Eigenfunctions normalization

It is convenient to use the set of eigenfunctions normalized in a box of dimensions (Lx, Ly, Lz) with periodic boundary conditions... 

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

The variation principle then says that the energy E0 of the ground state is the lower bound of the quantity Eq. II.6 for arbitrary normalized trial wave functions W and that further all eigenfunctions satisfy the relation... 

The spectrum of the operator q consists of the points in euclidean three space. The eigenfunctions x > are not normalizable in the usual way as they correspond to eigenvalues in the continuous spectrum, but are normalized to a 8-function... 

The eigenfunction 100, the electron density p = s10o, and the electron distribution function D = 4 x rJ p of the normal hydrogen atom as functions of the distance r from the nucleus. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

As an example we may calculate the energy of the helium atom in its normal state (24). Neglecting the interaction of the two electrons, each electron is in a hydrogen-like orbit, represented by equation 6 the eigenfunction of the whole atom is then lt, (1) (2), where (1) and (2) signify the first and the second electron. 

The radicals in the denominators are necessary in order that the new eigenfunctions be normalized. The wave equation (Equation 13) can now be written... 

Curve 1 shows the total energy for the normal state of the hydrogen molecule as given by the first-order perturbation theory curve 2, the naive potential function obtained by neglecting the resonance phenomenon and curve 3, the potential function for the antisymmetric eigenfunction, corresponding to elastic collision. 

Excited states of the hydrogen molecule may be formed from a normal hydrogen atom and a hydrogen atom in various excited states.2 For these the interelectronic interaction will be small, and the Burrau eigenfunction will represent the molecule in part with considerable accuracy. The properties of the molecule, in particular the equilibrium distance, should then approximate those of the molecule-ion for the molecule will be essentially a molecule-ion with an added electron in an outer orbit. This is observed in general the equilibrium distances for all known excited states but one (the second state in table 1) deviate by less than 10 per cent from that for the molecule-ion. It is hence probable that states 3,4, 5, and 6 are formed from a normal and an excited atom with n = 2, and that higher states are similarly formed. 

Substitution of this eigenfunction in an expression of the type of Equation 21 permits the evaluation of the perturbation energy W1, in the course of which use is made of the properties of orthogonality arid normalization of the spin eigenfunctions namely,... 

The Structures of Simple Molecules.—The foregoing considerations throw some light on the structure of very simple molecules in the normal and lower excited states, but they do not permit such a complete and accurate discussion of these questions as for more complicated molecules, because of the difficulty of taking into consideration the effect of several unshared and sometimes unpaired electrons. Often the bond energy is not great enough to destroy s-p quantization, and the interaction between a bond and unshared electrons is more important than between a bond and other shared electrons because of the absence of the effect of concentration of the eigenfunctions. 

This argument may be repeated in greater detail. Each atom in the molecule contains four L eigenfunctions, o-b, t+, it-, and cr0, of which the last is not important for bond formation. A normal nitrogen and a normal oxygen atom can combine to form a double-bonded II2 molecule. 

This quintic equation is easily reduced to three linear factors and one quadratic factor, the roots being -2a, -2a, 0, (-(13) -l)a, and ((13) -l)a. Since a is negative, the last of these roots, ((13) —l)a = 2.6055a, represents the normal state of the molecule. The eigenfunction corresponding to this is (before normalizing)... 

C 0 , whose contribution to the eigenfunction for the normal state is even larger than... 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

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