Particle indistinguishable

Because of the quantum mechanical Uncertainty Principle, quantum m echanics methods treat electrons as indistinguishable particles, This leads to the Paiili Exclusion Pnn ciple, which states that the many-electron wave function—which depends on the coordinates of all the electrons—must change sign whenever two electrons interchange positions. That IS, the wave function must be antisymmetric with respect to pair-wise permutations of the electron coordinates. 

An additional complication in the PIMC simulations arises when Bose or Fermi statistics is included in the formalism. The trace in the partition function allows for paths which may end at a particle index which is different from the starting index. In this way larger, closed paths may build up which eventually spread over the entire system. All such possible paths corresponding to the exchange of indistinguishable particles have to be taken into account in the partition function. For bosons these contributions are summed up for fermions the number of permutations of particle indices involved decides whether the contribution is added (even) or subtracted (odd) in the partition function. 

This discussion applies only to systems with distinguishable particles for example, systems where each particle has a different mass. The treatment of wave functions for systems with indistinguishable particles is more compli-... 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

The conclusion is then that the wavefunction representing a system composed of indistinguishable particles must be either symmetric or antisymmetric under the permutation operation. On purely physical grounds, this result is apparent, as the probability density must be independent of the permutation of indistinguishable particles or 1 (1,2) 2 = (2,1) 2. 

For non-localized indistinguishable particles the eigenfunctions are generalizations of the linear combinations tps and iPa- For particles with integral... 

For indistinguishable particles that obey Pauli s priciple, the wave function (g) takes the form... 

For indistinguishable particles, this has to by symmetrized appropriately. The 4He nucleus is a boson, and hence the total wave function does not change sign when interchanging the nuclei. One obtains for the properly symmetrized scattering amplitude... 

Here x) stands for the positional coordinates of all the particles in the system, E is the energy of the system, and 77 is the Hamiltonian operator. Since a symmetry operator merely rearranges indistinguishable particles so as to leave the system in an indistinguishable configuration, the Hamiltonian is invariant under any spatial symmetry operator R. Let tpi denote a set of eigenfunctions of H so that... 

Now, at sufficiently high temperatures, for N non-interacting identical and indistinguishable particles (e.g., an ideal gas), the partition function can be written in the form... 

Remembering that we count configurations for indistinguishable particles by dividing the number for distinguishable particles by N, we get... 

Quantum mechanics clearly denies the possibility of distinguishing between particles in translational motion.16 For the ideal gas entropy, we must therefore use Eq. (34) for indistinguishable particles ... 

Give an example of a system containing distinguishable particles and one containing indistinguishable particles. 

We now consider the case of indistinguishable particles, treating explicitly a system of Fermions. It is convenient to write the wave function for a system of free Fermions in the form,... 

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12] ... 

Boltzmann distribution — The Boltzmann distribution describes the number N, of indistinguishable particles that have energy , after N of them have been independently and identically distributed among a set of states i. The probability density function is... 

Fermi-Dirac statistics — Fermi-Dirac statistics are a consequence of the extension of the application of Pauli s exclusion principle, which states that no two electrons in an atom can be in the same quantum state, to an ensemble of electrons, i.e., that no two could have the same set of quantum numbers. Mathematically, in a set of indistinguishable particles, which occupy quantum states following the Pauli exclusion principle, the probability of occupancy for a state of energy E at thermal equilibrium is given by f(E) = —(A)—, where E is the... 

Since the TDKS equations (67-72) reproduce the exact nuclear densities, Eq. (76) yields the exact classical trajectory whenever species A contains only one nucleus. When species A contains more than one nucleus we have a system of indistinguishable particles and then, strictly speaking, the trajectories of single nuclei cannot be told apart Only the total density (R, t) and hence the center-of-mass trajectory of species A can be measured. In this case, trajectories of single nuclei can be defined by Eq. (76) within some effective single-particle theory. TDKS theory is particularly suitable for this purpose since the TDKS partial densities lead to the exact total density n. ... 

There is a great conceptual advantage in abandoning the missing information 1 P , 2) in favour of the fluctuation A N, 2) to determine the localization of electrons. Lennard-Jones (1952) pointed out that the extent to which a set of indistinguishable particles is spatially localized is determined by the system s pair density, the same distribution function which determines the... 

It should be noted that the exchange of electrons is not a physical phenomenon but a correction to any initial description which wrongly assumes that the electrons can be distinguished from each other. Since indistinguishable particles by definition cannot be labelled, electron exchange in wave-functions should be regarded as an exchange of the electron coordinates (ref. 10). 

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