Hamiltonian equations diagrams

Figure 21. Phase diagram for the ground-state energy of Hamiltonian equation (101), with the critical J as a function of a for N — 300. The solid line separates the bound-state energies from the resonance energies. The simon line is shown by the dashed line.

The procedure is demonstrated by a system with one degree of freedom, assumed to be conservative, with Hamiltonian H(q,p) = a. Solving for the momentum, the relation p = p(q, a.i) defines the equation of the orbit traced out by the system point in two-dimensional phase space for constant H = a.. The graphical details for two types of periodic motion are shown in the diagram. 

The various contributions to the effective Hamiltonian operator on the right-hand side of equation (7.43) can be represented diagramatically in a way which is very helpful in practice. These diagrams are shown in figure 7.1. For each diagram, the position at the bottom refers to the chosen zeroth-order manifold 0)° and the position at the top refers to the states in the complementary subspace. Each line joining these two positions stands for a connecting matrix element. 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

Most earlier theoretical work on multiple resonance used either the Bloch equations or the spin-Hamiltonian. That is, relaxation is considered separately from other effects. The tendency in recent years has been to adopt the density-matrix approach but on some occasions simpler methods have still been appropriate. In the interpretation of multiple resonance experiments it is often necessary to consider the labelling of the energy level diagram in detail. 

Thus far we have only considered one (boson) vector field, namely, the direct product field R Xn of creation and annihilation operators. The coefficients of the creation and annihilation operator pairs in fact also constitute vector fields this can be shown rigorously by construction, but the result can also be inferred. Consider that the Hamiltonian and the cluster operators are index free or scalar operators then the excitation operators, which form part of the said operators, must be contracted, in the sense of tensors, by the coefficients. But then we have the result that the coefficients themselves behave like tensors. This conclusion is not of immediate use, but will be important in the manipulation of the final equations (i.e., after the diagrams have contracted the excitation operators). Also, the sense of the words rank and irreducible rank as they have been used to describe components of the Hamiltonian is now clear they refer to the excitation operator (or, equivalently, the coefficient) part of the operator. 

What is not immediately evident from the discussion above is how to write the equation for the Invariant torus. This is hidden in the discussion of Section 3.1. We should recall that the Lie series actually defines a coordinate transformation, although thanks to the formula (4) we can perform the whole normalization procedure without even mentioning it. Actually, denoting by p(s qt s > the coordinates that give the Hamiltonian the normal form up to order s, i.e., the form analogous to that represented in the diagram of H(s above, we can calculate the transformation in explicit form as... 

From a practical point of view, we shall never be able to perform that whole normalization process for a generic Hamiltonian. However, we can perform a finite number, r say, of steps, and consider the Hamiltonian H r) truncated at the column r of the diagram above as the approximate normal form that we are interested in. Let us call H r p r q ) the truncated Hamiltonian. Then the canonical equations for H r p r q ) admit the simple solution... 

In Cizek s original paper [24], which derives from an even earlier dissertation, he reports results for semi-empirical Hamiltonians like PPP, but also even a partly ab initio result for Ni- However, his use of second-quantized based, diagrammatic techniques to derive the CC equations was unfamiliar to most quantum chemists (see [34]), likely delaying the appreciation of the CC method, although for simple cases like CCD, conventional Slater rule matrix evaluation can be applied [35] Also, explicit mles for diagrams were given and could have been used to derive more complicated CC equations. 

The eigenvectors of this Hamiltonian (for the H atom) have been familiar for about 50 years now the energy eigenvalues are shown in frequency units in the Breit-Rabi diagram of Figure 14. The hyperfine frequency of Mu in vacuum is vq = 4463.302 35(52) MHz,24 which can be compared with the corresponding value for the H atom, i o(H) = 1420.405751(1) MHz, to give almost exactly the ratio predicted by Equation 31 when g M i s replaced by OaH-p. for the proton. In fact, a more detailed calculation, combined with the experimental values of g and provides the best test of quantun electro-... 

In zero field, the Hamiltonian of Equation 43 predicts three frequencies, since the I F = 1 M = 0> substate is shifted in energy with respect to the degenerate case depicted in the Breit-Rabi diagram of Figure 14. These frequencies are given by and 1/2 ... 

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