Identical particles

Since indistinguishability is a necessary property of exact wavefiinctions, it is reasonable to impose the same constraint on the approximate wavefiinctions ( ) fonned from products of single-particle solutions. Flowever, if two or more of the Xj the product are different, it is necessary to fonn linear combinations if the condition P. i = vj/ is to be met. An additional consequence of indistinguishability is that the h. operators corresponding to identical particles must also be identical and therefore have precisely the same eigenfiinctions. It should be noted that there is nothing mysterious about this perfectly reasonable restriction placed on the mathematical fonn of wavefiinctions. 

The wavefimction of a system must be antisynnnetric with respect to interchange of the coordinates of identical particles y and 8 if they are fermions, and symmetric with respect to interchange of y and 5 if they are bosons. 

It would appear that identical particle pemuitation groups are not of help in providing distinguishing syimnetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very usefiil restrictions on the way we can build up the complete molecular wavefiinction from basis fiinctions. Molecular wavefiinctions are usually built up from basis fiinctions that are products of electronic and nuclear parts. Each of these parts is fiirther built up from products of separate uncoupled coordinate (or orbital) and spin basis fiinctions. Wlien we combine these separate fiinctions, the final overall product states must confonn to the pemuitation syimnetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis fiinctions. 

This chapter deals with qnantal and semiclassical theory of heavy-particle and electron-atom collisions. Basic and nsefnl fonnnlae for cross sections, rates and associated quantities are presented. A consistent description of the mathematics and vocabnlary of scattering is provided. Topics covered inclnde collisions, rate coefficients, qnantal transition rates and cross sections. Bom cross sections, qnantal potential scattering, collisions between identical particles, qnantal inelastic heavy-particle collisions, electron-atom inelastic collisions, semiclassical inelastic scattering and long-range interactions. 

The identical colliding particles, each with spin s, are in a resolved state with total spinin the range (0 2s). The spatial wavefiinction with respect to particle interchange satisfies = (—1 Wavefunctions for identical particles with even or odd total spin. S are therefore symmetric (S) or antisynnnetric (A) with respect to particle... 

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... 

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

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

Q is given by Equation (6.4) for a system of identical particles. We shall ignore any normalisation constants in our treatment here to enable us to concentrate on the basics, and so it does not matter whether the system consists of identical or distinguishable particles. We also replace the Hamiltonian by the energy, E. The internal energy is obtained via Equation (6.20) ... 

The characteristics of a powder that determine its apparent density are rather complex, but some general statements with respect to powder variables and their effect on the density of the loose powder can be made. (/) The smaller the particles, the greater the specific surface area of the powder. This increases the friction between the particles and lowers the apparent density but enhances the rate of sintering. (2) Powders having very irregular-shaped particles are usually characterized by a lower apparent density than more regular or spherical ones. This is shown in Table 4 for three different types of copper powders having identical particle size distribution but different particle shape. These data illustrate the decisive influence of particle shape on apparent density. (J) In any mixture of coarse and fine powder particles, an optimum mixture results in maximum apparent density. This optimum mixture is reached when the fine particles fill the voids between the coarse particles. 

Particulate systems composed of identical particles are extremely rare. It is therefore usefiil to represent a polydispersion of particles as sets of successive size intervals, containing information on the number of particle, length, surface area, or mass. The entire size range, which can span up to several orders of magnitude, can be covered with a relatively small number of intervals. This data set is usually tabulated and transformed into a graphical representation. 

Here N is the number of identical particles of solid, and y is the surface area to volume... 

Secondly, y must also be antisymmetric, meaning that it must change sign when two identical particles are interchanged. For a simple function, antisymmetry means that the following relation holds ... 

A population of identical particles residing on the sites of C, each having unit mass and moving from site to site with the same average speed. 

In general whether we are discussing symmetrical or antisymmetrical states, the numbers nx are usually termed the occupation numbers, and if we are given a complete spectrum of one-particle states, indicated by A, then the set of occupation numbers assigned to the values of A, specifies the state of the system of JY identical particles just as well as the assignment of a A-value to each particle. Thus we may use the notation just as well as A to specify the state, and so write... 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

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