Square-root operator

It should be noted that we have written E = +cVp2 + m2c2, rather than the more usual relation E2 — c2p2 + m2c4, so as to insure that the particles have positive energy. In equation (9-63), x(x,<) is a (2s + 1) component wave function whose components will be denoted by X (x,<) ( = 1,- -, 2s + 1) and the square root operator Vm2c2 — 2V2 is to be understood as an integral operator... 

We define the inverse of the Square root operator in order to deal with convergent expressions. 

When is a one component scalar function, one can take the square root of Eq. (9-237) and one thus obtains the relativistic equation describing a spin 0 particle discussed in Section 9.4. This procedure, however, does not work for a spin particle since we know that in the present situation the amplitude must be a multicomponent object, because in the nonrelativistic limit the amplitude must go over into the 2-component nonrelativistic wave function describing a spin particle. Dirac, therefore, argued that the square root operator in the present case must involve something operating on these components. 

The Holstein-Primakoff transformation also preserves the commutation relations (70). Due to the square-root operators in Eqs. (78a)-(78d), however, the mutual adjointness of S+ and 5 as well as the self-adjointness of S3 is only guaranteed in the physical subspace 0),..., i- -m) of the transformation [219]. This flaw of the Holstein-Primakoff transformation outside the physical subspace does not present a problem on the quantum-mechanical level of description. This is because the physical subspace again is invariant under the action of any operator which results from the mapping (78) of an arbitrary spin operator A(5i, 2, 3). As has been discussed in Ref. 100, however, the square-root operators may cause serious problems in the semiclassical evaluation of the Holstein-Primakoff transformation. 

Let s try another example. To solve this equation for x, we clear the square root operator by squaring both sides, and then solving for x. 

The log and antilog operators are inverses of each other, just like the x2 operator and the square root operator. Therefore, the operators on the right side of the equation cancel out. 

The calculation of m thus reduces to estimating values of kij for each pair of species. By convention, = 0 if i = j if i /, then ki is (usually) a small number, which may be either positive or negative. One possible difficulty with Equation 36 must be noted. Because of the square-root operation in the combination rule, values for the pure-component s must be so constrained as to always yield positive values for the i for all anticipated ranges of Tr and o>. 

As can be seen, UIC according to Eq. (5.22) does not lead to a symmetric matrix, unless F and G are themselves symmetric. Therefore, the square root operation must not be applied. It has been discussed that the lack of this operation causes a more significant presence of artefacts from the mathematical operations. As a remedy, generalized indirect covariance (GIG) according to Eq. (5.23) was proposed [14]. 

The square root operation on the covariance matrix, which is necessary to obtain the FT analogous spectrum, is calculated from Eq. (5.31) as U-D-U, hence the square root of the diagonal eigenvalue matrix. Since the corresponding UIC transformed spectrum is contained in the GIC matrix as an off-diagonal element, the GIC formalism provides a mean to apply the square root operation on the non-symmetrical matrix F-G". ... 

According to the definition of the covariance matrix, its square root corresponds to the FT NMR spectrum. Since the UIC treatment, in general, leads to an unsymmetrical matrix, the square root operation is not defined. The UIC covariance map will therefore lack the fuU equivalence to the FT counterpart. Snyder and Briischweiler devised the GIC formalism [14] where the UIC matrix is embedded into an array of matrices according to Eqs. (5.22) and (5.28). An illustration is given in Fig. 5.IF. The GIC array is symmetric and, by SVD, the square root and other power operations can be executed on the UIC submatrix. This is sketched in Fig. 5.IF and G for... 

GIC of the simplified models of the COSY and the multiplicity-edited HSQC. The spectrum resulting for A=1 resembles the UIC spectrum. Yet, the signal intensities are different when the square root operation >1 = 0.5 is applied, see Fig. 5.1 G. The positive effect of the GIC concerning the minimization of artefacts will be summarized in due course. It is not visible in this model. Fig. 5.1H symboHzes the transformation of a multiplicity-edited HSQC with an 1,1-ADEQUATE to yield a C—C correlation map, whose interpretation scheme will be given in Fig. 5.14. The construction of a 3R cube from two 2D spectra is illustrated in Fig. 5.11. It is the only covariance transformation among the examples that does not result from matrix multiphcation but from reconstruction according to Eq. (5.24). 

The energy operator of Eq. (5.4) is known as the square-root operator. It is apparent that the square root of the spatial differentiation, V, on the right-hand side of the resulting equation would be difficult to evaluate in position space. An expansion of the square root would lead to infinitely high derivatives with respect to the spatial coordinates so that time and spatial coordinates would be treated differently, again. 

The square-root operator is difficult to evaluate in position space because of the square root to be taken of a differential operator that would represent p. We have already discussed this issue in the context of the Klein-Gordon equation in section 5.1.1. Hence, the action of the X-operator is most conveniently studied in momentum space, where the inverse operator may be applied in closed form without expanding the square root. 

It should be recalled that, because of the presence of the external potential and the nonlocal form of Ep given by Eq. (11.11), all operators resulting from these unitary transformations are well defined only in momentum space (compare the discussion of the square-root operator in the context of the Klein-Gordon equation in chapter 5 and the momentum-space formulation of the Dirac equation in section 6.10). Whereas So acts as a simple multiplicative operator, all higher-order terms containing the potential V are integral operators and completely described by specifying their kernel. For example, the... 

Considering the derivation of DKH Hamiltonians so far, we are facing the problem to express all operators in momemtum space, which is somewhat unpleasant for most molecular quantum chemical calculations which employ atom-centered position-space basis functions of the Gaussian type as explained in section 10.3. The origin of the momentum-space presentation of the DKH method is traced back to the square-root operator in Sq of Eq. (12.54). This square root requires the evaluation of the square root of the momentum operator as already discussed in the context of the Klein-Gordon equation in chapter 5. Such a square-root expression can hardly be evaluated in a position-space formulation with linear momentum operators as differential operators. In a momentum-space formulation, however, the momentum operator takes a... 

Immediately we face the problem of interpreting the square-root operator on the right-hand side in Eq. [46]. Using, for example, a Taylor expansion would lead to an equation containing all powers of the derivative operator and thus to a nonlocal theory. Such theories are very difficult to handle, and they present an unattractive version of the Schrodinger equation with space and time coordinates appearing in an unsymmetrical form. In the interest of mathematical simplicity, we return to Eq. [40], making the transformation to a quantum mechanical operator representation ... 

Dirac s approach to quantization of the Hamiltonian above was to assume that the argument of the square root operator could be written as a perfect square. 

We recall that it was the desire to find an expansion of this square root operator that led to the development of the Dirac equation (see chapter 4). We see also that the assumption that X commutes with (a p) was justified. The free-particle Foldy-Wouthuysen transformation can now be written... 

The operators in the transformation now involve square roots of the square root operator, and it is much easier to use these in momentum space than in position space, where their interpretation is problematic. 

The first of the relativistic correction terms is called the mass-velocity operator. If we expand the square root operator in the classical relativistic Hamiltonian for a free particle, we find... 

There is a slight inconsistency in the preceding development of the perturbation, which is that the labeling of the orders of perturbation in powers of 1 / 2mc — V) does not strietly work because there are already terms of two different orders in the zeroth-order Hamiltonian. What we must do is to label the terms in each of the expansions of inverse or square root operators. It is in fact easier to define the perturbation series if we start from the unnormalized equation (18.1) and simply multiply El(2m( — V) by the formal perturbation parameter, to give... 

The form of equation (8) differs from the standardly used form (see, e.g.. Ref. 1). In equation (8), only the value of is needed (also when calculating the force), which saves a square root operation. 

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