Momentum operator space representation

The limelight now shifts to the time-energy phase space. A detailed description is beyond the scope of the present chapter has been recently reviewed (15). The present focus will therefore be on the interrelations between the position-momentum phase space representation and the propagator representing operators in the time-energy phase space. 

Because of the spherical symmetry of physical space, any realistic physical operator (such as the Schrodinger operator) must commute with the angular momentum operators. In other words, for any g e SO(3) and any f in the domain of the Schrodinger operator H we must have H o p(g ] = pig) o H, where p denotes the natural representation of 80(3 on L2(] 3 Exercise 8.15 we invite the reader to check that H does indeed commute with rotation. The commutation of H and the angular momentum operators is the infinitesimal version of the commutation with rotation i.e., we can obtain the former by differentiating the latter. More explicitly, we differentiate the equation... 

It is instructive to consider the momentum-space representation of the Gaussian wave packet. In this representation, the states are projected onto the eigenstates of the momentum operator, i.e., P p) = p p), which in the coordinate representation takes the form... 

In Section 4.2.2, we used the displacement (translation) operator exp(—ibpi/h). We consider here this operator and its action on the state ), i.e., we consider the momentum-space and coordinate-space representations of b) = exp(—ibpi/h) ). 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

The elements of the so, (3) algebra (the angular momentum operators) acting on the Fock space of the totally symmetric representations [N,0,0] of u,(3) (which is the space of the Q30) have been obtained by van der Jeugt [30]. In the space defined by Eq. (35), the elements... 

Now the DK Hamiltonian may be calculated to the desired level of accuracy. Within our finite basis set approximation the multiple integral expressions occurring at the evaluation of the momentum space operators are reduced to simple matrix multiplications, which are computationally not very demanding. As soon as /foKHn has been evaluated within the chosen p -representation, it can be transformed back to the usual configuration space representation by applying the inverse transformation This Hamiltonian is then available for every variational procedure without any further modifications. 

However, these benefits come at a price. Both Vgg and Vxc and their contributions to the transformations obviously change at each self-consistent iteration so the net effect is that some very complicated operator products, involving both momentum and direct space representations, must be done at every iteration. What Rosch and co-workers noticed [44] was that the singular part of the Hamiltonian Vxe of course does not change from iteration to iteration, so they attempted an incomplete DKH transformation which retained only V g and incorporated, therefore, the bare electron-electron interactions in the transformed Hamiltonian. 

This ensures that we can replace the p and p with the coordinate space representation —iV of the momentum operator p, since... 

Choosing Ho to be the kinetic energy operator for electrons, the single particle indices p, q in this equation should be read as the momentum space representation... 

Hence, one of the operators must be a differential operator while the other must be a simple multiplicative operator of the same variable. The first choice is called the position-space representation, while the second is called the momentum-space representation. Of course, one may add constants to these definitions but they are chosen to be zero since they would represent arbitrary shifts. Further, we must require that all arbitrary functions of position and momentum vanish. 

So far only the position-space formulation of the (stationary) Dirac Eq. (6.7) has been discussed, where the momentum operator p acts as a derivative operator on the 4-spinor Y. However, for later convenience in the context of elimination and transformation techniques (chapters 11-12), the Dirac equation is now given in momentum-space representation. Of course, a momentum-space representation is the most suitable choice for the description of extended systems under periodic boundary conditions, but we will later see that it gains importance for unitarily transformed Dirac Hamiltonians in chapters 11 and 12. We have already encountered such a situation, namely when we discussed the square-root energy operator in Eq. (5.4), which cannot be evaluated if p takes the form of a differential operator. 

The so-called real-space methods provide a viable alternative to the supercell approach for molecules and clusters. Real-space methods use only the position-space representation (position space is also known as real space ), which implies that molecules and clusters can be dealt with directly, without artificial supercells. The Laplacian operator V, exactly evaluated in momentum space (see equation 97), has to be approximated in real-space methods. The most popular approaches " use a finite-difference approximation for the Laplacian. For example, the second derivative with respect to xofa function y, z) can be approximated by the following finite difference. 

The solution of time evolution problems for classical systems is facilitated by introducing a classical phase space representation that plays a role in the description of classical systems in a matmer that is formally analogous to the role played by the coordinate and momentum representations in quantum mechanics. The state vectors T ) of this representation enumerate all of the accessible phase points. The phase function /(E ) is given by / (f ) = (f I/), which can be thought to represent a component of the vector f) in the classical phase space representation. The application of the classical Liouville operator (f ) to the phase function /(f ) is defined by (f )/(f ) = (f I/), where is an abstract op-... 

By representing the operator containing the potential energy in position state space and the one containing the kinetic energy in momentum space, one obtains the following phase space discretized path integral representation ... 

This operator, too, is a scalar in the space of total angular momentum for an electron. Tensors in this space are, for example, the operators of electric and magnetic multipole transitions (4.12), (4.13), (4.16). So, the operator of electric multipole transition (4.12) in the second-quantization representation is... 

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