Particle state vector

Consider another example. Suppose we are interested in following the total number of cells in a population of bacteria that are multiplying by binary division. Assume that the cells do not divide until after a certain age has been reached. In this case, it becomes essential to define cell age as the particle state although it is not of explicit interest originally. Thus, the identification of age as the particle state in this case was dictated by its influence on the birth rate. 

In general we may conclude that the choice of the particle state is determined by the variables needed to specify (i) the rate of change of those of direct interest to the application, and (ii) the birth and death processes. The particle state may generally be characterized by a finite dimensional vector, although in some cases it may not be sufficient. For example, in a diffusive mass transfer process of a solute from a population of liquid droplets to a surrounding continuous phase (e.g., liquid-liquid extraction) one would require a concentration profile in the droplet to calculate the transport rate. In this case, the concentration profile would be an infinite dimensional vector. Although mathematical machinery is conceivable for dealing with infinite dimensional state vectors, it is often possible to use finite dimensional approximations such as a truncated Fourier series expansion. Thus it is adequate for most practical applications to assume that the particle state can be described by a finite dimensional vector.  

The dependence of particle processes (i) and (ii) on the current particle alone of course implies that we are neglecting memory elfects. In other words, the choice of the particle state must be suitably made to support this assumption. 

The finite dimensional state vector can accommodate the description of particles with considerable internal structure. For example, consider a cell with m compartments. Each compartment may be considered as well mixed containing a total of n quantities. Suppose now the cell changes its state by interaction between its compartments and with the environment. The particle state can be described by a partitioned vector [x, X2. x ] where x represents the vector of n components in the ith compartment. 

It is also interesting to observe that a finite dimensional vector is adequate to describe particles with spatial, internal morphology where several discrete components may be located anywhere within the particle relative to, say, the centroid of the particle. In this case, the elements in the partitioned vector above may be interpreted as position vectors of such components. 

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The many-particle state vector used to be constructed from simpler units— the electronic and nuclear wave functions—and the many-electron wave function consists of the one-electron wave functions, the spinorbitals general expression for the many-electron wave function in terms of the products of spinorbitals is... 

Although it is possible to address this situation for the general particle state vector including internal and external coordinates, we shall take the route of establishing the results for the one-dimensional case and proceed to infer the generalization for the vectorial case without elaborate derivation. 

The particle state vector and its rate of change have now been identified. 

In the above, is an A-particle state vector satisfying the proper symmetry requirements with respect to particle permutation and Ej is the energy of state... 

Example, Free Particle. In this case the hamiltonian is H = Pa/2m and the state vector t> satisfies the equation... 

Symmetrizationof N-Particle States.—Let Pbe an arbitrary permutation13 among the eigenvalues occurring in the set A. By this we mean the following start with some arbitrary set of vectors A1>, A2>,- , AW>, one for each particle in its respective Hilbert space St then carry out an arbitrary permutation of the values... 

Exactly the same set of eigenvalues appears in this as before, but in a different order among the particles. There are Nl permutations, and if we add all the 2V vectors like that in Eq. (8-92) we shall have a vector that is completely symmetrical in all the particles.14 Thus we may define the symmetrical -particle state as... 

One speaks of Eqs. (9-144) and (9-145) as a representation of the operators a and o satisfying the commutation rules (9-128), (9-124), and (9-125). The states 1, - , ) = 0,1,2,- are the basis vectors spanning the Hilbert space in which the operators a and oj operate. The representation (9-144) and (9-145) is characterized by the fact that a no-particle state 0> exists which is annihilated by a, furthermore this representation is irreducible since in this representation a(a ) operating upon an n-particle state, results in an n — 1 ( + 1) particle state so that there are no invariant subspaces. Besides the above representation there exist other inequivalent irreducible representations of the commutation rules for which neither a no-particle state nor a number operator exists.8... 

The covariant amplitudes describing one-, two-, etc., particle systems can be defined in terms of the Heisenberg field operators (x) as follows oonsider a one-particle system described by the state vector IT) Since it describes a one-particle system, it has the property that... 

The commutations (9-416)-( 9-419) guarantee that the state vectors are antisymmetric and that the occupation number operators N (p,s) and N+(q,t) can have only eigenvalues 0 and 1 (which is, of course, what is meant by the statement that particles and antiparticles separately obey Fermi-Dirac etatistios). In fact one readily verifies that... 

In concluding this section we briefly establish the connection between the Dirac theory for a single isolated free particle described in the previous section and the present formalism. If T> is the state vector describing a one-particle state, iV T> = 1 T> consider the amplitude... 

This is verified by applying U(a,A) to the left and U(a,A) l to the right of both sides of Eqs. (11-142) and (11-97), mid making use of the transformation properties of (11-184) and (11-185) of the fields. Equations (11-241) and (11-242) were to be expected. They guarantee that four-vector, respectively. We are now in a position to discuss the transformation properties of the one-particle states. Consider the one-negaton state, p,s,—e>. Upon taking the adjoint of Eq. (11-241) and multiplying by ya we obtain... 

As illustrated in Fig. 2.8 of Section II, the general reflectivity lineshape shows (1) a sharp rise of the bulk 0-0 reflectivity (Section II.B.C) at E00, corresponding to the b coulombic exciton with a wave vector perpendicular to the (001) face (2) a dip, corresponding to the fission in the surface of a bulk polariton into one 46 -cm 1 phonon and one b exciton at E°° + 46 cm"1 (3) two vibrons E200 and E1 00 immersed in their two-particle-state continua with sharp low-energy thresholds. On this relatively smooth bulk reflectivity lineshape are superimposed sharp and narrow surface 0-0 transition structures whose observation requires the following ... 

In order to use the perturbation theory it is necessary that the state vectors in the matrix element Eq. (8) belong to the spectrum of the unperturbed Hamiltonian H0 only. However, this is usually not so, since, in p decay, the initial particles are not the same as the final products of the reaction the initial molecule containing the radioactive atom transforms into a different molecule besides, the ft electron and the neutrino appear. One of the ways to describe the initial and final states using only the H0 Hamiltonian is to use the isotopic spin formalism for both the nucleons and the leptons (/ electron and neutrino). In the appendix (Section V) we present the wave functions of the initial and the final states together with the necessary transformations, which one can use to factorize the initial matrix element Eq. (8) into the intranuclear and the molecular parts. Here we briefly discuss only the approximations necessary for performing such a factorization. 

According to the first postulate, the state of a physical system is completely described by a state function fifiq, /) or ket T1), which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q, q2, , so that the state function may also be written as q, q2, , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector XV is a function of x and t Tfix, /). For a particle or system in three dimensions, the components of q are x, y, z and I1 is a function of the position vector r and t Tfi r, /). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable. 

For a quantum particle the system is described by a state vector ip) which transforms into [ ip1) after symmetry translation, as described by the operator T(n), in the vector space of physical states ... 

Fig.2-33 presents results for 2Z = 8, i.e., 4 black and 4 white particles. Two initial state vectors S(0) were considered, given at the top of the figure. On the left-hand side, it is assumed that initially container A contains 2 black particles. On the right-hand side, there are 50% chances that container A will contain one black particle and 50% to contain four black particles. The results indicate that after ten steps, the system attains a steady state, independent of the initial conditions, where S(10) = [0.014, 0.229, 0.514, 0.229, 0.014]. Such a process is known as... 

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