Operators Laplacian

Pauli spin vector Dirac spin vector electron spin magnetic moment nuclear spin magnetic moment rotational magnetic moment electric dipole moment Ioldy Wouthuysen operator gradient operator Laplacian... 

A general symbol for a time-dependent wave function Thermodynamic probability - number of microstates Gradient operator Divergence operator Laplacian operator... 

Laplacian operator Surface tension Tensor of viscous tension... 

When /(<[)) takes the form of Eq. (21), the nuclear Laplacian operator is modified according to... 

Equation (6.12) cannot be solved analytically when expressed in the cartesian coordinates x, y, z, but can be solved when expressed in spherical polar coordinates r, 6, cp, by means of the transformation equations (5.29). The laplacian operator in spherical polar coordinates is given by equation (A.61) and may be obtained by substituting equations (5.30) into (6.9b) to yield... 

The symbols and are, respectively, the laplacian operators for a single nucleus and a single electron. The variable is the distance between nuclei a and / , Vai the distance between nucleus a and electron i, and the distance between electrons i and j. The summations are taken over each pair of particles. The quantity e is equal to the magnitude of the electronic charge e in CGS units and to e/(47reo) / in SI units, where eo is the permittivity of free space. 

The laplacian operators in equation (10.23) refer to the spaced-fixed coordinates Qa with components Qxa, Qya, Qza, so that... 

The designations employed in equation (5.6) are as follows D is the EEP diffusion coefficient in its own gas V is the Laplacian operator N is the concentration of EEPs in a gaseous phase N is the concentration of parent gas K is the rate constant of EEP de-excitation by own gas v is the rate constant of EEP radiative de-excitation ro is the cylinder radius v is the heat velocity of EEPs x, r are coordinates traveling along the cylinder axis and radius, respectively. 

Here, A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator V2 is defined as a sum of differential operators (in cartesian coordinates)... 

The operator V2, which is known as the Laplacian, takes on a particularly simple form in Cartesian coordinates, namely,... 

The quantity c = y/rjp is known as the jjjia e velocity, It is the speed a which waves travel along the string. Clearly, the left-hand side of Eq, (4) represents the one-dimensional Laplacian operating on the dependent variable. This expression can be easily generalized to represent wave phenomena in two or more dimensions in space. 

The corresponding operators in y and z are derived in the same way. The sum of these three operators yields the Laplacian as... 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

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