Operator number

a algebra makes it possible to simplify the operator forms of the various dynamics diagnostics discussed in Sections 9.1.4 and 9.1.8 the expectation value of a number operator (e.g., (a-a ), which tells us how many quanta or how much energy is in mode i as a function of time), the resonance energy term, O, the transfer rate term, Cl - 12, and Qi and P,. 

The equation of motion of the expectation value of any operator is given by the expectation value of the commutator of that operator with H, e.g., 

According to Eq. (9.1.95), (ci — is pure imaginary, and thus (aiai) = = y (aia2 ai42) 

Note also that the expectation value of the energy in mode 1 is given by 

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Matrices obey an algebra of their own that resembles the algebra of ordinary numbers in some respects and not in others. The elements of a matrix may be numbers, operators, or functions. We shall deal primarily with matrices of numbers in this chapter, but matrices of operators and functions will be important later. 

The occupation number operator for particles of momentum k can be defined as = k 1o ok, and the total number of particles operator 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 number operators N+,N. for the number of positively and negatively charged particles are then given by the following expressions... 

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... 

The number operators, when expressed in terms of the configuration space operators, assume the following form... 

For the physical interpretation of the theory it is convenient to he able to express the hamiltonian in the form Jkc (k)e(k), with c (k),c(k) satisfying 3-function commutation rales so that number operators can be introduced into the theory. The form (9-619) for the hamiltonian suggests that we define the operators ... 

Next we establish the connection of the previous formalism with the Fo< space description of photons. From the interpretation of a>(k) as the number operator for photons of momentum k polarization A, and of cA(k) and cA(k) as destruction and creation operators for... 

Observables, rate of change of, 477 Occupation number operator, 54 for particles of momentum k, 505 One-antiparticle state, 540 One-dimensional antiferromagnetic Kronig-Penney problem, 747 One-negaton states, 659 One-particle processes Green s function for computing amplitudes under vacuum conditions, 619... 

The new delightful book by Greenstein and Zajonc(9) contains several examples where the outcome of experiments was not what physicists expected. Careful analysis of the Schrddinger equation revealed what the intuitive argument had overlooked and showed that QM is correct. In Chapter 2, Photons , they tell the story that Einstein got the Nobel Prize in 1922 for the explaining the photoelectric effect with the concept of particle-like photons. In 1969 Crisp and Jaynes(IO) and Lamb and Scullyfl I) showed that the quantum nature of the photoelectric effect can be explained with a classical radiation field and a quantum description for the atom. Photons do exist, but they only show up when the EM field is in a state that is an eigenstate of the number operator, and they do not reveal themselves in the photoelectric effect. 

Here, n denotes a number operator, a creation operator, c an annihilation operator, and 8 an energy. The first term with the label a describes the reactant, the second term describes the metal electrons, which are labeled by their quasi-momentum k, and the last term accounts for electron exchange between the reactant and the metal Vk is the corresponding matrix element. This part of the Hamiltonian is similar to that of the Anderson-Newns model [Anderson, 1961 Newns, 1969], but without spin. The neglect of spin is common in theories of outer sphere reactions, and is justified by the comparatively weak electronic interaction, which ensures that only one electron is transferred at a time. We shall consider spin when we treat catalytic reactions. 

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. 

To determine the degeneracy of the energy levels or, equivalently, of the eigenvalues of the number operator N, we must first obtain the eigenvectors 0z) for the ground state. These eigenvectors are determined by equation (4.29). When equation (4.18a) is substituted for a, equation (4.29) takes the form... 

We next show that if the eigenvalue n of the number operator N is nondegenerate, then the eigenvalue n -h 1 is also non-degenerate. We begin with the assumption that there is only one eigenvector with the property that... 

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... 

Notice that the thermal average can be given by taking the vacuum average 0,0) of a thermal non-tilde variables. For instance for the particular case of the bosonic number operator, n = a a, the thermal distribution, as in Eq.(7) reads,... 

Another interesting website, http //matti.usu.edu/nlvm/nav/index.html, has a game, Circle 0, for practicing one-digit integer arithmetic. Click on Virtual Library. Then, click on the 9-12 box in the Numbers Operations row. Click on Circle 0, and play the integer game. 

The website http //matti.usu.edu/nlvm/nav/index.html has some helpful interactive exercises that explore the concept of fractional ordering and addition. Upon entering the website, click on Virtual Library. Click on the 6-8 box in the row entitled Numbers Operations. There are several activities to choose from that deal with fractions, including Fraction Pieces, Fractions—Adding, Fractions—Comparing, and Fractions—Equivalent. 

REQUESTED BY SHAFFORD INSTRUMENT NUMBER OPERATOR RUN DATE RUN TIME NUMBER OF DATA POINTS 34 AFK 22-FEB-83 13 04 47 600 SAMPLE VOLUME, UL DATA TIME, LOH DATA TIME, HIGH BASELINE TINE, LOM BASELINE TINE, HIGH BASELINE SLOPE 100 21.00 31.00 20.00 37.00 0.625000... 

It is easy to show that we have two invariants of transformations, namely number operator of fermions... 

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