This chapter assesses the mathematical theory of symmetry, which is one of the most remarkable in quantum mechanics. Not only does it simplify calculations, but it also reveals unexpected connections between apparently disparate phenomena. It considers angular momentum as a part of group theory and as properties of the harmonic oscillator. The chapter talks about the conservation of energy and of momentum that can be discussed in terms of group theory, which is used to classify the fundamental particles and the selection rules that govern what spectroscopic transitions are allowed. Integrals occur throughout quantum chemistry, for they include expectation values, overlap integrals, and matrix elements.
This chapter assesses the mathematical theory of symmetry, which is one of the most remarkable in quantum mechanics. Not only does it simplify calculations, but it also reveals unexpected connections between apparently disparate phenomena. It considers angular momentum as a part of group theory and as properties of the harmonic oscillator. The chapter talks about the conservation of energy and of momentum that can be discussed in terms of group theory, which is used to classify the fundamental particles and the selection rules that govern what spectroscopic transitions are allowed. Integrals occur throughout quantum chemistry, for they include expectation values, overlap integrals, and matrix elements.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH 3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
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.
This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrödinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.
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.