Real space wave functions

I. Real Space Wave Functions of f Electrons in Condensed Matter. 

A discussion of the real space wave functions for RE and actinides, showing the large differences with the atomic like wave functions can be found, for example, in Freeman s introductory paper to the Physics of Actinides and Related 4f Materials (11), (especially Figure 8 of that paper for y - U). See also (12). In our analysis here we have concentrated on the rare earths. 

The real space wave functions are the solutions of a Schrodinger like equation... 

Fig. 1. Perspective plot of a two-dimensional slice of a potential energy surface near a conical intersection. The degeneracy point is located at the origin in the uv plane. The radial distance from the intersection is denoted by r and the azimuthal angle around the intersection denoted by g. The adiabatic ground state real electronic wave function changes sign for any closed path in uv space which encircles the origin (such as the dashed curve C).

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

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. 

For the calculation of the hydrodynamic thickness we divide the profile artificially into elementary layers, the result being independent of the division chosen provided it is sufficiently fine. The s.a.n.s. data is obtained as a function of Q, the wave vector (4it/A sin(0/2), where X is the neutron wavelength and 0 the scattering angle. The Q resolution corresponds in real space to a fraction of a bond length which is small enough for defining an elementary layer. 

Chaos provides an excellent illustration of this dichotomy of worldviews (A. Peres, 1993). Without question, chaos exists, can be experimentally probed, and is well-described by classical mechanics. But the classical picture does not simply translate to the quantum view attempts to find chaos in the Schrodinger equation for the wave function, or, more generally, the quantum Liouville equation for the density matrix, have all failed. This failure is due not only to the linearity of the equations, but also the Hilbert space structure of quantum mechanics which, via the uncertainty principle, forbids the formation of fine-scale structure in phase space, and thus precludes chaos in the sense of classical trajectories. Consequently, some people have even wondered if quantum mechanics fundamentally cannot describe the (macroscopic) real world. 

A manner to do away with the problem is to introduce appropriate algorithms in the sense that mappings from real space to Hilbert space can be defined. The generalized electronic diabatic, GED approach fulfils this constraint while the BO scheme as given by Meyer [2] does not due to an early introduction of center-of-mass coordinates and rotating frame. The standard BO takes a typical molecule as an object description. Similarly, the wave function is taken to describe the electrons and nuclei. Thus, the adiabatic picture follows. The electrons instantaneously follow the position of the nuclei. This picture requires the system to be always in the ground state. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

The first assumption of quantum mechanics is that each state of a mobile particle in Euclidean three-space can be described by a complex-valued function of three real variables (called a wave function ) satisfying... 

---