The Physical Meaning of a Wave Function

The square of the function here means the square of the magnitude, 

I / 12. This distinction is important when orbitals with complex numbers are being considered I 12 = (real part)2 + (imaginary part)2. 

Now that we have examined some of the mathematical details of the quantum mechanical treatment of the hydrogen atom, we need to consider what it all means. What is a wave function, and what does it tell us about the electron to which it applies First, a warning There is always danger in taking a mathematical description of nature and using our human experiences to interpret it. Although our attempts to attach physical significance to mathematical descriptions are quite useful to us as we try to understand how nature operates, they must be viewed with caution. Simple pictorial models of a particular natural phenomenon always oversimplify the phenomenon and should not be taken too literally. With that caveat we will proceed to try to picture what the quantum mechanical atom is like. 

The quotient NdN2 is the ratio of the probabilities of finding the electron in the infinitesimally small volume elements dv around points 1 and 2. For example, if the value of the ratio N,/N2 is 100, the electron is 100 times more likely to be found at position 1 than at position 2. The model gives no information concerning when the electron will be at either position or how it moves between the positions. This vagueness is consistent with the concept of the Heisenberg uncertainty principle. 

Another way of representing the electron probability distribution for the Is orbital is to calculate the probability at points along a line drawn outward in any direction from the nucleus. The result is shown in Fig. 12.16(b), where R2 (the square of the radial part—the part that depends on r—of the Is orbital) is plotted versus r. Note that the probability of finding the electron at 

Another way of representing the electron probabiHty distribution for the Is orbital is to calculate the probability at points along a line drawn outward 

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It has been assumed, necessarily, that the reader has some prior familiarity with the basic notions of quantum theory. He is expected to know in a general way what the wave equation is, the significance of the Hamiltonian operator, the physical meaning of a wave function, and so forth, but no detailed knowledge of mathematical intricacies is presumed. Even the contents of a rather qualitative book such as Coulson s Valence should be sufficient, although, of course, further background knowledge will not be amiss. 

Given the limitations indicated by the uncertainty principle, what then is the physical meaning of a wave function for an electron That is, what is an atomic orbital Although the wave function itself has no easily visualized meaning, the square of the function does have a definite physical significance. The square of the function... 

The wave function T is a function of the electron and nuclear positions. As the name implies, this is the description of an electron as a wave. This is a probabilistic description of electron behavior. As such, it can describe the probability of electrons being in certain locations, but it cannot predict exactly where electrons are located. The wave function is also called a probability amplitude because it is the square of the wave function that yields probabilities. This is the only rigorously correct meaning of a wave function. In order to obtain a physically relevant solution of the Schrodinger equation, the wave function must be continuous, single-valued, normalizable, and antisymmetric with respect to the interchange of electrons. 

In further studies of chemistry and physics, you will learn that the wave functions that are solutions to the Schrodinger equation have no direct, physical meaning. They are mathematical ideas. However, the square of a wave function does have a physical meaning. It is a quantity that describes the probability that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probability distribution graphs for that orbital. These plots help chemists visualize the space in which electrons are most likely to be found around atoms. These plots are... 

The physical meaning of the wave functions is as follows The probability for finding the particle that the wave function describes in a small volume element dT is given by 2 dr. It is clear, then, that the probability of finding the particle in the complete region of configuration equals one (the condition of normalization).. 

To describe the electronic structure, the electronic wave function W(x, y, z, t) is used, which depends, in general, on both space and time. Here, however, only its spatial dependence will be considered, M (x, y, z). For detailed discussions of the nature of the electronic wave function, we refer to texts on the principles of quantum mechanics [9-12], For a one-electron system the physical meaning of the electronic wave function is expressed by the product of F with its complex conjugate F. The product F - M di gives the probability of finding an electron in the volume di = dx dydz about the point (x, y, z). 

Unsatisfactory features of these particular calculations are due to the limitations of the HF-RPA procedure to produce quantitatively correct spectroscopic patterns, and to the very small basis set that we adopted in order to carry out evaluations of several thousands of HF ground-electronic-state wave functions and their associated RPA spectra. Such a quantitative inadequacy of the computational model, however, does not alter the physical meaning of the conclusion that can be drawn the appearance of a single broad band in the optical spectrum of Na9 is simply a thermal broadening effect, which, in general, is particularly effective in the presence of several low-lying isomers or in the case of cluster structures characterized by low-frequency vibrations, accompanied by large nuclear displacements due to the low curvature of the BO surface. 

The vector k is real because the parameters kj are real. Since k is defined in terms of the reciprocal lattice vectors bj, it can be thought of as a wave-vector expfik r) represents a plane wave of wave-vector k. The physical meaning of this result is that the wavefunction does not decay within the crystal but rather extends throughout the crystal like a wave modified by the periodic function Mk(r). This fact was first introduced in chapter 1. 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

As required by (36), the variational parameter k is calculated to vary between k = 2 at R = 0 and k = 1 at R > 5ao- The parameter k is routinely interpreted as either a screening constant or an effective nuclear charge, as if it had real physical meaning. In fact, it is no more than a mathematical artefact, deliberately introduced to remedy the inadequacy of hydrogenic wave functions as descriptors of electrons in molecular environments. No such parameter occurs within the Burrau [84] scheme. 

It seems quite natural to describe the extended part of a quantum particle not by wavepackets composed of infinite harmonic plane waves but instead by finite waves of a well-defined frequency. To a person used to the Fourier analysis, this assumption—that it is possible to have a finite wave with a well-defined frequency—may seem absurd. We are so familiar with the Fourier analysis that when we think about a finite pulse, we immediately try to decompose, to analyze it into the so-called pure frequencies of the harmonic plane waves. Still, in nature no one has ever seen a device able to produce harmonic plane waves. Indeed, this concept would imply real physical devices existing forever with no beginning or end. In this case it would be necessary to have a perfect circle with an endless constant motion whose projection of a point on the centered axis gives origin to the sine or cosine harmonic function. This would mean that we should return to the Ptolemaic model for the Havens, where the heavenly bodies localized on the perfect crystal balls turning in constant circular motion existed from continuously playing the eternal and ethereal harmonic music of the spheres. 

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