Fundamental Assumptions of Quantum Mechanics

One major point of this book is to make deep predictions using only symmetry and very few assumptions about quantum mechanics. In this section we make explicit the assumptions we use and give some information about the experiments that justify these assumptions. 

To appreciate this section and, more broadly, to appreciate the importance of this book s topic as a justification for mathematics, one should understand the role of theory in the physical sciences. While in mathematics the intrinsic beauty of a theory is sufficient justification for its study, the value of a theory in the physical sciences is limited to the value of the experimental predictions it makes. For example, the theory of the double-helical structure of 

DNA (first proposed by Crick, Franklin and Watson in the 195O s [Ju, Part I]) suggested, and continues to suggest, experimental predictions in molecular biology. We hope, in the course of the book, to convince the reader that the mathematics we discuss (e.g., analysis, representation theory) is of scientific importance beyond its importance within mathematics proper. In order to succeed, we must use mathematics to pull testable experimental predictions from the physically-inspired assumptions of this section. 

The first assumption of quantum mechanics is that each state of a mobile particle in Euclidean three-space can be described by a complex-valued function / of three real variables (called a wave function ) satisfying 

To make use of this description, we must relate the function f to possible experiments. 

---

It is a bit of a lie to say, as we did in previous chapters, that complex scalar product spaces are state spaces for quantum mechanical systems. Certainly every nonzero vector in a complex scalar product space determines a quantum mechanical state however, the converse is not true. If two vectors differ only by a phase factor, or if two vectors normaUze to the same vector, then they will determine the same physical state. This is one of the fundamental assumptions of quantum mechanics. The quantum model we used in Chapters 2 through 9 ignored this subtlety. However, to understand spin we must face this issue. 

I have examined the corrections listed above, and / think that all of them fail the Schopenhauer test they adulterate the quantum-mechanical methodology as defined above. There is thus a need to go back to the fundamental assumptions of quantum mechanics and reinvent them to produce results that coincide with experimental data. I realize that this is going to be a difficult process, since the methods we criticize here have been part of the teaching of all aspects of the quantum theory for about 80 years. 

In many cases there is more than one way of stating the fundamental postulates. Thus either Lagrange s or Hamilton s form of the equations of motion may be regarded as fundamental for classical mechanics, and if one is so chosen, the other can be derived from it. Similarly, there are other ways of expressing the basic assumptions of quantum mechanics, and if they are used, the wave equation can be derived from them, but, no matter which mode of presenting the theory is adopted, some starting point must be chosen, consisting of a set of assumptions not deduced from any deeper principles. 

The above example shows how the time-dependent Schrodinger equation produces the time-independent Schrodinger equation, assuming a certain form of f) and a time-independent H. It is therefore correct to say that equation 10.29 is the fundamental equation of quantum mechanics, but given the separability assumption, more attention in textbooks is devoted to understanding the position-dependent part of the complete, time-dependent Schrodinger equation. It is easy to show that wavefunctions of the form in equation 10.31 are stationary states, because their probability distributions do not depend on time. Some wavefunctions are not of the form in equation 10.31, so the time-dependent Schrodinger equation must be used. 

Point 3, the assumption that Schriklinger s equation is exact, is very significant at a fundamental level. Einstein s Special Theory of Relativity proposed a modification of classical mechanics in a different direction from that of quantum mechanics. The corrections involved in Einstein s theory, which are not incorporated in Schrbdinger s equation, only become significant... 

It is immediately apparent that a theory like transition-state theory is making no pretensions at stating and describing the underlying principles of the behavior of the system. In any serious analysis in terms of the deeper and more fundamental laws of physics (of quantum mechanics, in particular) the further assumptions in its derivation are arbitrary, artificial, and somewhere between wildly simplistic and quite unsound. Nevertheless, the theory is typically introduced via a complex mathematical argument in which it is derived using a series of assumptions and approximations from the supposedly underlying equations of quantum theory and/or statistical mechanics. 

In contrast to force-field calculations in which electrons are not explicitly addressed, molecular orbital calculations, use the methods of quantum mechanics to generate the electronic structure of molecules. Fundamental to the quantum mechanical calculations that are to be performed is the solution of the Schrodinger equation to provide energetic and electronic information on the molecular system. The Schrodinger equation cannot, however, be exactly solved for systems with more than two particles. Since any molecule of interest will have more than one electron, approximations must be used for the solution of the Schrodinger equation. The level of approximation is of critical importance in the quality and time required for the completion of the calculations. Among the most commonly invoked simplifications in molecular orbital theory is the Bom-Oppenheimer [13] approximation, by which the motions of atomic nuclei and electrons can be considered separately, since the former are so much heavier and therefore slower moving. Another of the fundamental assumptions made in the performance of electronic structure calculations is that molecular orbitals are composed of a linear combination of atomic orbitals (LCAO). 

Key words Transition-state theory (TST) - Variational TST (VTST) - Fundamental assumption of TST -Quantum mechanical TST... 

The fundamental assumption of this theory is the Schrodinger equation, conceived by Erwin Schrodinger in 1926 from the wave and particle properties of matter. This equation, the basis of QUANTUM (WAVE) MECHANICS, is, in principle, applicable to the problems that arise when particles, such as electrons, nuclei, atoms, and molecules, are subject to a force. Many phenomena—for example, the stability of the covalent bond (Chapter 18) and the intensities of spectral lines—could not be explained prior to the application of quantum mechanics. 

Quantum mechanics is based on several statements called postulates. These postulates are assumed, not proven. It may seem difficult to understand why an entire model of electrons, atoms, and molecules is based on assumptions, but the reason is simply because the statements based on these assumptions lead to predictions about atoms and molecules that agree with our observations. Not just a few isolated observations Over decades, millions of measurements on atoms and molecules have yielded data that agree with the conclusions based on the few postulates of quantum mechanics. With agreement between theory and experiment so abundant, the unproven postulates are accepted and no longer questioned. In the following discussion of the fundamentals of quantum mechanics, some of the statements may seem unusual or even contrary. However questionable they may seem at first, realize that statements and equations based on these postulates agree with experiment and so constitute an appropriate model for the description of subatomic matter, especially electrons. 

The role—indeed, the existence—of quantum mechanics was appreciated only during the twentieth century. Until then it was thought that the motion of atomic and subatomic particles could be expressed in terms of the laws of classical mechanics introduced in the seventeenth century by Isaac Newton (see Fundamentals F.3), for these laws were very successful at explaining the motion of planets and everyday objects such as pendulums and projectiles. Classical physics is based on three obvious assumptions ... 

The Schrodinger equation does not provide a complete theory of quantum mechanics. Schrodinger, Heisenberg, and others devised several postulates (unproved fundamental assumptions) that form a consistent logical foundation for nonrelativistic quantum mechanics. In any theory based on postulates, the validity of the postulates is tested by comparing the consequences of the postulates with experimental fact. The postulates of quantum mechanics do pass this test. These postulates can be stated in slightly different ways. We will state five postulates in a form similar to that of Mandl and Levine. The first two postulates were introduced in Chapter 15 without calling them postulates. We now state them explicitly ... 

In its full generality, our third fundamental quantum-mechanical assumption says that the same kind of decomposition is possible with base states of the position observable, the momentum observable or indeed any observable. In other words, every observable has a complete set of base states. Typically the information about the base states and the value of the observable on each base state is collected into a mathematical object called a self-adjoint linear operator. The base states are the eigenvectors and the corresponding values of the observable are the eigenvalues. For more information about this point of view, see [RS, Section VIII.2]. 

Thus far our examination of the quantum mechanical basis for control of many-body dynamics has proceeded under the assumption that a control field that will generate the goal we wish to achieve (e.g., maximizing the yield of a particular product of a reaction) exists. The task of the analysis is, then, to find that control field. We have not asked if there is a fundamental limit to the extent of control of quantum dynamics that is attainable that is, whether there is an analogue of the limit imposed by the second law of thermodynamics on the extent of transformation of heat into work. Nor have we examined the limitation to achievable control arising from the sensitivity of the structure of the control field to uncertainties in our knowledge of molecular properties or to fluctuations in the control field arising from the source lasers. It is these subjects that we briefly discuss in this section. 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

These results mean that, once the GME coinciding with the CTRW has been built up, we cannot look at it as a fundamental law of nature. If this GME were the expression of a law of nature, it would be possible to use it to study the response to external perturbations. The linear response theory is based on this fundamental assumption and its impressive success is an indirect confirmation that ordinary quantum and statistical mechanics are indeed a fair representation of the laws of nature. But, as proved by the authors of Ref. 104, this is no longer true in the non-Poisson case discussed in this review. 

This assumption of less energy redistribution in the symmetric isotopomers, compared with asymmetric isotopomers, remains to be tested by ah initio quantum mechanical calculations, and as well as by a more direct experiment, as discussed later. Such fundamental quantum mechanical calculations would yield an a priori value of t]. In principle, p can also be inferred from the more direct but more difficult experiment. 

---