Configuration space operator, quantum

We may transform this to the quantum-mechanical Hamiltonian operator by substitution of the configuration space operators... 

The classical concept of object dissolves in so far the configuration space for the internal degrees of freedom is concerned. The material elements such as electrons and nuclei must be present to sustain quantum states, but locali-zability is not a requirement it may be a result of specific operators. The configuration space is an abstract mathematical space. Of course, one can force a representation as position vectors for particles. Consequently, one has to interpret the wavefunction. But again, Eqs. (3 and 4) demand amplitudes, energy gaps, and quantum numbers. This is spectroscopy of one type or another. The introduction of I-frames allows classical frameworks to be naturally incorporated. 

Introduction of Dirac notation [66,71] at this point helps us transform the trace of the density operator into an integral over configuration space, which ultimately gives rise to the path-integral representation. We let n> represent a state such that the system is found to have a particular set of quantum numbers n (it is an eigenstate of the measurement of n) similarly, we let x> represent a state in which the system is surely found at a particular position x. According to this picture, the wavefunction n(x) is a projection of the quantum number amplitude upon the position amplitude, specifically, i/rn(x) = and %( ) = partition function becomes... 

Quantum mechanics taJces many different guises. For instance, one can use a Hilbert space realination in terms of time-dependent wavefunctions ( 1, 2,- of an 7V-particle system, where = (r, C) is a compound configuration space and spin variable for a particle. Operators acting on this Hilbert space are obtained by making the common identifications p —ihV for the momentum and f— fioi position vector of a particle, which can be referred to as first quantization, producing quantum mechanical operators out of classical expressions. One can equally well use time-dependent field operators (so-called second quantization) tp, t) and their adjoints build Hilbert spaces (or rather... 

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]. 

Depending on gauge choices, Quiney et al. write two expressions became popular in atomic physics and quantum chemistry [213] the interaction operator in Lorenz (Feynman) gauge in configuration-space representation as... 

Basis of a representation Let X be the carrier space of the group G whose elements g e G are represented by unitary operators I7(g) e U G), which is typical when applying group theoretical methods to quantum mechanical problems. Assume that dim J =n and consider a set of n linearly independent functions defined in this configuration space with the property ... 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

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