Permutations of identical particles

For simplicity, we first consider a system composed of two identical particles 

If H( 1, 2) and 11(2. 1) were to differ, then the corresponding Schrodinger equations and their solutions would also differ and this difference could be used to distinguish between the two particles. 

The time-independent Schrodinger equation for the two-particle system is 

The eigenfunction V(l, 2) has the form of a wave in six-dimensional space. The quantity P (1, 2)V(1, 2) diq dr2 is the probability that particle 1 with spin function % is in the volume element diq centered at iq and simultaneously particle 2 with spin function is in the volume element dr2 at r2. The product tP (l, 2)XP(1, 2) is, then, the probability density. The eigenfunction x (2, 1) also has the form of a six-dimensional wave. The quantity x (2, l)x (2, 1) is the probability density for particle 2 being at iq with spin function X and simultaneously particle 1 being at r2 with spin function %i- In general, the two eigenfunctions E(1, 2) and XP(2, 1) are not identical. As an example, if 

the probability density of the pair of particles depends on how we label the two particles. Since the two particles are indistinguishable, we conclude that neither 0(1, 2) nor 0(2, 1) are desirable wave functions. We seek a wave function that does not make a distinction between the two particles and, therefore, does not designate which particle is at ri and which is at r2. 

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Under the conditions determined by the symmetric permutation of identical particles, any number of particles can be placed in each quantum state, lb distribute , of a given energy in gi quantum states, consider the fo Ibarriers separating the boxes are necessary. There are then nt particles and gi - 1 barriers. There fore, there are nf +g,- - 1 Bosons that can be permuted and (/i + gi -1) possible permutations. However, as the particles are indistinguishable, the elements of each ensemble of , can be permuted, leading to / possibilities. Clearly, the indistinguishability of the barriers corresponds to (gi - 1) permutations, The number of combinations is then given by... 

The energy of a quantum system is invariant to permutations of identical particles in the system. Thus, the Hamiltonian for a system with n identical particles can be said to commute with the elements of the nth-order symmetric group ... 

If N is the total number of molecules, and Nt = Nxt is the number of molecules of species i in the system, then the first factor is the ideal entropic contribution arising from permutations of identical particles. In the thermodynamic limit of large particle numbers this is well approximated by... 

The Longuet-Higgins Molecular Symmetry Group in this third example is the same as the previous one (56), because the number of feasible permutations of identical particles is the same. 

Another striking example of a group which leaves a physical system invariant is provided by the set of permutations of identical particles. If we have a molecule with 2 electrons, each of which may be found in either of 2 states, ip a or ipg, it would seem that the state of the whole system could be indicated in symbols by (1) (2) (electron 1 in the A state , electron 2 in the B state ). But if interchanging the electrons can make no observable difference surely u(2)t/>fi(l) (electron 2 in the A state , electron 1 in the B state ) would be equally acceptable - or even a combination of the two What group theory tells us is that the 2-electron system, described as being in state (1,2), must respond to an interchange of 1 and 2 by either (i) remaining... 

This mixing of the nominal nuclear and electronic variables that means that the separation of the Hamiltonian into electronic and nuclear parts such as is presented in the last section, is not maintained under the permutation of identical particles. In the absence of a settled division it is clearly problematic to know if the separated entities are precisely what they are claimed to be. Of course if there were some good reason for believing that the proton variables should not be permuted then there would be rather less trouble. 

Invariance with Respect to Permutation of Identical Particles (Fermions and Bosons)... 

Besides these strict conservation laws (energy, momentum, angular momentum, permutation of identical particles, charge, and baryon and lepton numbers), there are also some approximate laws. Two of these parity and charge conjugation, will be discussed below. They are rooted in these strict laws, but are valid only in some conditions. For example, in most experiments, not only the baryon number, but also the nirmber of nuclei of each kind, are conserved. Despite the importance of this law in chemical reaction equations, this does not represent any strict conservation law, as shown by radioactive transmutations of elements. 

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

Invariance with respect to permutations of identical particles (fermions and bosons)... 

INVARIANCE WITH RESPECT TO PERMUTATION OF IDENTICAL PARTICLES (FERMIONS AND BOSONS)... 

J, quantizes its component along the z axis, and II = 1 represents the parity with respect to the inversion. As to the invariance with respect to permutations of identical particles an acceptable wave function has to be antisymmetric with respect to the exchange of identical fermions, whereas it has to be s)mmetric when exchanging bosons. 

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. 

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