Single-particle Hilbert space

In the preceding discussion we have expanded the density in terms of N < M Hilbert space such that their norms are less than or equal to one and the trace of the density is equal to N. All these expansions could in principle be exact there is no need for M = r =, as is clearly demonstrated in the KS procedure, where M = N Ai M <°o and i- = < , then new forms of auxiliary states, i.e. different from single determinantal ones, are implicitly introduced. 

Using the Dirac notations a) = ipa( ) and assuming that ipa( ) are or-thonormal functions (a (3) = 5ap we can write the single-particle matrix (tight-binding ) Hamiltonian in the Hilbert space formed by 4>a ( )... 

Quantum Fractional Statistics relies on the fundamental assumption [2] that Ad=- [AN, where Ad is the change in the Hilbert-space dimension of a single particle when a number AN of ptarticles is added to the system confined into a finite region of the space, and g (exclusion statistical parameter) accounts for the number of states excluded by a single particle added to the system. If G denotes the total number of states available to a single particle, then. 

Just as the use of a finite basis set in independent-electron models restricts the domain of the relevant one-electron operator, h, so the algebraic approximation results in the restriction of the domain of the total Hamiltonian to a finite-dimensional subspace of the Hilbert space. In most applications of quantum mechanics to atoms and molecules which go beyond the independent-electron models, the JV-electron wavefunction is expressed in terms of the fVth-rank direct product space M generated by a finitedimensional single-particle space MlS that is... 

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