Boson operator

Spectral Representation.—As an application of the invariance properties of quantum electrodynamics we shall now use the results obtained in the last section to deduce a representation of the vacuum expectation value of a product of two fermion operators and of two boson operators. The invariance of the theory under time inversion and more particularly the fact that... 

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

Considering a bosonic oscillator characterized by the Hamiltonian H = wata, the state 0(/3)) can be constructed if for each bosonic operator a, another bosonic operator, say a, is introduced such that the tilde and non-tilde variables obey the following algebra... 

The operators of physical interest can be expanded as a power series in the bilinear products b bp of the boson operators.4 Special cases include the Hamiltonian H,... 

The (bilinear) expansion in the products of boson operators b]j serves to ensure the correspondance with quantum mechanics. To see this explicitly, say A, B, and C are operators familiar from wave mechanics and let A,B, and C be their corresponding matrix representations. If [A, 3] = C, then [A, B] = C. Now define... 

In the previous chapter we discussed the usual realization of many-body quantum mechanics in terms of differential operators (Schrodinger picture). As in the case of the two-body problem, it is possible to formulate many-body quantum mechanics in terms of algebraic operators. This is done by introducing, for each coordinate, r2,... and momentum p, p2,..., boson creation and annihilation operators, b ia, bia. The index i runs over the number of relevant degrees of freedom, while the index a runs from 1 to n + 1, where n is the number of space dimensions (see note 3 of Chapter 2). The boson operators satisfy the usual commutation relations, which are for i j,... 

All operators of the theory are then expanded in terms of the boson operators, b a, bia. This expansion can be written in a more compact form by noting that... 

As in the previous case of two bonds, discussed in Chapter 4, we introduce boson operators for each bond... 

For these vibrations, the quantization scheme of Section 4.2 can be carried over without any modification (Iachello and Oss, 1991a). The potentials in each stretching coordinate 5 are in an anharmonic force field approximation represented by Morse potentials. The boson operators (Ot,xt) correspond to the quantization of anharmonic Morse oscillators, with classical Hamiltonian... 

Any algebraic operator, written in terms of the boson operators a, T of Chapter 2, can be converted into a classical operator, written in terms of the variables % (or p, q). There are several (equivalent) ways of deriving the classical limit of boson operators. We describe here that due to van Roosmalen (1982) and van Roosmalen and Dieperink (1982). A classical limit corresponding to any algebraic operator is obtained by considering its expectation value in the state... 

The expectation value of H in the coherent state (7.17) can be evaluated explicitly for any Hamiltonian. However, an even simpler construction of Hd (valid to leading order in N) can be done (Cooper and Levine, 1989) by introducing intensive boson operators (Gilmore, 1981). In view of its simplicity, we report here this construction. If one divides the individual creation and annihilation operators by the square root of the total number of bosons, the relevant commutation relations become... 

We also note that had we not used the method of intensive boson operators but rather evaluated Eq. (7.18) exactly, we would have obtained (van Roosmalen, 1982)... 

This is a function of the complex variables E,j, ,2- Once more, by making use of intensive boson operators, one can easily obtain Hd(pi,q, p2, q2). The potential functions can then be defined as... 

The expectation value of H can again be simply evaluated if one introduces intensive boson operators such that... 

For three-dimensional problems, there are in total four boson operators. There must be three more boson operators in addition to Eq. (7.136). They are given by (Leviatan and Kirson, 1988)... 

The boson operator b is called the vibration (or fluctuation) boson since, when applied to the ground state, it generates the vibrational modes... 

As in the previous case of a single U(4), we introduce projective coherent states, constructed with condensate boson operators (Leviatan and Kirson, 1988 Shao, Walet, and Amado, 1992, 1993)... 

We have deleted from Eq. (7.159) all rotational terms which will be discussed in the following section. In Eq. (7.159) the boson operators b (bB) create (destroy) a bending mode... 

These boson operators are related to those introduced in [20] as shown in Table 7. They transform under D4/, as the representation indicated by the appropriate letter. [An extended s-wave i can also be introduced.] One then constructs the Hamiltonian by expanding it into bilinear products of creation and annihilation operators, with the constraint that H must transform as the representation of... 

Table IV 1 7. Boson operators in the method of interacting bosons...

We consider now the complete basis set of vibronic functions I f m, n) obtained as the direct product of the electronic functions l/+) and 1/1) by the phonons wave functions Im, n), where the integer numbers m and n (positive or zero) label the occupation numbers of the boson operators b b+ and b b, respectively. The Renner-Teller Hamiltonian (14) when applied to any vibronic function I f m,n) couples it with three states (at most) in fact it holds that ... 

We also noted [15] that the OAI and OAIT [11] theories are crucially different. In OAI, the coefficients multiplying s and d boson operators are constant thus OAI can be an IBA. In OAIT, however, the coefficients depend on v, and thus OAIT is not an IBA. (With OAIT, the SU(6) symmetry, which is the key ingredient of the phenomenological IBA, may largely be lost.) In any case, the (somewhat better looking) OAIT numerical results, rather than the (poorer) OAI results, were presented in OAI tables, without mentioning at all that this was done. 

The kinetic energy of the mode Q being invariant, we may write for the molecular hamiltonian, using the excitation boson operators B, B ... 

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