Postulates, of quantum mechanics

Having solved only one problem in quantum mechanics we allow for the fact that there are only a very few known solved problems and try to write down the postulated mles and then we will apply the rules to another problem to reinforce the concepts. 

Postulate I The state of a quanmm-mechanical system is completely specified by a function (r, i) that depends on the coordinates of the particle and on time. This function, called the wave fimction or the state function, has the important property that the product of (r, f) (r, t) dxdydz is the probability that the particle lies in the volume dxdydz located at tix,y,z) at time t. 

Corollary In order for the wave function to be used in the Schrodinger equation, it must have 

It must be defined for at least first and second derivatives. 

Postulate II To every observable laboratory measurement in classical mechanics there corresponds an operator in quantum mechanics. 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

We first state the postulates succinctly and then elaborate on each of them with particular regard to the mathematical properties of linear operators. The postulates are as follows. 

The state of a physical system is defined by a normalized fimction P of the spatial coordinates and the time. This function contains all the information that exists on the state of the system. 

Every physical observable A is represented by a linear hermitian operator A. 

Every individual measurement of a physical observable A yields an eigenvalue of the corresponding operator A. The average value or expectation value A) from a series of measurements of A for systems, each of which is in the exact same state 

The quantum mechanical model of atomic structure is based on a set of postulates that can only be justified on the basis of their ability to rationalize experimental behavior. However, the foundations of quantum theory have their origins in the field of classical wave mechanics. The fundamental postulates are as follows  

Postulate 2 For every physically observable variable in classical mechanics, there exists a corresponding linear, Hermitian operator in quantum mechanics. Examples are shown in Table 3.2, where the symbol indicates a quantum mechanical operator and h = h/2n. A Hermitian operator is one which satisfies Equation (3.28). 

Postulate 3 In any measurement where an exact solution can be obtained, the only values that will ever be observed are the eigenvalues a that satisfy the eigenvalue equation given by Equation (3.30). This postulate is the one that is responsible for the quantization of energy. 

Although measurements must alw s yield an eigenvalue, the state does not initially have to be an eigenstate of A. Any arbitrary state can be expanded 

TABLE 3.2 Classical mechanical observables and their quantum mechanical operators. 

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Since it is not possible to generate antisynnnetric combinations of products if the same spin orbital appears twice in each tenn, it follows that states which assign the same set of four quantum numbers twice cannot possibly satisfy the requirement P.j i = -ij/, so this statement of the exclusion principle is consistent with the more general symmetry requirement. An even more general statement of the exclusion principle, which can be regarded as an additional postulate of quantum mechanics, is... 

By learning the solutions of the Schrodinger equation for a few model systems, the student can better appreciate the treatment of the fundamental postulates of quantum mechanics as well as their relation to experimental measurement because the wavefunctions of the known model problems can be used to illustrate. 

Hilbert Space and Quantum Mechanics.—In this section we shall express the fundamental postulates of quantum mechanics in terms of the concepts developed in the previous sections. 

The postulates of quantum mechanics discussed in Section 3.7 are incomplete. In order to explain certain experimental observations, Uhlenbeck and Goudsmit introduced the concept of spin angular momentum for the electron. This concept is not contained in our previous set of postulates an additional postulate is needed. Further, there is no reason why the property of spin should be confined to the electron. As it turns out, other particles possess an intrinsic angular momentum as well. Accordingly, we now add a sixth postulate to the previous list of quantum principles. 

Chapter 3 is the heart of the book. It presents the postulates of quantum mechanics and the mathematics required for understanding and applying the postulates. This chapter stands on its own and does not require the student to have read Chapters 1 and 2, although some previous knowledge of quantum mechanics from an undergraduate course is highly desirable. 

In order to systematize the procedures and basic premises of quantum mechanics, a set of postulates has been developed that provides the usual starting point for studying the topic. Most books on quantum mechanics give a precise set of rules and interpretations, some of which are not necessary for the study of inorganic chemistry at this level. In this section, we will present the postulates of quantum mechanics and provide some interpretation of them, but for complete coverage of this topic the reader should consult a quantum mechanics text such as those listed in the references at the end of this chapter. 

The Hartree-Fock approach derives from the application of a series of well defined approaches to the time independent Schrodinger equation (equation 3), which derives from the postulates of quantum mechanics [27]. The result of these approaches is the iterative resolution of equation 2, presented in the previous subsection, which in this case is solved in an exact way, without the approximations of semiempirical methods. Although this involves a significant increase in computational cost, it has the advantage of not requiring any additional parametrization, and because of this the FIF method can be directly applied to transition metal systems. The lack of electron correlation associated to this method, and its importance in transition metal systems, limits however the validity of the numerical results. 

It is one of the postulates of quantum mechanics that for every observable quantity there is a corresponding operator such that an average or expectation value of the observable may be obtained by evaluating the expression... 

The postulates and theorems of quantum mechanics form the rigorous foundation for the prediction of observable chemical properties from first principles. Expressed somewhat loosely, the fundamental postulates of quantum mechanics assert dial microscopic systems are described by wave functions diat completely characterize all of die physical properties of the system. In particular, there aie quantum mechanical operators corresponding to each physical observable that, when applied to the wave function, allow one to predict the probability of finding the system to exhibit a particular value or range of values (scalar, vector. 

We begin with a brief recapitulation of some of the key features of quantum mechanics. The fundamental postulate of quantum mechanics is that a so-called wave function, P, exists for any (chemical) system, and that appropriate operators (functions) which act upon h return the observable properties of the system. In mathematical notation. 

The preceding statements together with the postulates of spin and the Pauli principle (Sections 1.15 and 1.16) can be taken as the postulates of quantum mechanics. 

Special attention must be paid in systems of identical particles, where we have to take into account the symmetry postulate of quantum mechanics. This means that the space of states for fermions is the antisymmetric subspace of while the symmetric subspace dK+N refers to bosons. 

Do you think it is reasonable to describe the Schrodinger equation as a postulate of quantum mechanics What is a postulate ... 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

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) ... 

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