Fermionic degenerate state

Consider N fermions in a box with fixed energy levels E, (including rest-mass which itself includes internal excitation energy). To each E, there correspond co, distinct (degenerate) states, made up of gi internal states and 4n Vp2dp/h3 kinetic degrees of freedom, where p is momentum and V is the volume. 

The structure of the parametric UA for the 4-RDM satisfies the fourth-order fermion relation (the expectation value of the commutator of four annihilator and four creator operators [26]) for any value of the parameter which is a basic and necessary A-representability condition. Also, the 4-RDM constructed in this way is symmetric for any value of On the other hand, the other A-representability conditions will be affected by this value. Hence it seems reasonable to optimize this parameter in such a way that at least one of these conditions is satisfied. Alcoba s working hypothesis [48] was the determination of the parameter value by imposing the trace condition to the 4-RDM. In order to test this working hypothesis, he constructed the 4-RDM for two states of the BeHa molecule in its linear form Dqo/,. The calculations were carried out with a minimal basis set formed by 14 Hartree-Fock spin orbitals belonging to three different symmetries. Thus orbitals 1, 2, and 3 are cr orbitals 4 and 5 are cr and orbitals 6 and 7 are degenerate % orbitals. The two states considered are the ground state, where... 

For the degenerate ground state, the wave function (10) is relevant only for bosons (deuterons), for the two particles in the same states are indistinguishable. For fermions (protons), spins are correlated in such a way that the total wave function is antisymmetrical with respect to particle permutations, according to the Pauli principle. The spatial wave function can be rewritten as... 

The origin of the nonvauishing Joule-Thomson effect is the effective repulsive (Fermions) and attractive (Bosons) potential exerted on the gas molecules, which arises from the different ways in which quantum states can be occupied in sy.stems obeying Fermi-Dirac and Boso-Einstein statistics, respectively [17]. In other words, the effective fields are a consequence of whether Pauli s antisymmetry principle, which is relativistic in nature [207], is applicable. Thus, a weakly degenerate Fermi gas will always heat up ((5 < 0), whereas a weakly degenerate Bose gas will cool down (5 > 0) during a Joule-Thomson expansion. These conclusions remain valid even if the ideal quantum gas is treated relativistically, which is required to understand... 

The two sets of detenmnants therefore form a basis for the doubly degenerate Kramers pairs of the iV-electron states. For the fermion irreps of the binary groups, the only other element of symmetry that can be exploited is inversion, which may be handled in the same way as in nonrelativistic Cl theory. 

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