Differentiability eigenvalue equation

The last section discussed angular momentum from the viewpoint of solving the differential eigenvalue equations that result from expressing L2... 

This reduces the Schrodinger equation to = 4/. To solve the Schrodinger equation it is necessary to find values of E and functions 4/ such that, when the wavefunction is operated upon by the Hamiltonian, it returns the wavefunction multiplied by the energy. The Schrodinger equation falls into the category of equations known as partial differential eigenvalue equations in which an operator acts on a function (the eigenfunction) and returns the... 

From the definitions (3.316) and (3.318) of the two Fock operators / and /, we can see that the two integro-differential eigenvalue equations (3.312) and (3.313) are coupled and cannot be solved independently. That is,/ depends on the occupied orbitals, through Jj, and depends on the occupied a orbitals, through JJ. The two equations must thus be solved by a simultaneous iterative process. 

There is a linear algebra equivalent of the differential eigenvalue equation. Instead of a differential operator acting on a function, a square matrix multiplies a column vector, and this is equal to a constant, the eigenvalue, times the vector ... 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

Frost and Pearson treated Scheme XV by the eigenvalue method, and we have solved it by the method of Laplace transforms in the preceding subsection. The differential rate equations are... 

The benefit is now that the HF equations are turned from complicated integro-differential equations into pseudo-eigenvalue equations for the unknown expansion coefficients c. 

The eigenvalue equation (S9.1-15) therefore presents an intuitive geometrical picture of how a matrix A operates on a general vector u by differentially stretching its components in different eigen-directions. 

To find the stationary points of the RF model we differentiate Eq. (3.18) and set the result equal to zero. We arrive at the eigenvalue equations... 

The second complication is that the equation, as traditionally interpreted, only handles point particles, but produces eigenfunction solutions of more complex geometrical structure. By analogy with electromagnetic theory the square of the amplitude function could be interpreted as matter intensity, but this is at variance with the point-particle assumption. The standard way out is to assume that ip2 represents a probability density rather than intensity. Historical records show that this interpretation of particle density was introduced to serve as a compromise between the rival matrix and differential operator theories of quantum observables, although eigenvalue equations, formulated in either matrix or differential formalism are known to be mathematically equivalent. 

Substituting (7.27) into (7.26) and using the angular-momentum eigenvalue equation (6.35), we obtain an ordinary differential equation for the radial function R(r) ... 

Usually, the problem is to find simultaneously V and the values o that satisfy the eigenvalue equation (2.1), the form of the operator having been previously established. An operator is a symbol telling us to carry out a certain mathematical operation on anything following it. An example is the differential operator O = d/dx. It is easily found that ip=x is not an eigenfunction of this operator... 

Substitution of (1-3) in the eigenvalue equation leads to the following coupled differential equations for the expansion coefficients / (Q) [48]... 

The AOs are obtained by solving some kind of differential Schrodinger-type eigenvalue equation, which for a single electron can be written ... 

Both works [2] and [3] show the separations of the eigenvalue equations for H and H, and H and H, in their respective spheroconal coordinates, into Lame differential equations in the individual elliptical cone angular coordinates. The corresponding solutions are Lam6 spheroconal polynomials included in the classic book of Whittaker and Watson [12]. In practice, the numerical evaluation of such Lame functions was not developed in an efficient manner so that the exact formulation of Ref. [2] did not prosper. Consequently, the analysis of rotations of asymmetric molecules took the route of perturbation theory using the familiar basis of spherical harmonics. 

Since we first require eigenvalues only, we begin with the unforced set of differential-difference equations for w = 1 since the spherical harmonic of interest will be Pj(cos )e . From Eq. (4.14), this set is... 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

It immediately follows from the hermitian character of the respective subsystem hamiltonians and their eigenvalue equations that these energy functions satisfy the following differential Hellmann-Feynman theorems ... 

As an example, in Fig. 5.1 we return to our favored ammonia molecule and list all nuclear permutations, with and without the all-particle inversion operator, that leave the full Hamiltonian invariant. Nuclear permutations are defined here in the same way as in Sect. 3.3. A permutation such as (ABC) means that the letters A, B, and C are replaced by B, C, and A, respectively. The inversion operator, E, inverts the positions of all particles through a common inversion center, which can be conveniently chosen in the mass origin. In total, 12 combinations of such operations are found, which together form a group that is isomorphic to Ds. How is this related to our previous point group At this point it is very important to recall that the state of a molecule is not only determined by its Hamiltonian but also, and to an equal extent, by the boundary conditions. The eigenvalue equation is a differential equation that has a very extensive set of mathematical solutions, but not all these solutions are also acceptable states of the physical system. The role of the boundary conditions is to define constraints that Alter out physically unacceptable states of the system. In most cases these constraints also lead to the quantization of the energies. 

This is no longer an eigenvalue equation but an inhomogeneous differential (or in general integro-differential) equation. One way of solving such equations is to expand in terms of the eigenfunctions of... 

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