The Postulates of Quantum Mechanics

There exists a function, F(x, y, z, t), of the coordinates and time that we call a wave function and describe as a probability amplitude. This wave function is in general a complex function that is, 

The value of the integral in Eq. (20.4) must be independent of the time, t. This implies that the time dependence of the wave function must have the form 

The requirement expressed by Eqs. (20.4) and (20.6), namely that the wave function be quadratically integrable, imposes severe restrictions on ij/. The wave function must be single-valued, continuous, and may not have singularities anywhere of a character that result in the nonconvergence of the integral in Eq. (20.6). In particular, at the extremes of the Cartesian coordinates, x = + co, y = + co, and z = + oo, the wave function, as well as well as its first derivative, must vanish. 

The expectation value, A, of any observable is related to the wave function of the system 

Since we are dealing with wave functions that are functions only of coordinates, then, as was pointed out in Section 19.14, to obtain the expectation value of any function of the coordinates we multiply that function by the probability density, and integrate over the entire space. Thus, for a function, /(x, y, z), we have 

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If the electron is in the Is state, the hydrogen atom is in its lowest state of energy. In a polyelectronic atom such as carbon (six electrons) or sodium (eleven electrons) it would not seem unreasonable if all the electrons were in the Is level, thereby giving the atom the lowest possible energy. We might denote such a structure for carbon by the symbol Is and for sodium, ls . This result is wrong, but from what has been said so far there is no apparent reason why it should be wrong. The reason lies in an independent and fundamental postulate of the quantum mechanics, the Pauli exclusion principle no two electrons... 

These will be s mthesized in the so-called Postulates of the quantum mechanics. Here we just overview them, without excessively detailing the mathematical device and physical consequences, for which the Volume I of the present five-volume work is fully dedicated (Putz, 2016a). 

The first postulate of the quantum mechanics refers thus to the correspondence principle by which the functions and the classical quantities (as coordinate, momentum, energy, or orbital momentmn) become operators, having as the connection element (of correspondence) the Planck constant h or its reduced form h = h/(27r). The list of these correspondences is shown in Table 3.1. 

Then, in the electronic states, characterized by wave function (2.14) the vibrational deviation (amplitude) is calculated through the average values formula of the observable (by quantum averaging) postulate of the quantum mechanics ... 

The approach to be used here is, to be sure, well known in parts of theoretical physics, but is novel as far as chemistry is concerned. It is based on the view that macroscopic matter is to be described by a suitably generalized formulation of quantum mechanics, namely Quantum Field Theory the traditional postulate that matter is made up or composed of microscopic elementary constituents (in the classical building-block sense) is given up, and instead the fundamental postulate of the quantum theory of matter is, to paraphrase Gertrude Stein, Matter is Matter is Matter. Then if our interest is chemistry we have of course to confront the obvious question as to how we may construct the particles we call atoms and molecules i.e. we must establish how the notions of atom and molecule emerge from quantum theory construed in a general and modem way as the theory of matter. This is the subject matter of the next section of the review... 

The main postulate of quantum mechanics is that the whole information of a general system under study is contained in the total wave functions Tjt). This is known as completeness of the quantum-mechanical description. Such a circumstance is coherent with Eq. (6) since 2 is assumed to carry the whole information of the investigated system. This is formally expressed through the closure relation for the exact orthonormalized basis Tfc) ... 

Van t Hoff postulated free rotation round a single bond in order to explain the lack of cis and Irons isomers in molecules of the type of di-chlorethane. In the light of the quantum mechanical theory of the chemical bond, the free rotation is explained by the axial symmetiy of the a bond between the two carbon atoms. Thus the a bond is not in itself a hindrance to free rotation, but as the rotation occurs the relative configurations of the atoms will be changed, so that the distances between the non-bonded atoms and consequently their energies of interaction will alter. 

The index of measurement statistics corresponding to a given preparation can be expressed in the form of a density operator 0. Some preparations result in states described by density operators that are pure (density matrices are idempotent), and some in states described by density operators that are mixed (density matrices are not Idempotent). In the context of the quantum mechanical postulates, the preceding sentence is all that need be said about any given preparation and, therefore, any given state. 

The main lessons to be kept for the further theoretical and practical investigations of the quantum mechanics postulates and basic applications that are presented in the present chapter pertain to the following ... 

Postulate 4 If is not an eigenfunction of the quantum mechanical operator, then a series of measurements on identical systems of particles will yield a distribution of results, such that Equation (3.32) will describe the average (or expectation ) value of the observable (assuming the wave function is normalized). 

Based on this commutativity, according to the postulate [P3] of the quantum mechanics, further follows that the Hamiltonian of the system and the translation operator have the same set of eigen-function P, which implies, for the translation operator, the existence of the eigen-values equation (according to the quantum mechanics postulate [PI]) ... 

First of all it is shown that the laws of thermodynamics are a consequence of the postulates of the quantum theory, together with one other postulate which is of a statistical character. Secondly statistical mechanics makes it possible to obtain important new theorems not known in pure thermodynamics, including methods of calculating heat capacities, free energies, etc., from spectroscopic data. 

The problem of the quantum mechanical description of a harmonic oscillator is our first example of applying the postulates of quantum mechanics. It also provides a valuable comparison with the classical description considered earlier in this chapter. The picture of the system is the same as that in Figure 7.1, a mass m connected to an immovable wall by a harmonic spring with force constant k. The steps for quantum mechanical treatment of this problem, as well as any other problem, are the following ... 

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

Up until now, little has been said about time. In classical mechanics, complete knowledge about the system at any time t suffices to predict with absolute certainty the properties of the system at any other time t. The situation is quite different in quantum mechanics, however, as it is not possible to know everything about the system at any time t. Nevertheless, the temporal behavior of a quantum-mechanical system evolves in a well defined way drat depends on the Hamiltonian operator and the wavefiinction T" according to the last postulate... 

Suppose that the system property A is of interest, and that it corresponds to the quantum-mechanical operator A. The average value of A obtained m a series of measurements can be calculated by exploiting the corollary to the fifth postulate... 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

The observed structure of the spectra of many-electron atoms is entirely accounted for by the following postulate Only eigenfunctions which are antisymmetric in the electrons , that is, change sign when any two electrons are interchanged, correspond to existant states of the system. This is the quantum mechanics statement (26) of the Pauli exclusion principle (43). 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

The second postulate states that a physical quantity or observable is represented in quantum mechanics by a hermitian operator. To every classically defined function A(r, p) of position and momentum there corresponds a quantum-mechanical linear hermitian operator A(r, (h/i)V). Thus, to obtain the quantum-mechanical operator, the momentum p in the classical function is replaced by the operator p... 

The relationship between the geometry of the saddle point of index one (SPi-1) and the accessibility to the quantum transition states cannot be proved, but it can be postulated [43,172], To some extent, invariance of the geometry associated with the SPi-1 would entail an invariance of the quantum states responsible for the interconversion. Thus, if a chemical process follows the same mechanism in different solvents, the invariance of the geometry of the SPi-1 to solvent effects would ensure the mechanistic invariance. This idea has been proposed by us based on computational evidence during the study of some enzyme catalyzed reactions [94, 96, 97, 100-102, 173, 174, 181-184],... 

To test this conclusion we need values of the energies of normal covalent bonds between unlike atoms. These values might be calculated by quantum-mechanical methods it is simpler, however, to make a postulate and test it empirically. Since a normal covalent bond A—B is similar in character to the bonds A —A and B—B, we expect the value of the bond energy to be intermediate between the values for A—A and B—B result follows from the postulate of the... 

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