Creation operator Hamiltonian

D. V. Averim and K. K. Likharev developed a theory for describing the behavior of small tunneling junctions based on electron interactions. They had started from previous work on Josephson junctions (Likharev and Zorin 1985, Ben-Jacob 1985, Averin and Likharev 1986b) and established the fundamental features of the single-charging phenomena. Their work is based on a quantization theory and handles the tunneling phenomenon as a perturbation, described by annihilation and creation operators of a Hamiltonian. 

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. 

The functions fk and are the counterparts of the so-called destruction (annihilation) and creation operators in the Heisenberg-Dirac picture. It is noted in anticipation that these operators occur as the solutions a,k(t) = lulkt of the Hamiltonian equation... 

The main idea of TFD is the following (Santana, 2004) for a given Hamiltonian which is written in terms of annihilation and creation operators, one applies a doubling procedure which implies extending the Fock space, formally written as Ht = H H. The physical variables are described by the non-tilde operators. In a second step, a Bogolyubov transformation is applied which introduces a rotation of the tilde and non-tilde variables and transforms the non-thermal variables into temperature-dependent form. This formalism can be applied to quite a large class of systems whose Hamiltonian operators can be represented in terms of annihilation and creation operators. 

Because the creation operators (4 (E) for the eigenfunctions of El diagonalize the Hamiltonian, we have... 

In the single-mode Hamiltonian H, the quantities (fit) are the photon annihilation (creation) operators, respectively to is the frequency of the... 

The reference state of A-electron theory becomes a reference vacuum state 4>) in the field theory. A complete orthonormal set of spin-indexed orbital functions fip(x) is defined by eigenfunctions of a one-electron Hamiltonian Ti, with eigenvalues ep. The reference vacuum state corresponds to the ground state of a noninteracting A-electron system determined by this Hamiltonian. N occupied orbital functions (el < pi) are characterized by fermion creation operators a such that a] ) =0. Here pt is the chemical potential or Fermi level. A complementary orthogonal set of unoccupied orbital functions are characterized by destruction operators aa such that aa < >) = 0 for ea > p and a > N. A fermion quantum field is defined in this orbital basis by... 

These can be very explicitly identified as annihilation and creation operators for photons, the quanta of the electromagnetic field, with frequency a>x, propagation vector kx and polarization ex - The Hamiltonian (5.49) becomes... 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

This is often referred to as the connected cluster form of the similarity-transformed Hamiltonian. This expression makes the truncation of the Haus-dorff expansion even clearer since the Hamiltonian contains at most four annihilation and creation operators (in n) can connect to at most four cluster operators at once. Therefore, the Hausdorff expansion must truncate at the quartic terms. 

The rank 2 Hamiltonian contains two identical creation operators and two identical annihilation operators, which introduces the factor (2 )(2 ) = 4. This compensates the numerical factor associated with a V vertex. 

The subscript i labels the principle quantum number and angular momentum quantum numbers (njlm). Here a,- and bj denote electron-annihilation and positron-creation operators, respectively, defined via diagonaiization of the unperturbed Hamiltonian (1.11)... 

Having the Hamiltonian treatment, it is easy to proceed to a quantum mechanical formulation of the Fermi resonance problem. The classical variables (9.15) and their complex conjugates correspond to the annihilation and creation operators of a quantum oscillator ... 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

Equations (34) and (35) tell us how many-electron states, constructed by letting electron creation operators act on the vacuum state, transform under the elements of the symmetry group of the core Hamiltonian. The vacuum state is assumed to be invariant under the action of these symmetry operations, i.e., R10) = 10). 

As a second example, we can think of the case where the one-electron Hamiltonian has spherical symmetry. Then the Kramers pair creation operator corresponding to the shell n and subshell l is given by... 

In other words, when a Kramers pair creation operator acts on an (N — 2)-electron state 1,4). which is an eigenfunction of the core Hamiltonian, it produces an /V-electron state which is also an eigenfunction of //, with an eigenvalue increased... 

The above considerations may be put in mathematical form by referring the following formulas 100). Let a be the creation operator for an electron in the single-particle state j>) and Uj the corresponding annihilation operator. Then the Hamiltonian operator of N electrons can be written ... 

To make further progress, the zero-order Hamiltonian and the perturbation must be written in second quantized form. Recall that the annihilation operator, a and the creation operator, a], satisfy the following anticommutation relations... 

Construct the matrices corresponding to the Hamiltonian, Ha, of the new left block A, and the site creation operators of all the necessary sites. That of the annihilation operators are simply the adjoints of the creation operators. It is some times computationally efficient to store matrices of operators such as occupation number and site spin operators, although these can be computed from the matrices for the site creation operators. 

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