Schrodinger interpretation

The most attractive feature of the hydrodynamic model is that it obviates the statistical interpretation of quantum theory, by eliminating the need of a point particle. However, even Einstein, despite his famous insistence that "the old one does not play dice", and despite the convincing physical picture of the Schrodinger interpretation, remained convinced that an electron had to be a point particle. His first allegiance was, after all, with his own special theory of relativity that imposes an upper limit of c on the speed at which any signal can be transmitted. 

Although Eq. (139) looks like a Schrodinger equation that contains a vector potential x, it cannot be interpreted as such because t is an antisymmetric matrix (thus, having diagonal terms that are equal to zero). This inconvenience can be repaired by employing the following unitary bansformation ... 

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

It is not the intention that this book should be a primary reference on quantum mechanics such references are given in the bibliography at the end of this chapter. Nevertheless, it is necessary at this stage to take a brief tour through the development of the Schrodinger equation and some of its solutions that are vital to the interpretation of atomic and molecular spectra. 

After the discovery of quantum mechanics in 1925 it became evident that the quantum mechanical equations constitute a reliable basis for the theory of molecular structure. It also soon became evident that these equations, such as the Schrodinger wave equation, cannot be solved rigorously for any but the simplest molecules. The development of the theory of molecular structure and the nature of the chemical bond during the past twenty-five years has been in considerable part empirical — based upon the facts of chemistry — but with the interpretation of these facts greatly influenced by quantum mechanical principles and concepts. 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

In recent years the old quantum theory, associated principally with the names of Bohr and Sommerfeld, encountered a large number of difficulties, all of which vanished before the new quantum mechanics of Heisenberg. Because of its abstruse and difficultly interpretable mathematical foundation, Heisenberg s quantum mechanics cannot be easily applied to the relatively complicated problems of the structures and properties of many-electron atoms and of molecules in particular is this true for chemical problems, which usually do not permit simple dynamical formulation in terms of nuclei and electrons, but instead require to be treated with the aid of atomic and molecular models. Accordingly, it is especially gratifying that Schrodinger s interpretation of his wave mechanics3 provides a simple and satisfactory atomic model, more closely related to the chemist s atom than to that of the old quantum theory. 

Schrodinger has interpreted the quantity being the conjugate complex... 

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

The essential features of the particle-wave duality are clearly illustrated by Young s double-slit experiment. In order to explain all of the observations of this experiment, light must be regarded as having both wave-like and particlelike properties. Similar experiments on electrons indicate that they too possess both particle-like and wave-like characteristics. The consideration of the experimental results leads directly to a physical interpretation of Schrodinger s wave function, which is presented in Section 1.8. 

The KG equation is Lorentz invariant, as required, but presents some other problems. Unlike Schrodinger s equation the KG equation is a second order differential equation with respect to time. This means that its solutions are specified only after an initial condition on bothand d /dt has been given. However, in contrast to k, d /dt has no direct physical interpretation [61]. Should the KG equation be used to define an equation of continuity, as was done with Schrodinger s equation (4), it is found to be satisfied by... 

Although Dirac s equation does not directly admit of a completely self-consistent single-particle interpretation, such an interpretation is physically acceptable and of practical use, provided the potential varies little over distances of the order of the Compton wavelength (h/mc) of the particle in question. It allows, for instance, first-order relativistic corrections to the spectrum of the hydrogen atom and to the core-level densities of many-electron atoms. The latter aspect is of special chemical importance. The required calculations are invariably numerical in nature and this eliminates the need to investigate central-field solutions in the same detail as for Schrodinger s equation. A brief outline suffices. 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

A numerical solution of the Schrodinger equation in Eq. [1] often starts with the discretization of the wave function. Discretization is necessary because it converts the differential equation to a matrix form, which can then be readily handled by a digital computer. This process is typically done using a set of basis functions in a chosen coordinate system. As discussed extensively in the literature,5,9-11 the proper choice of the coordinate system and the basis functions is vital in minimizing the size of the problem and in providing a physically relevant interpretation of the solution. However, this important topic is out of the scope of this review and we will only discuss some related issues in the context of recursive diagonalization. Interested readers are referred to other excellent reviews on this topic.5,9,10... 

---