Hamiltonian boson

This model, called the spin-boson Hamiltonian, is probably the only fully manageable problem of this kind (with the possible exception of some very artificial problems) with a transparent solution. 

When the potential V Q) is symmetric or its asymmetry is smaller than the level spacing (Oq, then at low temperature (T cuo) only the lowest energy doublet is occupied, and the total energy spectrum can be truncated to that of a TLS. If V Q) is coupled to the vibrations whose frequencies are less than coq and co, it can be described by the spin-boson Hamiltonian... 

The analytic results for the spin-boson Hamiltonian with fluctuating tunneling matrix element (5.67) are investigated in detail by Suarez and Silbey [1991a]. Here we discuss only the situation when the qi vibration is quantum, i.e., (o P P 1. When the bath is classical, cojP, j 1, the rate... 

Leggett et al. [1987] have set forth a rigorous scheme that reduces a symmetric (or nearly symmetric) double well, coupled linearly to phonons, to the spin-boson problem, if the temperature is low enough. However, in the case of nonlinear coupling (which is necessary to introduce in order to describe the promoting vibrations), no such scheme is known, and the use of the spin-boson Hamiltonian together with (3.67) relies rather on intuition, and is not always Justifiable. 

A transition linearly coupled to the phonon field gradient will experience, from the perturbation theory perspective, a frequency shift and a drag force owing to phonon emission/absorption. Here we resort to the simplest way to model these effects by assuming that our degree of freedom behaves like a localized boson with frequency (s>i. The corresponding Hamiltonian reads... 

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... 

In order to test this experimental finding we are comparing with the eigenvalues of a Hamiltonian for a realistic quark model, namely the Goldstone-boson-exchange (GBE) constituent quark model (Glozman et al, 1998). It includes the kinetic energy in relativistic form... 

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,... 

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... 

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... 

The method of Section 7.6 can be used to find the potential functions corresponding to the boson Hamiltonians of Chapter 2. According to Eq. (2.30), one has in this case two possible chains... 

I) Potentials corresponding to chain (I), U(3). If one takes as boson Hamiltonian... 

Kirson, M. W., and Leviatan, A. (1985), Resolution of Any Interacting-Boson-Model Hamiltonian into Intrinsic and Collective Parts, Phys. Rev. Lett. 55, 2846. 

Here a and are the usual oscillator creation and annihilation operators with bosonic commutation relations (73), and 0i,..., 1 ,..., 0Af) denotes a harmonic-oscillator eigenstate with a single quantum excitation in the mode n. According to Eq. (80a), the bosonic representation of the Hamiltonian (79) is given by... 

It is interesting to note that different bosonic Hamiltonians Jif may correspond to the same original Hamiltonian H. This ambiguity reflects the fact that a transformation of the bosonic Hamiltonian M" M which corresponds to the identity transformation in the physical subspace does not change the dynamics in this subspace. For example, the two bosonic Hamiltonians... 

DoF by introducing the variables A = A2 = Atot — N, Q = Qi Q [89]. Assuming furthermore that Atot = (A tot) = 1, we obtain the classical spin-boson Hamiltonian... 

Another ambiguity in defining the classical mapping Hamiltonian is related to the fact that different bosonic quantum Hamiltonians may correspond to the same original quantum Hamiltonian H. This problem was already discussed in Section VI.A.2 for A-level systems. In the context of nonadiabatic dynamics, a different version of the mapping Hamiltonian is given by... 

Another subtlety is that the assumption nuclei behave as Dirac particles, amounts to assuming that all nuclei have spin 1/2. However, it is not uncommon to have nuclei with spin as high as 9/2 worse nuclei with integer spins are bosons and do not obey Fermi-Dirac statistics. The only justification to use equation (75) for such a case is that the resulting theory agrees with experiment. Under the assumption, we are in a position to extend our many-fermion Hamiltonian to molecules assuming that the nuclei are Dirac particles with anomalous spin. The molecular Hamiltonian may then be written as... 

Or is the frequency of the harmonic oscilator and b) are boson (phonon) creation (annihilation) operators. In order to use the perturbation theory we have to split the Hamiltonian (16) onto the unperturbed part Hq and the perturbation H ... 

The next step is that we find inverse transformations to (25-28) and substitute these inverse transformations into eq. (22) and then applying Wick theorem, we requantize the whole Hamiltonian (16) in a new fermions and bosons [14]. This leads to new V-E Hamiltonian (we omit sign on the second quantized operators)... 

The bosonic part of Hamiltonian Hp is not given in a diagonal form. To bring it to diagonal form as in eq. (59) we can proceede as follows. 

As we show later, the energy of the state of any system of N indistinguishable fermions or bosons can be expressed in terms of the Hamiltonian and D (12,1 2 ) if its Hamiltonian involves at most two-particle interactions. Thus it should be possible to find the ground-state energy by variation of the 2-matrix, which depends on four particles. Contrast this with current methods involving direct use of the wavefunction that involves N particles. A principal obstruction for this procedure is the A-representability conditions, which ensure that the proposed RDM could be obtained from a system of N identical fermions or bosons. 

The G-reduced Hamiltonians are necessary and sufficient for at least three important classes of Hamiltonians (i) all one-particle Hamiltonians, (ii) bosons or fermions with harmonic interactions [24], and (iii) all Hamiltonians with... 

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