Eigen function

Eigen function In wave mechanics, the Schrodinger equation may be written using the Hamiltonian operator H as... 

Only certain energy values ( ) will lead to solutions of this equation. The corresponding values of the wave functions are called Eigen functions or characteristic wave functions. 

Consider two well-separated atoms A and B with electron wave functions and which are eigen functions of the atoms, with energies and ei. If we bring these atoms closer, the wave functions start to overlap and form combinations that describe the chemical bonding of the atoms to form a molecule. We will neglect the spin of the electrons. The procedure is to construct a new wave function as a linear combination of atomic orbitals (LCAO), which for one electron has the form... 

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Let y be one of the eigen function of the operator and let a be the corresponding eigen value. We will have... 

The quantum mechanical operator for the linear momentum in one direction is d/dx. The operator applied on the eigen functions of a particle in a one dimensional box and thus shown that these functions are not eigen functions of the momentum operator and suggest a possible reason for this. 

The expression of eigen function for one dimensional box is given by the expression... 

If two operators and commute then they have the same set of eigen functions... 

This shows that y, is an eigen function of the operator with the eigen value This is possible only if (Yi) is a multiple of y, i.e. 

Let f be the eigen function, of with the eigen value 1. Thus. 

Much interest has developed on approximate techniques of solving quantum mechanical problems because exact solutions of the Schrodinger equation can not be obtained for many-body problems. One of the most convenient of such approximations for the solution of many-body problems is the application of the variational method. For instance, with approximate eigen-functions p , the eigen-values of the Hamiltonian H are En... 

And the best estimate of the eigen-functions ipn is obtained by minimising the variational integral, and is given by E . This well-known procedure can be extended from the Schrodinger equation to the diffusion equation, since, in effect, both are diffusion equations [499]. 

From the diffusion equation (9), the lowest eigen-function in the spherical volume is... 

The rate coefficient can be directly related to the current of the diffusing species up to the reaction radius R, but it is much more conveniently related to the relaxation of the lowest eigen-function... 

This is the simplest version of the rate coefficient which can be derived. Ham suggested that a better approximation to the eigen-function A0 can be obtained from the variational integral... 

Following the usual procedure (see Morse and Feshbach [499]), the Green s function can be expanded in the eigen-functions [Pg.313]

Here, the electron wave function J/M(r R) is the eigen function of Hamiltonian... 

Generally speaking, the adiabatic wave function (2) is not a stationary one because it is not the eigen function of total Hamiltonian of the system (1). In reality, the electron wave function J/M(r R) depends on R and so the differential operator rR acts not only on / (R), but also on i/q/r R). It results in appearance of non-adiabatic correction operator in the basis of functions (2)... 

Here, Wv( p ) is the eigen function of Hamiltonian H( p ) with the eigen energy ev and the summation occurs on all states including the integration on the continuous spectra, 8 is the infinitesimal defining the rule of the pole bypass in expression (51). 

In the expression (53), i/r coul>s the eigen function of Hamiltonian Coui(fi) and zn (q, — 8qs) is the wave function of, sth mode of the system without electron. Note that the potential of the electron interaction with the vibrations E/,-(r, q ) does not include in Hamiltonian H( p ), therefore the equilibrium position of the vibration mode is shifted to the value Sqs with respect to one in the system with electron (48). It is supposed that the vibrations are the phonons of the media, and the shift of the equilibrium positions are only taken into consideration. The possible shift of the frequencies may be easily taken into account by the corresponding changes in the phonon energy (see Chapter 2, formulas (20b), (27) and (27a)). The set of quantum numbers in (51) is v = (p, ns ), and the corresponding energies are... 

The eigen values and eigen functions of the Hamiltonian in equation (8.28) have been tabulated for J = 5/2,3,..., 8 in the literature [13] by using the equation... 

Above investigation for the behaviour of the solution at the free stream can be extended to other equations of (2.4.15) to check their usefulness in obtaining the eigen function. At the free stream the characteristic roots for Eqn. (2.4.15a) are —a, —Q. Equation (2.4.15b) being a third order equation has three roots given by [—a, —Q, a- -Q ]. Thus, this equation is also violently unstable, even for low wave numbers and Reynolds numbers. Equation (2.4.15c) has the asymptotic behaviour for large y s as dictated by the characteristic roots given by [—a, —Q, Finally, the characteristic... 

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