Identical particles permutation operators

A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. 

Any permutation (of identical particles) operator (due to the non-distinguishability of identical particles) and operators (for the non-relativistic Hamiltonian, p. 66, due to the absence of spin variables in it)... 

The recipe for constructing a two-electron second-quantization operator is therefore given by expressions (1.4.15) and (1.4.37). For any interaction between identical particles, the operator g (xi,X2) is symmetric in xi and X2- The integrals (1.4.37) therefore automatically exhibit the permutational symmetry in (1.4.17). We also note the following usefiil permutational symmetries for real spin orbitals ... 

In equation (8.32) the operator P is any one of the N operators, including the identity operator, that permute a given order of particles to another order. The summation is taken over all N permutation operators. The quantity dp is always - -1 for the symmetric wave function Ps, but for the antisymmetric wave function Wa, dpis-l-l(—l)if the permutation operator P involves the... 

It should be emphasized that in the above presentation of permutation operations, they were carried out on symbols, rather than physical objects. One 1 was exchanged for another as a result of a paper operation . The ication of this principle in physical systems must be made with cate. When it is said that the "exchange of two identical particles yields the following results , it must be understood that it is the exchange of identity of the par-tides, stich as labels or coordinate s that has been made. 

The total Hamiltonian operator H must commute with any permutations I among identical particles (X) due to their indistinguishability. For example, for a system including three types of distinct identical particles (including electrons) like 7Li2 6Li2 with a 1 conformation, one must satisfy the following commutative laws ... 

When the system is made up of identical particles (e.g. electrons in a molecule) the Hamiltonian must be symmetrical with respect to any interchange of the space and spin coordinates of the particles. Thus an interchange operator P that permutes the variables qi and (denoting space and spin coordinates) of particles i and j commutes with the Hamiltonian, [.Pij, H] = 0. Since two successive interchanges of and qj return the particles to the initial configuration, it follows that P = /, and the eigenvalues of are e = 1. The wave functions corresponding to e = 1 are such that... 

Determination of a wave function for a system that obeys the correct permutational symmetry may be ensured by projection onto the irreducible representations of the symmetry groups to which the systems in question belong. For each subset of identical particles i, we can implement the desired permutational symmetry into the basis functions by projection onto the irreducible representation of the permutation group, for total spin 5, using the appropriate projection operator T,. The total projection operator would then be a product ... 

The projection operator takes the form of a sum of all of the possible permutations of the identical particles. Pi, each multiplied by an appropriate constant, a,- ... 

The concept of exchange degeneracy is of appreciable import herein (see Section 7.1 below). Baym (Ref. 16, p. 391) defines it by considering the permutations P among a set of identical particles as applied to a Hamiltonian operator opH symmetric in these. Various of the different energy eigenstates of opP opH will be degenerate, with such superpositions not affected by Zeeman interactions.13... 

Altmann considered two types of operations that belong to the Schrodinger subgroup the Euclidean and the discrete symmetry operations. Euclidean operations are those that change the laboratory axes, leaving the Hamiltonian operator invariant. They are translations and rotations of the whole molecule, in free space, in which the x,y,z molecular axes are kept constant. A discrete symmetry operation is a change of the molecular axes in such away as to induce permutations of the coordinates of identical particles [10]. 

In Eqs. (98)—(102), cyclic notation was used, with / denoting the identity permutation and x, representing either particles /3, or holes w,-. The trivial permutation operator, P = /, will not be explicitly used. It should be noted that the signs are in agreement with the parities of the permutation cycles, so that the criterion of correct phase for noncanonical diagrams is implemented on the single index set permutation operator level. 

Ptj, the symmetry operation involving interchange of identical particles (nuclei or electrons). All particles axe either Bosons or Fermions, and the total wavefunction must, respectively, be even or odd upon interchange of any pair of identical particles. The total wavefunction of a homonuclear molecule, exclusive of the nuclear spin part, is classified s or a according to whether it is even or odd with respect to nuclear exchange. Since electrons are Fermions, the total molecular wavefunction must be odd with respect to permutation of any two electrons. This requirement is satisfied by the determinantal form of the electronic wavefunction (see Section 3.2.4). 

Permutation operations that interchange identical particles. 

Before considering the symmetry under permutations of identical particles it is necessary first to say a little ab out the spin of particles. Each particle is specified not only by space variables but also by spin variables. These have not been considered so far because there are no spin operators in the Hamiltonians discussed in the previous sections. Nevertheless spin is, indirectly, very important in the construction of approximate wavefunctions. 

If our analysis of the problem is correct, however, these approaches cannot be considered fully satisfactory. The arguments given above imply that in order to construct the potential well in terms of which feasible operations can be defined, it is necessary largely to ignore the permutational symmetry that feasible operations are invoked to restore at least in part. There is also something of a logical difficulty in such approaches. The permutations of the variables of identical particles are simply mathematical operations in the quantum theory they do not correspond to physical operations. However, once the idea of feasibility is associated with a permutation then some physical effect seems inevitably to be implied. So it is... 

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