Fermions, antisymmetrized states

Since [P, H] = 0, P is a constant of the motion, which means that a system of particles represented by either ips or will keep that symmetry for all time. The particles of Nature that fall into the two classes with either symmetrical or antisymmetrical states, are known as bosons and fermions respectively. 

Although the previous discussion has focused on ground states, the DMC method can also be applied to the calculation of electronically excited states. This is most simply achieved using the fixed-node approximation. Note that the ground state of a fermion system is itself an excited state. It is the lowest antisymmetric state of the system. 

Particles with antisymmetric wave function are called fermions - they have to obey the Pauli exclusion principle. Apart from the familiar electron, proton and neutron, these include the neutrinos, the quarks (from which protons and neutrons are made), as well as some atoms like helium-3. All fermions possess "half-integer spin", meaning that they possess an intrinsic angular momentum whose value is hbar = li/2 pi (Planck s constant divided by 27i) times a half-integer (1/2, 3/2, 5/2, etc ). In the theory of quantum mechanics, fermions are described by "antisymmetric states", which are explained in greater detail in the article on identical particles. 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Systems containing symmetric wave function components ate called Bose-Einstein systems (129) those having antisymmetric wave functions are called Fermi-Ditac systems (130,131). Systems in which all components are at a single quantum state are called MaxweU-Boltzmaim systems (122). Further, a boson is a particle obeying Bose-Einstein statistics, a fermion is one obeying Eermi-Ditac statistics (132). 

Pauli s original version of the exclusion principle was found lacking precisely because it ascribes stationary states to individual electrons. According to the new quantum mechanics, only the atomic system as a whole possesses stationary states. The original version of the exclusion principle was replaced by the statement that the wavefunction for a system of fermions must be antisymmetrical with respect to the interchange of any two particles (Heisenberg [1925], Dirac [1928]). 

The Hilbert space of pure A -particle fermion states. It is an iV-foId antisymmetric tensor product of the Hilbert space of pure one-particle states. 

These four antisymmetric wave functions are normalized if the single-particle spatial wave functions singlet state occurs... 

These restrictions, imposed above on electrons, apply equally to all pariiqles that are represented by antisymmetric wavefunctions, the so-called Fermions. The condition that no more than one particle can occupy a given quantum state leads immediately to the expression for the number of possible combinations. If C nhgi) is the number of combinations that can be made with g, particles taken tii at a time,... 

The requirement that electrons (and fermions in general) have antisymmetric many-particle wave functions is called the Pauli principle, which can be stated as follows ... 

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

As an example, consider H2. The nuclear spin of H is and we have three symmetric nuclear spin functions and one antisymmetric function. The symmetric spin functions are of the form (1.251)—(1.253), and correspond to the two nuclear spins being parallel. Designating the quantum number of the vector sum of the two nuclear spins as 7, we have 7= 1 for the symmetric spin functions. The antisymmetric spin function has the form (1.254), and corresponds to 7 0. The ground electronic state of H2 is a 2 state, and the nuclei are fermions hence the symmetric (7=1) nuclear spin functions go with the J= 1,3,5,... rotational levels, whereas the 7=0 spin function goes with the7=0,2,4,... levels. 

Special attention must be paid in systems of identical particles, where we have to take into account the symmetry postulate of quantum mechanics. This means that the space of states for fermions is the antisymmetric subspace of while the symmetric subspace dK+N refers to bosons. 

The condition for the states to be antisymmetrized for fermions and symmetrized for bosons, respectively, may be expressed by the following commutation rules ... 

From all this one must conclude that the determinantal and second-quantized formulations should be regarded as a poor man s group theory which, while convenient, hides the basic freeon dynamics. These fermion methods have the additional disadvantage that their antisymmetric fermion functions are not normally pure spin (freeon) states so that spin-projection may be required. A method for avoiding (approximately) spin projection is the employment of the variation principle to approximate the ground state e. g., unrestricted Hartree-Fock theory. Finally the use of the fermion formulations has lead to the spin paradigm as a replacement for the more fundamental freeon dynamics. 

Although H is nonrelativistic, relativistic effects manifest themselves in the realizable eigenstates through permutational symmetry. Spectroscopic states must be totally antisymmetric under the permutation of the labels of any two electrons. In addition, total anti symmetry/symmetry with respect to the permutations of the labels of fermionic/bosonic nuclei must be exhibited. For this reason, the set ]y jMn r,R s not deternuned by mj,Zj> alone but by mj,Zj,Sj>, where Sj> is the set of nuclear spins. 

There are two characteristic difficulties of multichannel many-fermion problems. The first is that computational methods can of course directly address only a finite number of channels whereas the physical problem has an infinite number of discrete channels and the ionisation continuum. The second is that the electrons are identical so that the formulation in terms of one-electron states must be explicitly antisymmetric in the position (or momentum) and spin coordinates. 

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... 

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