Particle Number Operators

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

As a logical extension, particle number operators are next defined such that N = 2j Nj = bpj. The commutation rules require that... 

Incorporation of particle number operators defines the density operator of the grand canonical ensemble as10... 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

In analytical investigations it is often desirable to leave the particle number free and consider operators that fix only the parity, but in applications to electronic structure theory one deals with fixed particle number and one may restrict A to have a definite action on the particle number N, so that A+A is particle conserving. There are then two cases for the one-body operator A consideration of A = with undetermined coefficients gives rise to the... 

Q < I — y, here the a,- form a (finite-dimensional) basis of annihilation operators, y is the 1-RDM, and I is the identity matrix of the appropriate size. If the density matrices are known to be real symmetric then the may be assumed real, otherwise they should be assumed complex. For fixed particle number... 

These operators are particle-number conserving, that is, action of any excitation operator on an -electron wavefunction (with n arbitrary) leads again to an -electron wavefunction (or deletes it). 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

A proof of the bounds for the occupation numbers will be given in Section II. F. Expectation values of (particle-number conserving) operators are easily expressed in terms of the density matrices. For example, for the energy... 

Particle Number Concentration and Size Distribution. The development of aerosol science to its present state has been directly tied to the available instrumentation. The introduction of the Aitken condensation nuclei counter in the late 1800s marks the beginning of aerosol science by the ability to measure number concentrations (4). Theoretical descriptions of the change in the number concentration by coagulation quickly followed. Particle size distribution measurements became possible when the cascade impactor was developed, and its development allowed the validation of predictions that could not previously be tested. The cascade impactor was originally introduced by May (5, 6), and a wide variety of impactors have since been used. Operated at atmospheric pressure and with jets fabricated by conventional machining, most impactors can only classify particles larger... 

The sums of products of CFP obey some additional relations. In fact, the operators of particle number N, of orbital L and spin S momenta are expressed in terms of tensorial products of electron creation and annihilation operators - relationships (14.17), (14.15) and (14.16), respectively. We can expand the submatrix elements of such tensorial products using (5.16) and then go over, using (15.21) and (15.15), to the CFP. On the other hand, these submatrix elements are given by the quantum numbers of the states of the lN configuration. Then we obtain... 

Here t may represent Is or j, whereas z-projection of the quasispin operator is determined by the difference between the particle number operator and the hole number operator in the pairing state (a, / )... 

According to (14.17), the z-projection of the quasispin operator is related to the operator of the particle number in the shell... 

All these operators commute with the particle number operator in the pairing state... 

In analogy with (18.18H18.20) we conclude that the operator t)1 annihilates an electron in the first subshell and creates it in the second (and the operator T l vice versa), so that the total number of electrons in the subshells will remain unchanged (N = Ni + N2 = N[ + Nf). The operator has the form 1( 2 — iVi)/2, where Ni and N2 are the particle number operators for the first and second subshells, respectively. 

Using the operators AaP and C xxx2) we may introduce the particle number operators... 

Taking the diagonal elements we may obtain the connection between the particles number operators. For this purpose let us now derive some relations that are important for further consideration. First, we introduce the operator of the density of pairs pab by... 

Comparison between Experimental Results and Model Predictions. As will be shown later, the important parameter e which represents the mechanism of radical entry into the micelles and particles in the water phase does not affect the steady-state values of monomer conversion and the number of polymer particles when the first reactor is operated at comparatively shorter or longer mean residence times, while the transient kinetic behavior at the start of polymerization or the steady-state values of monomer conversion and particle number at intermediate value of mean residence time depend on the form of e. However, the form of e influences significantly the polydispersity index M /M of the polymers produced at steady state. It is, therefore, preferable to determine the form of e from the examination of the experimental values of Mw/Mn The effect of radical capture mechanism on the value of M /M can be predicted theoretically as shown in Table II, provided that the polymers produced by chain transfer reaction to monomer molecules can be neglected compared to those formed by mutual termination. Degraff and Poehlein(2) reported that experimental values of M /M were between 2 and 3, rather close to 2, as shown in Figure 2. Comparing their experimental values with the theoretical values in Table II, it seems that the radicals in the water phase are not captured in proportion to the surface area of a micelle and a particle but are captured rather in proportion to the first power of the diameters of a micelle and a particle or less than the first power. This indicates that the form of e would be Case A or Case B. In this discussion, therefore, Case A will be used as the form of e for simplicity. 

Figure 5 represents a typical example of the variation of the number of polymer particles with mean residence time 0. The solid line shows the theoretical value predicted by the Nomura and Harada model with e= 1.28x 10 . The dotted line is that predicted by the Gershberg model(or the Nomura and Harada model with Case C for ), where Eq. (23) was used instead of Eq.(16) for Ap. The value of Nt produced at longer mean residence time differs, therefore, by a factor of T(5/3) between the solid and dotted lines in Figure 5. From the comparison between the experimental and theoretical results shown in Figure 5, it is confirmed that the steady state particle number can be maximized by operating the first stage reactor at a certain low value of mean residence time max which is considerably lower than that in the succeeding reactors. This is so-called "pre-reactor principle". It is, therefore, desirable to operate the first reactor at such mean residence time as producing something like a maximum number of polymer particles in order to increase the rate of polymerization in the succeeding reactors. This will result in a decrease in the number of necessary reactors and hence, in the capital cost.

Figure 9. Variation of particle number with reaction time in batch operation (calculation conditions S = 6.25 g/L H20 I0 = 1.25 g/L H20 Scue = 0.50 g/L...

To calculate the current we find the time evolution of the particle number operator Ns = dfada due to tunneling from the left (i = L) or right (i = R) contact. 

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