Wave equation symmetrical

In this case, too, molecule formation results from the symmetric eigenfunction. The corresponding perturbation energy W1 is obtained from an equation of the type of Equation 20 involving I hj and the wave equation 28. It is... 

Until now we have described electrons states as obeying the wave equation without specifying which wave equation. In general it is necessary to solve the Dirac equation even for a local potential. If the local potential is spherically symmetric the Dirac equation reduces to ... 

This net symmetric regauging operation successfully separates the variables, so that two inhomogeneous wave equations result to yield the new Maxwell... 

For the H system and its isotopomers after separating the CM motion from the Schrodinger equation, the problem is reduced to a two pseudo-particle problem. In the basis functions defined in eqn.40 r = (r[, r 2) is the 6x1 vector of relative coordinates defined above. The ground state spatial wave function (symmetric with respect to exchange of electrons) is then given as the symmetry projected linear combination of the 0, ... 

The procedure is to solve the wave equation for electrons in a suitable spherically symmetric positive potential. The solutions depend on quantum numbers, much like those of atoms, and even more like those of nuclei. The magic numbers correspond to the filling of shells. For the alkali metals the magic numbers are A 2, 8, (18,) 20, 34, 40 and 58. ... 

Now it can be shown that if a helium atom is initially in a symmetric state no perturbation whatever will suffice to cause it to change to any except symmetric states (the two electrons being considered to be identical). Similarly, if it is initially in an antisymmetric state it will remain in an antisymmetric state. The solution of the wave equation has provided us with... 

The equations of quantum statistical mechanics for a system of non-identical particles, for which all solutions of the wave equations are accepted, are closely analogous to the equations of classical statistical mechanics (Boltzmann statistics). The quantum statistics resulting from the acceptance of only antisymmetric wave functions is considerably different. This statistics, called Fermi-Dirac statistics, applies to many problems, such as the Pauli-Sommerfeld treatment of metallic electrons and the Thomas-Fermi treatment of many-electron atoms. The statistics corresponding to the acceptance of only the completely symmetric wave functions is called the Bose-Einstein statistics. These statistics will be briefly discussed in Section 49. 

Since we have considered only assemblages of point particles heretofore, we have not given the rules for setting up the wave equation for a rigid body. We shall not discuss these rules here1 but shall take the wave equation for the symmetrical top... 

Since the dynamics of rigid bodies is based on the dynamics of particles, these rules must be related to the rules given in Chapter IV. For a discussion of a method of finding the wave equation for a system whose Hamiltonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys. Rev. 32, 812 (1928), and for the specific application to the symmetrical top see the references below. 

Inasmuch as the known solutions of the wave equation for a symmetrical-top molecule form a complete set of orthogonal functions (discussed in the preceding section), we can expand the wave function p in terms of them, writing... 

There is a close connection between the coordinate system in which a given wave equation is separated and the dynamical quan dties which are the constants of the motion for the resulting wave functions. Thus for a single particle in a spherically symmetric field the factor S(d, wave function which depends only on the angles satisfies the equation (see Sec. 18a)... 

Once again it must be emphasized that we are here dealing not with a phenomenon predicted by quantum mechanics, but with a convenient mode of description, in terms of approximations, of matters to which those approximations ought really never to have been applied. That they have been so applied in an imperfect world is a necessity imposed by the absence of methods which are at the same time precise and manageable. A correct solution of the wave equation for a combined carbon atom (in methane) would presumably predict four symmetrically disposed axes of maximum electric density for the configuration of minimum energy, and not the existence of 2s and 2p wave functions. The latter apply to isolated atoms in any case. Interaction with other atoms modifies the density distribution, as is seen from the fact that two hydrogen atoms, each with... 

Diameter of nanoelements or nanodroplets and nanoparticles that constitute substmcture of the given aerosol, that is, the secondary particle obtained by its solidification, can be determined by the wave equation of the centrally symmetric standing wave formed as the... 

This allows splitting the wave equation into symmetrical terms... 

As stated above, the transverse mode components determine the intensity distribution over the cross-section of the beam. The simplest (basic) mode solution when solving the wave equation for a cylindrical-symmetric resonator is the so-called Gaussian mode, which yields for the beam intensity distribution... 

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