Interpretation of the Wave Function

It follows that the contribution of electrons to the dipole moment jl is 

X = ejxp(r)dxdydz j,y = ejyp(r)dxdydz = ejzp(r)dxdydz, (1.24) 

The wave function is thus probability amplitude. The remarkable thing is that by superposing plane waves (adding probability amplitudes), we can obtain a new wave function whose absolute square also means a probability density. 

If plane waves are added, they add up in some regions and cancel in others  

One may show that if n - oo, is concentrated in a region roughly of magnitude Ax = h/Ap. Ap is the uncertainty in p, since the sum in Equation 1.26 contains terms with c 0 in an interval Ap. It is possible to show the uncertainty relation of Heisenberg  

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

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. 

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 / ... 

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... 

We will soon encounter the enormous consequences of this antisymmetry principle, which represents the quantum-mechanical generalization of Pauli s exclusion principle ( no two electrons can occupy the same state ). A logical consequence of the probability interpretation of the wave function is that the integral of equation (1-7) over the full range of all variables equals one. In other words, the probability of finding the N electrons anywhere in space must be exactly unity,... 

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) ... 

The probability interpretation of the wave function in quantum mechanics obtained by forming the square of its magnitude leads naturally to a simple idea for the weights of constituent parts of the wave function when it is written as a linear combination of orthonormal functions. Thus, if... 

The local approach may be extended, as Hiberty[44] suggests, by allowing the AOs to breathe . This is accomplished in modem times by writing the orbitals in as linear combinations of more primitive AOs, all at one nuclear center, and optimizing these linear combinations along with the coefficients in Eq. (7.1). The breathing thus contributes a nonlinear component to the energy optimization. This latter is, of course, only a practical problem it contributes no conceptual difficulty to the interpretation of the wave function. 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

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,... 

The paper by Max Born on Quantummechanik des Stossvorgange, in which he had proposed the statistical interpretation of the wave function, had appeared in 1926.32 Niels Bohr had presented his principle of complementarity at the Como Conference in September 192733 and Heisenberg had formulated the uncertainty principle shortly before the Solvay Conference.34... 

The BOVB method has several levels of accuracy. At the most basic level, referred to as L-BOVB, all orbitals are strictly localized on their respective fragments. One way of improving the energetics is to increase the number of degrees of freedom by permitting the inactive orbitals to be delocalized. This option, which does not alter the interpretability of the wave function, accounts better for the nonbonding interactions between the fragments and is referred to... 

An important feature of the BOVB method is that the active orbitals are chosen to be strictly localized on a single atom or fragment, without any delocalization tails. If this were not the case, a so-called "covalent" structure, defined with more or less delocalized orbitals like, e.g., Coulson-Fischer orbitals, would implicitly contain some ionic contributions, which would make the interpretation of the wave function questionable [27]. The use of pure AOs is therefore a way to ensure an unambiguous correspondence between the concept of Lewis structural scheme and its mathematical formulation. Another reason for the choice of local orbitals is that the breathing orbital effect is... 

The VB structures can be defined in different ways according to the desired level of accuracy, but all levels agree on the principle that the active orbitals should be strictly localized on their specific atom or fragment, and not allowed to delocalize in the course of the orbital optimization process. This latter condition is important for keeping the interpretability of the wave function in terms of Lewis structures, but also for a correlation-consistent description of the system throughout a potential surface. 

By analogy, we interpret the square of the wave function i/r for a particle as a probability density for that particle. That is, [i/r(x, y, z)] dV is the probability that the particle will be found in a small volume dY = dxdydz centered at the point (x, y, z). This probabilistic interpretation of the wave function, proposed by the German physicist Max Born, is now generally accepted because it provides a consistent picture of particle motion on a microscopic scale. 

State the conditions that a function must satisfy in order to be a solution of the Schrodinger equation. Explain how these conditions provide the probability interpretation of the wave function (Section 4.5). 

Recalling the quantum-mechanical interpretation of the wave function as a probability-amplitude, we see that a product form of the many-body wave function corresponds to treating the probability amplitude of the many-electron system as a product of the probability amplitudes of individual electrons (the orbitals). Mathematically, the probability of a composed event is only equal to the probability of the individual events if the individual events are independent (i.e., uncorrelated). Physically, this means that the electrons described by the product wave function are independent. Such wave functions thus neglect the fact that, as a consequence of the Coulomb interaction, the electrons try to avoid each other. The correlation energy is the additional energy lowering obtained in a real system due to the mutual avoidance of the interacting electrons. 

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