elements of the second one are of the form I 3 × M n where the n × n matrix M n acts on the vectors for the different particles. To the first subgroup there belong the rotations giving rise to the rotation group SO (3, IR). The second subgroup contains permutations and transformations to various types of relative coordinates. In the present section we shall deal with this second subgroup and hence shall drop the reference to the three vector components. Moreover we shall need at present only orthogonal n × n transformation matrices.
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.
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).