ELEMENTS OF QUANTUM MECHANICS

Quantum mechanics is the theory that captures the particle-wave duality of matter. Quantum mechanics applies in the microscopic realm, that is, at length scales and at time scales relevant to subatomic particles like electrons and nuclei. It is the most successful physical theory it has been verified by every experiment performed to check its validity. It is iso the most counter-intuitive physical theory, since its premises are at variance with our everyday experience, which is based on macroscopic observations that obey the laws of classical physics. When the properties of physical objects (such as solids, clusters and molecules) are studied at a resolution at which the atomic degrees of freedom are explicitly involved, the use of quantum mechanics becomes necessary. 

In this Appendix we attempt to give the basic concepts of quantum mechanics relevant to the study of solids, clusters and molecules, in a reasonably self-contained form but avoiding detailed diseussions. We refer the reader to standard texts of quantum mechanics for more extensive discussion and proper justification of the statements that we present here, a couple of which are mentioned in the Further reading section. 

Although the Schrddinger equation, the fundamental equation of quantum mechanics, may be derived from the classical wave equation by a heuristic approach, it is currently more common to present quantum mechanics as a set of postulates, such as those tabulated below (Levine, 1983). 

Postulate 1. The state of a system is described by a function i of the coordinates and the time. This function, called the state function or wave function, contains all the information that can be determined about the system. i j is single-valued, continuous, and quadratically integrable. 

Postulate 2. To every physical observable there corresponds a linear Hermitian operator. To find this operator, write down the classical-mechanical expression for the observable in terms of Cartesian coordinates and corresponding linear-momentum components, and then replace each coordinate x by the operator x and each momentum component p, by the operator — ih didx. 

Postulate 3. The only possible values that can result from measurements of the physical observable G are the eigenvalues g, of the equation  

Postulate 4. If G is any linear Hermitian operator that represents a physical observable, then the eigenfunctions 4 , of the eigenvalue equation above form a complete set. 

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The following undergraduate texts discuss the historical development of quantum concepts and introduce the elements of quantum mechanics. 

Although one could consider the electron density as just one of the many quantum chemical descriptors available, it deserves special attention. In QSM, it is the only descriptor used for a number of reasons. The idea of using the electron density as the ultimate molecular descriptor is founded on the basic elements of quantum mechanics. First of all, it is the all-determinmg quantity in density functional theory (DFT) [11] and also holds a very close relation to the wave function. Convincing arguments were given by Handy and are attributed to Wilson [12], although initial ideas can also be traced back to Bom [13] and von... 

The concept of quantum states is the basic element of quantum mechanics the set of quantum states I > and the field of complex numbers, C, define a Hilbert space as being a linear vector space the mapping < P P> introduces the dual conjugate space (bra-space) to the ket-space the number C(T )=< I P> is a... 

We summarize the relevant elements of quantum mechanics. A particle or any other system, whose state is classically described by coordinates q and momenta p, is described in quantum mechanics by a wave function ij/(q). 

Fayer, M.D. Elements of Quantum Mechanics. Oxford University Press, Inc., New York, NY, 2000. 

Jean Barriol, Elements of Quantum Mechanics with Chemical Applications, Barnes and Noble, New York, 1971. 

Following the turbulent developments in classical chaos theory the natural question to ask is whether chaos can occur in quantum mechanics as well. If there is chaos in quantum mechanics, how does one look for it and how does it manifest itself In order to answer this question, we first have to realize that quantum mechanics comes in two layers. There is the statistical clicking of detectors, and there is Schrodinger s probability amplitude -0 whose absolute value squared gives the probability of occurrence of detector clicks. Prom all we know, the clicks occur in a purely random fashion. There simply is no dynamical theory according to which the occurrence of detector clicks can be predicted. This is the nondeterministic element of quantum mechanics so fiercely criticized by some of the most eminent physicists (see Section 1.3 above). The probability amplitude -0 is the deterministic element of quantum mechanics. Therefore it is on the level of the wave function ip and its time evolution that we have to search for quantum deterministic chaos which might be the analogue of classical deterministic chaos. 

Pauling, L. C. and Wilson, E. R. Introduction to Quantum Mechanics New York, 1935 Dushman, S. The Elements of Quantum Mechanics New York, 1938... 

F. Strocchi, Elements of Quantum Mechanics of Infinite Systems. World Scientific, Singapore, 1985. 

The basic semiclassical idea is that one uses a quantum mechanical description of the process of interest but then invokes classical mechanics to determine all dynamical relationships. A transition from initial state i to final state f, for example, is thus described by a transition amplitude, or S-matrix element Sfi, the square modulus of which is the transition probability Pf = Sfi 2. The semiclassical approach uses classical mechanics to construct the classical-limit approximation for the transition amplitude, i.e. the classical S-matrix the fact that classical mechanics is used to construct an amplitude means that the quantum principle of superposition is incorporated in the description, and this is the only element of quantum mechanics in the model. The completely classical approach would be to use classical mechanics to construct the transition probability directly, never alluding to an amplitude. 

The stochastic Bloch equation is a semiphenomenological equation with some elements of quantum mechanics in it. To understand better whether our results are quantum mechanical in origin, we analyze a classical model. Lorentz invented the theory of classical, linear interaction of light with matter. Here, we investigate a stochastic Lorentz oscillator model. We follow Allen and Eberley [108] who considered the deterministic model in detail. The classical model is also helpful because its physical interpretation is clear. We show that for weak laser intensity, the stochastic Bloch equations are equivalent to classical Lorentz approach. 

This is a particularly simple expression that still captures much of the essence of quantum mechanical bonding. Hence it can be used in large-scale simulations to introduce basic elements of quantum mechanics into the interatomic forces. 

For a proof of (15.152), see J. Barriol, Elements of Quantum Mechanics with Chemical Applications, Barnes and Noble, 1971, pages 281-282.] H2O has 4 /2 3 = 2 canonical covalent structures, and these are A and B. (Actually, which structures are taken as the canonical ones is arbitrary, since the orbitals can be arranged in various ways on the ring.) To use Rumer s method when the number of orbitals to be paired is odd, we add a phantom orbital, whose contribution is subtracted at the end of the calculation. 

The computational problem is formally the same whether a Gaussian, plane wave or polynomial basis is used - calculate matrix elements of quantum mechanical operators over basis functions and solve the variational problem by an iterative procedure - but the nature of the functions results in some differences. With a GTO basis the matrix elements are calculated directly, while with a plane wave basis the matrix elements involving the potential energy can be generated by simple multiplication, as long... 

Schematic of an optical interference apparatus. [ Michael D Fayer, Elements of Quantum Mechanics, 2001, by permission of Oxford University Press, USA.]...

In this section, we shall provide a discussion of elements of quantum mechanics, which have been selected mainly for the purpose of serving the development of relativistic quantum chemistry in later chapters. Unfortunately, in view of the complexity of the subject and the limited space in this book, this conceptual discussion of quantum theory cannot be discussed in satisfactory depth. Also, some aspects will be fully developed only in later chapters. 

Barriol, J. 1971. Elements of quantum mechanics with chemical applications. New York Barnes Nobles. 

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