Particle state space defined

We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

We postulate that there exists an average number density function defined on the particle state space,... 

Since velocities through both internal and external coordinate spaces are defined, it is now possible to identify particle (number) fluxes, i.e., the number of particles flowing per unit time per unit area normal to the direction of the velocity. Thus / (x, r, t)R(x, r, Y, t) represents the particle flux through physical space and / (x, r, t)X x, r, Y, t) is the particle flux through internal coordinate space. Both fluxes are evaluated at time t and at the point (x, r) in particle state space. Indeed these fluxes are clearly important in the formulation of population balance equations. 

The equilibrium system A (with N particles) is possible to be identified with the (state) space it occupies, being defined by "one-particle" railing of cells ["one-particle (state) space], with the... 

The wave function is an irreducible entity completely defined by the Schrbdinger equation and this should be the cote of the message conveyed to students. It is not useful to introduce any hidden variables, not even Feynman paths. The wave function is an element of a well defined state space, which is neither a classical particle, nor a classical field. Its nature is fully and accurately defined by studying how it evolves and interacts and this is the only way that it can be completely and correctly understood. The evolution and interaction is accurately described by the Schrbdinger equation or the Heisenberg equation or the Feynman propagator or any other representation of the dynamical equation. 

In principle, one can extract from G(ti)) the complete series of the primary (one-hole, Ih) and excited (shake-up) states of the cation. In practice, one usually restricts the portion of shake-up space to be spanned to the 2h-lp (two-hole, one-particle) states defined by a single-electron transition, neglecting therefore excitations of higher rank (3h-2p, 4h-3p. ..) in the ionized system. In the so-called ADC[3] scheme (22), elertronic correlation effects in the reference ground state are included through third-order. In this scheme, multistate 2h-lp/2h-lp configuration interactions are also accounted for to first-order, whereas the couplings of the Ih and 2h-lp excitation manifolds are of second-order in electronic correlation. 

The relationship between iV-particle states, in which we include mixed states, represented by A -particle operators as defined in Eq. (2.1), and the space-spin density p(y) is not 1-1. Here and throughout the following development, y... 

First we define the linear map that produces the densities from N-particle states. It is a map from the space of A -particle Trace Class operators into the space of complex valued absolute integrable functions of space-spin variables... 

A partial differential equation is then developed for the number density of particles in the phase space (analogous to the classical Liouville equation that expresses the conservation of probability in the phase space of a mechanical system) (32>. In other words, if the particle states (i.e. points in the particle phase space) are regarded at any moment as a continuum filling a suitable portion of the phase space, flowing with a velocity field specified by the function u , then one may ask for the density of this fluid streaming through the phase space, i.e. the number density function n(z,t) of particles in the phase space defined as the number of particles in the system at time t with phase coordinates in the range z (dz/2). 

The classical problem is no less daunting The initial state is defined in terms of the 3N coordinates and 3N momenta of the N particles, which together define a 6iV-dimensional phase space. Each point in this phase space defines a specific microstate. Evolution of the system is described by the trajectory of the starting point through phase space. Not only is it beyond the capabilities of the most powerful computer to keep track of these fluctuations, but should only one of the starting parameters be defined incorrectly, the model fatally diverges very rapidly from the actual situation. 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

The state of a classical system can be completely described by specifying the positions and momenta of all particles. Space being three-dimensional, each particle has associated with it six coordinates - a system of N particles is thus characterized by 6N coordinates. The 6N-dimensional space defined by these coordinates is called the phase space of the system. At any instant in time, the system occupies one point in phase space... 

In order to generalize the HF equations to n-particle states, we perform a variational procedure. In this procedure, it is convenient to identify clearly and uniquely the particle space in which the two-particle wave functions are acting. This identification is considerably easier when the particle pairings are uniquely and distinctly defined, as is done in constructing the wave function or in partitioning the Hamiltonian. We can now obtain an eigenvalue-like... 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

Because H is dynamically dependent on spin and space variables, the expression in parentheses in the r.h.s. of Eq. (3) involving integration over the latter defines a spin operator. This is just the effective Hamiltonian of interest to us. By virtue of point (iii), when the integrations are to be performed for the H" term in the Hamiltonian, only the unit operator in A need to be retained. The resulting expression will thus have the form (Ap H"l ). If one takes into account that the space state 1 ) is a product (or a combination of products, see above) of localized, one-particle states, one can immediately see that upon integrating over the spatial variables r , n= 1,2,...,AI, the spatial parts of the individual spin-dependent terms will be replaced by the corresponding quantum mechanical averages. Thus, for the entire expression in Eq. (3) is none other than one of the matrix element of the standard NMR Hamiltonian, Wnmr, between two spin-product basis states,... 

We have seen that electrons are distributed into different atomic orbitals (Table 1.2). An orbital is a three-dimensional region around the nucleus where there is a high probability of finding an electron. But what does an orbital look like Mathematical calculations indicate that the 5 atomic orbital is a sphere with the nucleus at its center, and experimental evidence supports this theory. The Heisenberg uncertainty principle states that both the precise location and the momentum of an atomic particle cannot be simultaneously determined. This means that we can never say precisely where an electron is—we can only describe its probable location. Thus, when we say that an electron occupies a s atomic orbital, we mean that there is a greater than 90% probability that the electron is in the space defined by the sphere. 

These are elements of a Hilbert space Y which is the direct sum of four subspaces related to the four entries of the above defined column vector. The first component Qata, pt) is element of of the physical N fermion Hilbert speice (excluding the ground state due to Q) and can be understood as an approximate particle-hole type excitation. The second entry a OsQ is a bra state and thus an element of the dual space of the N particle space. This second entry can also be seen as an approximate excited state but the hole and particle role of the indices r and s has interchcmged. Each of the remaining two entries is the direct sum of tensor product states of bras or kets in the N — 1 and N + 1 particle Hilbert spaces, e. g. ( i J l e... 

The size of the chemical system is small. In this case we cannot apply continuous state models even as an approximation. (For the case of a finite particle number, a precise continuous model cannot be defined, since in the strict sense the notion of concentration can be used only for infinite systems.) Discrete state space, but deterministic, models are also out of question, since fluctuations cannot be neglected even in the zeroth approximation , because they are not superimposed upon the phenomenon, but they represent the phenomenon itself. 

Clearly, in the foregoing discussion, the change of particle state has been viewed as a deterministic process. It is conceivable, however, that in some situations the change could be occurring randomly in time. In other words the velocities just defined may be random processes in space and time. It will be of interest for us to address problems of this kind. For the present, however, we postpone discussion of this issue until later in this chapter. 

Next, it is necessary to define the average number of pairs of particles at each instant t with specified states. Accordingly, we define /2(x, r x, r, t) to represent the average number of distinct pairs of particles at time t per unit volumes in state space located about (x, r) and (x, r ), respectively. The source term for the rate of production of particles in volume (x, r) of state... 

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