Dirac momentum-space representation

For both the elimination and transformation techniques the momentum-space representation of the Dirac equation. 

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 technology for solving the Schrddinger equation is so much farther advanced in r space than in p space that it is most practical to obtain the momentum-space from its position-space counterpart The transformation theories of Dirac [118,119] and Jordan [120,121] provide the hnk between these representations ... 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

In classical mechanics, positions and momenta are treated on an equal footing in the Hamiltonian picture. In quantum mechanics, they become operators, but it is true that the position r and momentum p of a particle are appropriate conjugate variables that can entirely equivalently describe a state of a system under the commutation relation [r, p] = i (Dirac, 1958). This equivalence is usually demonstrated by the example of the onedimensional harmonic oscillator. The choice of the most appropriate representation depends on convenient description of the phenomenon considered. Generally, the position representation is useful for most bound-state problems such as atomic and molecular electronic structures as well as for many scattering problems. The momentum-space treatment... 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

Equation of quantum state. The Dirac bra-ket formalism of quantum mechanics. Representation of the wave-momentum and coordinates. The adjunct operators. Hermiticity. Normal and adjunct operators. Scalar multiplication. Hilbert space. Dirac function. Orthogonality and orthonormality. Commutators. The completely set of commuting operators. 

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