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Cambridge University Press eBooks

Symmetry Transformations of States and Operators

Rocco Schiavilla

Published Dec 5, 2024• 0 citations
Publisher

Abstract

Construction ofunitary operators inducing space and time translations, and rotations; the anti-unitary operator inducing time reversal; consequences of invariance under a symmetry transformation; periodic potentials and Bloch waves; the Kronig-Penney model; the ammonia molecule and broken parity symmetry; consequences of time reversal invariance on the scattering amplitude of spinless particles; Kramers degeneracy.

DOI
10.1017/9781009473637.017

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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).
Abstract The delta function potential is a simple model of zero-range contact interaction in non-relativistic quantum mechanics in one dimension. The Kronig–Penney model is a one-dimensional periodic array of delta functions and provides a simple illustration of energy bands in a crystal. Here we investigate contact interactions that generalize the delta function potential and corresponding generalizations of the Kronig–Penney model within conventional and symmetric quantum mechanics. In conventional Hermitian quantum mechanics we determine the most general contact interaction compatible with self-adjointness and in quantum mechanics we consider interactions that respect symmetry under the transformation where denotes parity and denotes time reversal. In both cases we find that the most general interaction has four independent real parameters and depending on the values of those parameters the contact interaction can support zero, one or two bound states. By contrast the conventional delta function can only support zero or one bound state. In the symmetric case moreover the two bound state energies can be both real or a complex conjugate pair. The transition from real to complex bound state energies corresponds to the spontaneous breaking of symmetry. The scattering states for the symmetric case are also found to exhibit spontaneous breaking of symmetry wherein the eigenvalues of the non-unitary S -matrix depart the unit circle in the complex plane. We also investigate the energy bands when the generalized contact interactions are repeated periodically in space in one dimension. In the Hermitian case we find that the two bound states result in two narrow bands generically separated by a gap. These bands intersect at a single point in the Brillouin zone as the interaction parameters are varied. Near the intersection the bands form a massless Dirac cone. In the symmetric case we find that as the parameters of the contact interaction are varied the two bound state bands undergo a symmetry breaking transition wherein the two band energies go from being real to being a complex conjugate pair. The symmetric Kronig–Penney model provides a simple soluble example of the transition which has the same form as in other models of symmetric crystals.
Transformations, conservation laws, and symmetries have played an important role in the development of modern quantum mechanics. This chapter includes translation in space and time, rotation in space and rotation operator, parity operations and space inversion, time reversal operator and its properties for both zero and non-zero spin particles, solved examples, and unsolved exercises. The chapter discusses in detail how the rotation in space leads to important relations between various components of orbital and total angular momenta, which play an important role in formulations of quantum mechanics. Parity conservation and the most striking properties of the parity operator, relevance of time reversal in classical and quantum mechanical descriptions, and the antiunitary and antilinear properties of the time reversal operator are also featured details in the chapter.