Wave functions, physical interpretation

Space integrals of expressions quadratic in the wave function are interpreted as expectation values of the corresponding physical quantities. This interpretation suggests that (47) should be interpreted as the expectation value of the photon energy, which would mean that... 

This equation expresses one of the most fundamental postulates of quantum mechanics, and is one of the most remarkable discoveries of physics during the 20th century. The absolute square of the electronic wave-function, is interpreted as the probability of finding the electrons at positions r. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

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... 

If the basic set xpk is chosen complete, the virial theorem will be automatically fulfilled and no scaling is necessary. In such a case, the wave function under consideration may certainly be expressed in the form of Eq. III. 18, but, if the basis is chosen without particular reference to the physical conditions of the problem, the series of determinants may be extremely slowly convergent with a corresponding difficulty in interpreting the results. It therefore seems tempting to ask whether there exists any basic set of spin orbitals. which leads to a most "rapid convergency in the expansion, Eq. III. 18, of the wave function for a specific state (Slater 1951). 

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. 

Another experiment that relates to the physical interpretation of the wave function was performed by O. Stem and W. Gerlach (1922). Their experiment is a dramatic illustration of a quantum-mechanical effect which is in direct conflict with the concepts of classical theory. It was the first experiment of a non-optical nature to show quantum behavior directly. 

Young s double-slit experiment and the Stem-Gerlaeh experiment, as described in the two previous sections, lead to a physical interpretation of the wave function associated with the motion of a particle. Basic to the concept of the wave function is the postulate that the wave function contains all the... 

According to Bom s statistical interpretation, the wave function completely describes the physical system it represents. There is no information about the system that is not contained in i). Thus, the state of the system is determined by its wave function. For this reason the wave function is also called the state function and is sometimes referred to as the state F(x, t). 

The wave function W(x, i) may be represented as a Fourier integral, as shown in equation (2.7), with its Fourier transform A p, t) given by equation (2.8). The transform A p, i) is uniquely determined by F(x, t) and the wave function F(x, t) is uniquely determined by A p, i). Thus, knowledge of one of these functions is equivalent to knowledge of the other. Since the wave function F(x, /) completely describes the physical system that it represents, its Fourier transform A(p, t) also possesses that property. Either function may serve as a complete description of the state of the system. As a consequence, we may interpret the quantity A p, f)p as the probability density for the momentum at... 

The wave function for this system is a function of the N position vectors (ri, r2,. .., r v, i). Thus, although the N particles are moving in three-dimensional space, the wave function is 3iV-dimensional. The physical interpretation of the wave function is analogous to that for the three-dimensional case. The quantity... 

The wave function P itself is not observable. A physical interpretation can only be associated with the square of the wave function in that... 

If we multiply the probability density P(x, y, z) by the number of electrons N, then we obtain the electron density distribution or electron distribution, which is denoted by p(x, y, z), which is the probability of finding an electron in an element of volume dr. When integrated over all space, p(x, y, z) gives the total number of electrons in the system, as expected. The real importance of the concept of an electron density is clear when we consider that the wave function tp has no physical meaning and cannot be measured experimentally. This is particularly true for a system with /V electrons. The wave function of such a system is a function of 3N spatial coordinates. In other words, it is a multidimensional function and as such does not exist in real three-dimensional space. On the other hand, the electron density of any atom or molecule is a measurable function that has a clear interpretation and exists in real space. 

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

There needs to be some physical interpretation of the wave function and its relationship to the state of the system. One interpretation is that the square of the wave function, ip2, is proportional to the probability of finding the parts of the system in a specified region of space. For some problems in quantum mechanics, differential equations arise that can have solutions that are complex (contain (-l)1/2 = i). In such a case, we use ip ip, where ip is the complex conjugate of ip. The complex conjugate of a function is the function that results when i is replaced by — i. Suppose we square the function (a + ib) ... 

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. 

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... 

But in fact, Y (or the more general time-dependent wave function Y, which is the product of two functions, one involving the time alone and the other the coordinate alone) was difficult to interpret physically, because the idea of the... 

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

If, also, each of the has a certain physical interpretation or significance, then one says the wave function or the state represented by it, consists of a fraction... 

As discussed in Section 3.2.3.6, the major difficulty in wave function based calculations is that correlation between electrons of opposite spin must somehow be introduced into a theory that starts with the physically unrealistic premise that electrons of opposite spin move independently of each other. However, this tacit assumption not only provides a mathematically tractable starting point (i.e., HF theory) for wave function based calculations, but this assumption also underpins the entire concept of orbitals (i.e., wave functions for single electrons). The existence of MOs may be a construct, but it is a construct that has proven to be very useful in interpreting the results of both calculations and experiments. 

Complex scalar products arise naturally in quantum mechanics because there is an experimental interpretation for the complex scalar product of two wave functions (as we saw in Section 1.2). Students of physics should note that the traditional brac-ket notation is consistent with our complex scalar product notation—just put a bar in place of the comma. The physical importance of the bracket will allow us to apply our intuition about Euclidean geometry (such as orthogonality) to states of quantum systems. 

Bom coined the term "Quantum mechanics and in 1925 devised a system called matrix mechanics, which accounted mathematically for the posidon and momentum of the electron in the atom. He devised a technique called the Born approximation in scattering theory for computing the behavior of subatomic particles which is used in high-energy physics. Also, interpretation of the wave function for Schrodinger s wave mechanics was solved by Born who suggested that the square of the wave function could be understood as the probability of finding a particle at some point in space, For this work in quantum mechanics. Max Bom received the Nobel Prize in Physics in 1954,... 

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