These six dimensions are here focused on and they rather naturally divide into three pairs, two involving time, two involving space, and two recognizing the conti-nuity/discontinuity dichotomy. For ease in visualization, the pairs can be presented as opposite faces of a cube (Figure 1) on which ideal gas theory requires only half the faces, one face from each pair, the three meeting at one vertex of the cube. [Pg.96]

To model the two-particle system mathematically we need to find a mathematical projective space whose basis corresponds to the list of six states. We want more than just dimensions to match we want the physical representations on the individual particle phase spaces to combine naturally to give the physical representations on the combined phase space. The space that works is [Pg.341]

The notion of an abstiact volume in the space of the coordinates of a Hamiltonian (i.e., position and momentum coordinates) is formalized by the introduction of phase space, of which there are at least two kinds. One kind should be regarded as a six-dimensional particle space. The six dimensions are those of the three spatial coordinates and the three momentum coordinates needed in the mechanical description of a single particle. At any instant in time, a particle is at one point in this six-dimensional phase space. If there were several particles in the system, each could be associated with a distinct point in this space at every instant in time. A six-dimensional box in phase space can be referred to as having a particular volume in that space. Then, the postulate of equal probabilities is a statement that the probability that the phase space point represents a single particle is the same in any one among all like-sized boxes. [Pg.345]

The imaginary character of the waves in the wave particle model becomes clearer when the theory of two particles which repel each other, like two electrons, is considered. It is then found necessary to consider a set of waves in an imaginary space of six dimensions. Such waves are certainly imaginary, so that there is little doubt that if there is anything at all like reality in the wave particle model, it is the particles, and not the waves. [Pg.67]

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6N dimensions three components of p and the three components of q for each atom. If we use the symbol r to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of T and is often written as A(r). As the system evolves in time then T will change and so will A(r). [Pg.59]

Let us cortsider a collection of N iderrtical qrrantum objects each with three degrees of freedom (e.g. one punctual gas molectrle). To define the movemerrt of these objects, we must first define the three spatial coordinates and the three qrrantities of movement (or three velocity componerrls), i.e. a total of six coordinates in a space with six dimensions. In this space, called the quarrtum space, an object is defined by a poirrt. For N objects, we rrse an hyperspace with 6N dimensions, called the Gibbs space of phase. In this hyperspace, it is a system state which is represented by a poirrt Each ensemble defined constitutes a complexion of the system. [Pg.98]

The three basis vectors (a, b, c) and all derived vectors (q) represent translations in the lattice. They translate the unit cell, including every atom and/or molecule located inside the unit cell, in three dimensions, thus filling the entire space of a crystal. The unit cell can be completely described by specifying a total of six scalar quantities, which are called the unit cell dimensions or lattice parameters. These are (see also Figure 1.4) [Pg.7]

It may be, also, that orbits that are not circular would give better values than circular orbits. Computations of the frequences on this basis present formidable difficulties. The fact, however, that the two quantum and three quantum orbits lie not in a plane, but in space of three dimensions may explain the appearance of three critical absorption wave-lengths in the L series, and six critical absorption wave-lengths in the M series, etc. [Pg.7]

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