Wave function interpretation

Schrodinger s equation appears incomplete in the sense of lacking an operator for spin, only because its eigenfunction solutions are traditionally considered complex variables. The wave function, interpreted as a colunrn vector, operated on by square matrices, such that... 

The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly rather, its existence is inferred from the various experiments whose outcome is most rationally explained using a wave function interpretation of quantum mechanics. Further, the A -particle wave function is a rather complicated construction, depending on 3N spatial coordinates as well as N spin coordinates, correlated in a manner that almost defies description. By contrast, the electron density of an N-electron system is a much simpler quantity, described by three spatial coordinates and even accessible to experiment. In terms of the wave function, the electron density is expressed as... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

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. 

M. Born (Edinburgh) fundamental research in quantum mechanics, especially for the statistical interpretation of the wave function. 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

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. 

The interpretation of the square of the wave function as a probability distribution, the Heisenberg uncertainty principle and the possibility of tunnelling. 

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). 

The structural interpretation of the principal quantum number of nucleonic orbital wave functions and the structural basis provided by the close-packed-spheron theory for the neutron and proton magic numbers are discussed in notes submitted to Phys. Rev. Letters and Nature (L. Pauling, 1965). 

The wave functions for the two inner-core spherons can, of course, be described as the symmetric and antisymmetric combinations of l.t and Ip-functions. The Nilsson (19) treatment of neutron and proton orbitals in deformed nuclei is completely compatible with the foregoing discussion, which provides a structural interpretation of it. 

A final example is the concept of QM state. It is often stated that the wave function must be square integrable because the modulus square of the wave function is a probability distribution. States in QM are rays in Hilbert space, which are equivalence classes of wave functions. The equivalence relation between two wave functions is that one wave function is equal to the other multiplied by a complex number. The space of QM states is then a projective space, which by an infinite stereographic projection is isomorphic to a sphere in Hilbert space with any radius, conventionally chosen as one. Hence states can be identified with normalized wave functions as representatives from each equivalence class. This fact is important for the probability interpretation, but it is not a consequence of the probability interpretation. 

III. Experimental observation of Quantum Mechanics. Only this final section should address the rules that govern interpretations of experiments measuring properties of QM systems with macroscopic devices. This includes probability interpretation, uncertainty relations, complementarity and correspondence. Then experiments can be discussed to show how the wave functions manipulated in section I can be used to predict the probabilistic outcome of experiments. 

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... 

An analysis of the Stem-Gerlach experiment also contributes to the interpretation of the wave function. When an atom escapes from the high-temperature oven, its magnetic moment is randomly oriented. Before this atom interacts with the magnetic field, its wave function is the weighted sum of two possible states a and / ... 

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... 

Suppose we wish to measure the position of a particle whose wave function is W(jc, i). The Bom interpretation of F(x, as the probability density for finding the associated particle at position x at time t implies that such a measurement will not yield a unique result. If we have a large number of particles, each of which is in state /) and we measure the position of each of these particles in separate experiments all at some time t, then we will obtain a multitude of different results. We may then calculate the average or mean value x) of these measurements. In quantum mechanics, average values of dynamical quantities are called expectation values. This name is somewhat misleading, because in an experimental measurement one does not expect to obtain the expectation value. 

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... 

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