Creation and annihilation

214 hydration can be also obtained from the reverse process, in which solute-solvent interactions are turned off. This corresponds to moving the solute from the aqueous solution to the gas phase. Then the calculated quantity is the negative of 214hydration- If the same order parameter, A, is used for the forward and the reverse transformations, the changes in the free energy with A should be reversible, and, consequently, the sum of the calculated free energies differences should be zero. This is shown in Fig. 2.7. Discrepancies between the forward and the reverse 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of 

As before, we can perform reverse simulations. Instead of annihilating the solute, we can create it by turning on the perturbation part of the Hamiltonian. The resulting free energy differences are connected through the relation Z A reation — creation = Annihilation - annihilation- Comparison of this creation scheme with the transformation described by the horizonal arrow reveals two important differences. First, the vertical transformations require two sets of simulations instead of one, although one of them involves only solute in the gas phase and, is, therefore, much less computationally intensive. Second, the two methods differ in their description of the solute in the reference state. In both cases the solute does not interact with the solvent. For the vertical transformations, however, all interactions between atoms forming the solute vanish, whereas in the horizontal transformation, the molecule remains intact. 

---

Monte Carlo simulations require less computer time to execute each iteration than a molecular dynamics simulation on the same system. However, Monte Carlo simulations are more limited in that they cannot yield time-dependent information, such as diffusion coefficients or viscosity. As with molecular dynamics, constant NVT simulations are most common, but constant NPT simulations are possible using a coordinate scaling step. Calculations that are not constant N can be constructed by including probabilities for particle creation and annihilation. These calculations present technical difficulties due to having very low probabilities for creation and annihilation, thus requiring very large collections of molecules and long simulation times. 

The somewhat awkward antisymmetiizing operator necessarily in first quantization is replaced by formal rules for manipulating creation and annihilation operators. 

In the context of fra/u-polyacetylene cjia and c are, respectively, the creation and annihilation operators of an electron with spin projection a in the n-orbital of the nth carbon atom (n= l,...,N) that is perpendicular to the chain plane (see Fig. 3-3). Furthermore, u is the displacement along the chain of the nth CH unit from its position in the undimerized chain, P denotes the momentum of this unit, and M is its mass. 

Creation and Annihilation Operators.—In the last section there was a hint that the theory could handle problems in which populations do not remain constant. Thus < , < f>s 2 is the probability density in 3A -coordinate space that the occupation numbers are , and the general symmetrical state, Eq. (8-101), is one in which there is a distribution of probabilities over different sets of occupation numbers the sum over sets could easily be extended to include sets corresponding to different total populations N. 

When dealing with systems described by antisymmetrical states, the creation and annihilation operators are defined in such a way that the occupation numbers can never be greater than unity. Thus we have a creation operator af defined by... 

At the end of Section 8.16 we mentioned that the Fock representation avoids the use of multiple integrations of coordinate space when dealing with the many-body problem. We can see here, however, that the new method runs into complications of its own To handle the immense bookkeeping problems involved in the multiple -integrals and the ordered products of creation and annihilation operators, special diagram techniques have been developed. These are discussed in Chapter 11, Quantum Electrodynamics. The reader who wishes to study further the many applications of these techniques to problems of quantum statistics will find an ample list of references in a review article by D. ter Haar, Reports on Progress in Physics, 24,1961, Inst, of Phys. and Phys. Soc. (London). 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

We shall denote the creation and annihilation operators for a negaton of momentum p energy Ep = Vp2 + m2 and polarizations by 6 (p,s) and 6(p,s) respectively. In the following, by the polarization we shall always mean the eigenvalue of the operator O-n, where O is the Stech polarization operator and n some fixed unit vector. We denote the creation and annihilation operators for a positon (the antiparticle) of momentum q energy = Vq2 + m2, polarization t, by d (q,t) and... 

This confirms our interpretation of the operators 6,6 and d,d as creation and annihilation operators for particles of definite momentum and energy. Similar consideration can be made for the angular momentum operator. The total electric charge operator is defined as... 

Definition of Normal Product.—Given a product of free field creation and annihilation operators U,X,- -, FF, we define the operator N as... 

We now introduce a notation involving a normal product with one or more contracted pairs of factors. If U, V, denote a set of free-field creation and annihilation operators, we define the mixed product by... 

Normal product of free-field creation and annihilation operators, 606 Normal product operator, 545 operating on Fermion operators, 545 N-particle probability distribution function, 42... 

When the operators A and B in Eq. (2.7) are sin q)le creation and annihilation operators the resulting propagator is called electron (nopagator or one-particte Green s function, and = -t-1. Collecting all these creation and amiihi-lation operators in a row vector a, the electron propagator can be expressed as. 

Here the indices a and b stand for the valence orbitals on the two atoms as before, n is a number operator, c+ and c are creation and annihilation operators, and cr is the spin index. The third and fourth terms in the parentheses effect electron exchange and are responsible for the bonding between the two atoms, while the last two terms stand for the Coulomb repulsion between electrons of opposite spin on the same orbital. As is common in tight binding theory, we assume that the two orbitals a and b are orthogonal we shall correct for this neglect of overlap later. The coupling Vab can be taken as real we set Vab = P < 0. 

It is obvious that during deformation of the sample due to mechanical loading the creation and annihilation defects will also take place. Similar to preceding experiments in this case the value of deformation would determine the concentration of defects. However, in case of mechanical loading the defects will be evenly spread over the whole volume of samples, whereas in case of silver oxidation they remain localized only in the surface-adjacent layers. Therefore, emission of oxygen atoms under conditions of mechanical deformation of samples in oxygen atmosphere has low probability due to intensive annihilation of defects in surface-adjacent layers. Special experiments confirmed this conclusion. 

We introduce the eigenstates and eigenvalues for the creation and annihilation operators (coherent states86) ... 

The creation and annihilation operators defined in Table II obey the following eigenvalues equations ... 

Here /(R) and pflX) denote the shift and generalized momentum for the molecular vibration of the low frequency a>9 and reduced mass m, at the Rth site of the adsorbate lattice bi+(K) and K) are creation and annihilation operators for the collectivized mode of the adsorbate that is characterized by the squared frequency /2(K) = ml + d>, a,(K)/m , with O / iat(K) representing the Fourier component of the force constant function /jat(R). Shifts i//(R) for all molecules are assumed to be oriented in the same arbitrary direction specified by the unit vector e they are related to the corresponding normal coordinates, ue (K), and secondary quantization operators ... 

To invoke the perturbation theory for a small anharmonic coupling coefficient, we use the Wick theorem for the coupling of the creation and annihilation operators of low-frequency modes in expression (A3.19). Retaining the terms of the orders y and y2, we are led to the following expressions for the shift AQ and the width 2T of the high-frequency vibration spectral line 184... 

Particles whose creation and annihilation operators satisfy these relationships are called fermions. It is found that [119] these commutation relations lead to wave functions in space that are antisymmetric. 

The definition of a grand canonical density operator requires that the system Hamiltonian be expressed in terms of creation and annihilation operators. 

We first find the Green function Go for H0 and then obtain pertur-batively the wave functional for the total Hamiltonian. In fact, each mode of the quadratic part Ho can be solved exactly in terms the time-dependent creation and annihilation operators (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003)... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

Here and hereafter the tilde conjugation rule, (cA) = c A, will be used. The time-dependent creation and annihilation operators for the fictitious Hamiltonian (33) are obtained by applying the tilde conjugation rule to Eq. (24)... 

---